potential and flux field landscape theory of spatially wei...

Potential and Flux Field Landscape Theory of SpatiallyInhomogeneous Non-Equilibrium Systems

A Dissertation presented

by

Wei Wu

to

The Graduate School

in Partial Fulfillment of the

Requirements

for the Degree of

Doctor of Philosophy

in

Physics

Stony Brook University

December 2014


The Graduate School

Wei Wu

We, the dissertation committee for the above candidate for the

Doctor of Philosophy degree, hereby recommend

acceptance of this dissertation

Jin Wang - Dissertation AdvisorAssociate Professor, Departments of Chemistry and Physics

Philip B. Allen - Chairperson of DefenseProfessor, Department of Physics and Astronomy

Thomas K. AllisonAssistant Professor, Departments of Chemistry and Physics

Huilin LiProfessor, Department of Biochemistry and Cell Biology


This dissertation is accepted by the Graduate School

Charles TaberDean of the Graduate School

ii

Abstract of the Dissertation

Potential and Flux Field Landscape Theory of SpatiallyInhomogeneous Non-Equilibrium Systems

by

Wei Wu

Doctor of Philosophy

in

Physics


2014

In this dissertation we establish a potential and flux field landscape theoryfor studying the global stability and dynamics as well as the non-equilibriumthermodynamics of spatially inhomogeneous non-equilibrium dynamical systems.The potential and flux landscape theory developed previously for spatially homo-geneous non-equilibrium stochastic systems described by Langevin and Fokker-Planck equations is refined and further extended to spatially inhomogeneous non-equilibrium stochastic systems described by functional Langevin and Fokker-Planckequations. The probability flux field is found to be crucial in breaking detailedbalance and characterizing non-equilibrium effects of spatially inhomogeneoussystems. It also plays a pivotal role in governing the global dynamics and for-mulating a set of non-equilibrium thermodynamic equations for a generic class ofspatially inhomogeneous stochastic systems. The general formalism is illustratedby studying more specific systems and processes, such as the reaction diffusionsystem, the Ornstein-Uhlenbeck process, the Brusselator reaction diffusion model,and the spatial stochastic neuronal model. The theory can be applied to a varietyof physical, chemical and biological spatially inhomogeneous non-equilibriumsystems abundant in nature.

iii

Contents

List of Figures vii

List of Tables viii

Acknowledgements ix

1 Introduction 11.1 Global Stability and Dynamics . . . . . . . . . . . . . . . . . . . 11.2 Non-Equilibrium Thermodynamics . . . . . . . . . . . . . . . . . 51.3 Scope and Structure . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Global Stability and Dynamics of Non-Equilibrium Systems 82.1 Global Stability and Dynamics of Spatially Homogeneous Non-

Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 Deterministic and Stochastic Dynamics of Spatially Ho-

mogeneous Systems . . . . . . . . . . . . . . . . . . . . 92.1.2 Potential and Flux Landscape Theory for Spatially Homo-

geneous Non-Equilibrium Systems . . . . . . . . . . . . 122.1.3 Lyapunov Function Quantifying the Global Stability of S-

patially Homogeneous Non-Equilibrium Systems . . . . . 162.1.4 An Illustrative Example . . . . . . . . . . . . . . . . . . 24

2.2 Global Stability and Dynamics of Spatially Inhomogeneous Non-Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . 332.2.1 The Method of Formal Extension . . . . . . . . . . . . . 342.2.2 Deterministic and Stochastic Dynamics of Spatially Inho-

mogeneous Systems . . . . . . . . . . . . . . . . . . . . 352.2.3 Potential and Flux Field Landscape Theory for Spatially

Inhomogeneous Non-Equilibrium Systems . . . . . . . . 40

iv

2.2.4 Lyapunov Functional Quantifying the Global Stability ofSpatially Inhomogeneous Non-Equilibrium Systems . . . 45

2.3 Reaction Diffusion Systems . . . . . . . . . . . . . . . . . . . . . 492.3.1 Dynamics of Reaction Diffusion Systems . . . . . . . . . 492.3.2 Potential and Flux Field Landscape for Reaction Diffu-

sion Systems . . . . . . . . . . . . . . . . . . . . . . . . 552.3.3 Lyapunov Functional Quantifying the Global Stability of

Reaction Diffusion Systems . . . . . . . . . . . . . . . . 562.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Non-Equilibrium Thermodynamics of Stochastic Systems 613.1 Non-Equilibrium Thermodynamics for Spatially Homogeneous S-

tochastic Systems with One State Transition Mechanism . . . . . 613.1.1 Stochastic Dynamics . . . . . . . . . . . . . . . . . . . . 623.1.2 Generalized Potential-Flux Landscape Framework . . . . 643.1.3 Non-Equilibrium Thermodynamic Context . . . . . . . . 693.1.4 State Functions of Non-Equilibrium Isothermal Processes 713.1.5 Thermodynamic Laws of Non-Equilibrium Isothermal Pro-

cesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.1.6 Summary and Discussion . . . . . . . . . . . . . . . . . . 913.1.7 Extension to Systems with One General State Transition

Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 953.2 Non-Equilibrium Thermodynamics for Spatially Homogeneous S-

tochastic Systems with Multiple State Transition Mechanisms . . 993.2.1 Stochastic Dynamics for Multiple State Transition Mech-

anisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.2.2 Potential-Flux Landscape Framework for Multiple State

Transition Mechanisms . . . . . . . . . . . . . . . . . . . 1023.2.3 Non-Equilibrium Thermodynamics for Multiple State Tran-

sition Mechanisms . . . . . . . . . . . . . . . . . . . . . 1043.2.4 Necessary and Sufficient Condition for the Collective Def-

inition Property . . . . . . . . . . . . . . . . . . . . . . . 1093.2.5 Ornstein-Uhlenbeck Processes of Spatially Homogeneous

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.3 Non-Equilibrium Thermodynamics for Spatially Inhomogeneous

Stochastic Dynamical Systems . . . . . . . . . . . . . . . . . . . 1213.3.1 Description of Spatially Inhomogeneous Systems . . . . . 1223.3.2 Stochastic Dynamics of Spatially Inhomogeneous Systems 125

v

3.3.3 Potential-Flux Field Landscape Framework for SpatiallyInhomogeneous Systems . . . . . . . . . . . . . . . . . . 128

3.3.4 Non-Equilibrium Thermodynamics of Spatially Inhomo-geneous Systems . . . . . . . . . . . . . . . . . . . . . . 131

3.3.5 Ornstein-Uhlenbeck Processes of Spatially InhomogeneousSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

3.3.6 Spatial Stochastic Neuronal Model . . . . . . . . . . . . . 1493.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4 Conclusion 166

Bibliography 168

A Proof of Several Equations 179

B Proof of the Necessary and Sufficient Condition for the Collective Def-inition Property 181

C Ornstein-Uhlenbeck Processes for Spatially Homogeneous Systems 183

D Abstract Representation and Representation Transformation 189D.1 Abstract Representation . . . . . . . . . . . . . . . . . . . . . . . 189D.2 Representation Transformation . . . . . . . . . . . . . . . . . . . 194

E Ornstein-Uhlenbeck Processes for Spatially Inhomogeneous Systems 202E.1 Dynamical Equations in the Abstract Representation . . . . . . . 202E.2 Thermodynamic Expressions in the Space Configuration Repre-

sentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

F Spatial Stochastic Neuronal Model 208

vi

List of Figures

2.1 Streamlines of the deterministic driving force . . . . . . . . . . . 252.2 Steady state probability distribution for D = 1 . . . . . . . . . . . 272.3 Potential landscape for D = 1 . . . . . . . . . . . . . . . . . . . 282.4 Streamlines of the negative gradient of potential landscape for

D = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5 Streamlines of the flux velocity . . . . . . . . . . . . . . . . . . . 302.6 The intrinsic potential landscape as the Lyapunov function . . . . 312.7 Probability transport dynamics . . . . . . . . . . . . . . . . . . . 40

3.1 Temporal profile of the system’s (renormalized) transient entropyS , cross entropy U and relative entropy A in the process of re-laxing to the equilibrium state in the spatial stochastic neuronalmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

vii

List of Tables

1.1 Comparison of different approaches . . . . . . . . . . . . . . . . 4

2.1 Types of (non-)equilibrium dynamics . . . . . . . . . . . . . . . . 142.2 Lyapunov function and stability of a fixed point . . . . . . . . . . 172.3 (non-)equilibrium dynamics . . . . . . . . . . . . . . . . . . . . 442.4 Characterization of chemical reactions in the Brusselator model . 52

viii

Acknowledgements

I am grateful for my advisor Prof. Jin Wang’s assistance, support and adviceduring my academic endeavor as a Ph.D student. I am very glad to have been giventhe opportunity to work on a series of related research projects in a systematic way.What I have learned in these experiences has no doubt become a treasure for meand will continue to be cherished in my future scientific pursuit.

I am thankful to the members (present and former) in our research group, whohave provided an inviting and stimulating research environment. They includeNathan Borggren, Yuan Yao, Zhedong Zhang, Chunhe Li, Cong Chen, Zaizhi Lai,Haidong Feng, Zhiqiang Yan, Bo Han, Xiaosheng Luo, Ruonan Lin, Fang Liu,Weixin Xu, Ronaldo Olviera, Qiang Lu, and Jeremy Adler.

I am also deeply indebted to my family for their unconditional support.

The text of this dissertation in part is a reprint of the materials as it appears inthe following publications.

W. Wu and J. Wang, Potential and flux field landscape theory. I. Global sta-bility and dynamics of spatially dependent non-equilibrium systems, The Journalof Chemical Physics, 139, 121920 (2013). Copyright 2013, AIP Publishing LLC.

W. Wu and J. Wang, Potential and flux field landscape theory. II. Non-equilibriumthermodynamics of spatially inhomogeneous stochastic dynamical systems, TheJournal of Chemical Physics, 141, 105104 (2014). Copyright 2014, AIP Publish-ing LLC.

Permission to use these publications in this dissertation is granted by AIP Pub-lishing LLC. The co-author listed in these publications directed and supervised theresearch that forms the basis for this dissertation.

ix

Chapter 1

Introduction

We live in a non-equilibrium world.1 Non-equilibrium dynamical systemsconstantly exchanging matter, energy and information with the environment areubiquitous in nature. Our earth is a non-equilibrium system constantly receivingenergy from the Sun. Our human body is a non-equilibrium system constantlyconsuming energy for survival. Uncovering the principles and physical mecha-nisms underlying these activated processes is vital for understanding the physical,chemical and biological non-equilibrium systems. In this dissertation we focus ontwo aspects of non-equilibrium systems, namely the global stability and dynamicsas well as the non-equilibrium thermodynamics.

1.1 Global Stability and DynamicsSignificant efforts and progresses have been made on the study of the glob-

al stability and dynamics of non-equilibrium systems [1–14]. Many studies havebeen focused on spatially homogeneous systems, where spatial dependence dueto spatial inhomogeneity is ignored. The deterministic dynamics of these systemscan be described by an ordinary differential equation of the system’s finite degreesof freedom. When extrinsic or intrinsic stochastic fluctuations have to be takeninto account, the dynamics governing these non-equilibrium systems becomes s-

1Most of the material in this chapter was originally co-authored with Jin Wang in two journalarticles. Reprinted with permission from W. Wu and J. Wang, The Journal of Chemical Physics,139, 121920 (2013). Copyright 2013, AIP Publishing LLC. Reprinted with permission from W.Wu and J. Wang, The Journal of Chemical Physics, 141, 105104 (2014). Copyright 2014, AIPPublishing LLC.

1

tochastic. Stochastic Markovian dynamics, governed by Langevin, Fokker-Planckand (discrete-state) master equations [15, 16], has found extensive applications instudying such systems. For example, Langevin equations have been used to studycritical dynamics [17], chemical reactions [18–20], soft colloidal systems [21] andcell migration [22]. Fokker-Planck equations have been employed in the study ofanomalous transport [23], plasma physics [24], chemical reaction processes [25]and polymer dynamics [26]. Master equations have been utilized to model, forinstance, systems weakly coupled to heat baths [27], gas phase chemical kinetic-s [28] and complex reaction networks [29]. Besides Markovian dynamics, non-Markovian dynamics with memory effects [30, 31] has also been used in some s-tudies, among which is the generalized Langevin equation [32] employed to studynon-equilibrium molecular dynamics [33] and its further generalization utilized toinvestigate chemical reactions taking place in changing environments [34].

It is a well-known and readily observable fact that spatial inhomogeneity ischaracteristic of non-equilibrium systems in the physical world. Some typical ex-amples of non-equilibrium spatially inhomogeneous systems with spatial-temporaldynamics of pattern formation and self-organization [1,35–37] are Rayleigh-Benardconvection in fluids [38], Turing pattern in chemical morphogenesis [39], Drosophi-la embryo differentiation in developmental biology [40], and plant distributions inthe population dynamics of ecological systems [41]. These systems have beenstudied using the reaction diffusion equation, where the spatial dependence of thesystem is accounted for on the mean field level, ignoring intrinsic and extrinsicstochastic fluctuations. The mean field description can give local dynamics andlocal stability analysis for these systems. However, it cannot address the globalnature of the system, such as global stability, dynamics and robustness. Besides,it is well known that in some cases fluctuations can play an important role wherethe mean field description breaks down. A typical example is when the systemis approaching the critical point of a continuous phase transition where the cor-relation length diverges. Therefore a stochastic description is needed to accountfor the effect of fluctuations. Large classes of spatially inhomogeneous system-s with stochastic fluctuations can be described by functional Langevin equations(including stochastic partial differential equations) [1,15,31,35–37,42–47], func-tional Fokker-Planck equations [1, 15, 31, 35–37, 45–51] and spatial master equa-tions [1, 15, 51–53]. They have been used to study various problems in physics,chemistry and biology [1,35–37,43–60], such as surface growth [58], electromag-netic field propagation [59], reaction diffusion of proteins in Drosophila embryodevelopment [51] and signal transduction of neurons [60].

For stochastic spatially inhomogeneous systems described by functional Langevin

2

equations and spatial master equations, standard procedures such as the Janssen-De Dominicis formalism [45–47] and Doi-Peliti formalism [61–65] have been de-veloped to map these equations to field theory formalisms. Hence, field-theoretictechniques developed in quantum field theory and statistical mechanics, such asdiagrammatic perturbation expansion and renormalization group [66–70], havebeen employed to study these systems, in complementary to other methods suchas exact solutions [71–74], the Smoluchowski theory [75,76] and computer simu-lations [77–80]. The standard and usually perturbative field-theoretic approaches,although are able to incorporate the effect of fluctuations systematically, still can-not address the issues of global stability and global dynamics of the system.

An alternative approach is based on the concept of non-equilibrium potential.This concept is pioneered by R. Graham and coworkers for stochastic systems de-scribed by Fokker-Planck equations in the small fluctuation limit [81, 82], furtherdeveloped by G. Hu for stochastic systems described by master equations in thesmall fluctuation limit [83,84], and constructed alternatively by P. Ao for stochas-tic systems governed by Langevin equations (no requirement of small fluctuationlimit but with certain restrictions) [85]. The non-equilibrium potential in the s-mall fluctuation limit is a Lyapunov function of the deterministic system, whichcharacterizes the global stability of the deterministic system in terms of the to-pography of the potential landscape (e.g., basin of attractions and barrier heights).But it is not a Lyapunov function of the stochastic system with finite fluctuationsthat quantifies the global stability of the stochastic system. The non-equilibriumpotential approach was extended and applied by R. Graham and H. S. Wio et al tosome specific spatially inhomogeneous systems, where the exact form of the non-equilibrium potential functional can be obtained [48, 86, 87]. This has proven tobe a useful complementary approach to other field-theoretic approaches. Recent-ly, the non-equilibrium potential landscape framework has been established forgeneral spatially inhomogeneous systems (including general reaction diffusionsystems) with intrinsic fluctuations governed by spatial master equations [51].

What has also been realized for non-equilibrium systems is that the steady s-tate probability distribution (closely related to the potential landscape) alone is notcomplete in characterizing the non-equilibrium steady state [88], nor is it com-plete in determining the dynamics of the system [9, 81, 82, 89, 90]. The steadystate probability flux (also called the curl flux due to its divergence-free property)must be considered together with the probability distribution (or equivalently thepotential landscape) to complete the whole picture for non-equilibrium system-s. Energy input or pump to the system has been found to be one way to createnonzero probability curl flux [91]. For spatially homogeneous non-equilibrium

3

systems, we have established a potential and flux landscape theory, which inte-grates the potential landscape and the curl flux together, to quantify the globalstability and global dynamics of the system [9–12, 14]. We have shown that theeffective driving force of the spatially homogeneous non-equilibrium dynamicscan be explicitly decomposed into a gradient-like term of the potential landscapeand a curl-like term of the probability flux. While the potential landscape attractsthe system down along the gradient similar to electrons moving in an electric field,the probability flux drives the system in a curly way much like a magnetic fieldacting on electrons. In the small fluctuation limit nonzero probability flux allowsfor the existence of continuous attractors such as limit cycles and even possiblychaotic strange attractors in the corresponding deterministic dynamics [9–14]. Forspatially inhomogeneous non-equilibrium systems, the indispensable probabilityflux also needs to be brought out explicitly together with the potential landscape,to give a complete characterization of the global stability and global dynamics ofspatially inhomogeneous non-equilibrium systems. This provides an alternativeand complementary approach to the current field-theoretic methods. It also formsan extension of the non-equilibrium potential approach by explicitly introducingand exploiting the probability curl flux, capable of studying the global dynamic-s of the system. Thus it may offer insights not accessible by other approaches.In Table 1.1 we list the relevant methods and approaches mentioned and indicatewhether they are able to incorporate stochastic fluctuations and capable of study-ing the global stability and global dynamics of the system.

MeanField

Theory

PerturbativeField-Theoretic

Techniques

Non-EquilibriumPotential

Approach2

Potential andFlux Landscape

TheoryFluctuations N Y Y Y

GlobalStability

N N Y Y

GlobalDynamics

N N N Y

Table 1.1: Comparison of different approaches

2The non-equilibrium potential is based on stochastic dynamics. Yet in the small fluctuationlimit stochastic fluctuations are completely suppressed. Thus the “Y(es)” for stochastic fluctua-tions is not clear-cut. Also, the “N(o)” for global dynamics is a reserved one, since there do existformulations in this respect even though it is not fully brought out in this approach.

4

The study of global stability and dynamics in the context of the potential andflux landscape theory forms one major topic of this dissertation. Detailed ex-planations, discussions and illustrations are given in chapter 2. In particular, theBrusselator reaction diffusion model is studied in Sec. 2.3 using the potential andflux landscape theory, demonstrating its advantage over other approaches.

1.2 Non-Equilibrium ThermodynamicsComplementary to the dynamic aspect is the thermodynamic aspect of non-

equilibrium systems. The relation between dynamics and thermodynamics hasbeen at the heart of controversy since the establishment of thermodynamics asa scientific theory. Following Onsager’s seminal work [92], a theory of non-equilibrium thermodynamics in the linear regime based on the local equilibriumhypothesis was systematically developed by Prigogine [93] and many other re-searchers [94], which was further applied and extended to a variety of directionsand areas [95–102]. In the past few decades a non-equilibrium thermodynam-ic framework based on stochastic Markovian dynamics, termed stochastic ther-modynamics generally, has emerged [1, 2, 7, 11, 12, 103–137]. Much effort hasbeen devoted to constructing a consistent non-equilibrium thermodynamics, us-ing methods of statistical mechanics, from the underlying stochastic dynamics,described by master equations as well as Langevin and Fokker-Planck equations,for steady states and also transient processes [1, 2, 7, 11, 12, 103–125]. The re-sulting work has been applied, for instance, in the study of gene network [126] inbiochemical systems [127]. The realization that thermodynamic quantities can bedefined for stochastic trajectories has directed the development of this field ontothe more refined single trajectory level [128–130]. (The term ‘stochastic ther-modynamics’ also refers specifically to this refined non-equilibrium thermody-namics.) Its combination with the fluctuation theorem proves fruitful [131, 132].It has been realized that the second law of thermodynamics has multiple facets[114, 116–119]. Stochastic thermodynamics is now emerging as a powerful toolfor nanosciences [134], with applications in molecular motors [135] and chemicaloscillation systems [136] among others.

Stochastic thermodynamics is a work still in progress, with some remainingissues to be addressed. We mention three of them that are relevant to the presentwork. First, stochastic thermodynamics deals with both stochastic dynamics andnon-equilibrium thermodynamics. Yet the connection between these two levelsdoes not seem to be transparent. It would be nice to have a ‘bridge’ connect-

5

ing these two levels explicitly. Second, it has been noted that entropy produc-tion could be underestimated if physically different state transition mechanisms(constituent processes, reservoirs, reaction channels etc.) are not identified cor-rectly [116–118, 124]. Although a systematic presentation has been given forFokker-Planck equations with multiple (state) transition mechanisms on an essen-tially one-dimensional state space [118], a more general formulation is not yetavailable. This is not just a trivial extension. Also, it is not yet clear under whatconditions the results for one transition mechanism may also apply to multipletransition mechanisms. Third, spatially inhomogeneous systems have been large-ly unexplored in the context of stochastic thermodynamics, although there do exista few studies in this context [129, 137]. Since spatial inhomogeneity is typical ofnon-equilibrium systems, this issue is obviously important.

This brings out the other major topic of this dissertation to be discussed inchapter 3, where we address the above three issues related to non-equilibriumthermodynamics within the context of the potential and flux landscape theory, bysynthesizing, extending and transcending some of the essential results develope-d previously in the framework of stochastic thermodynamics. The key feature inour approach is that we use the potential and flux landscape framework as a bridgeto construct the non-equilibrium thermodynamics from the underlying stochasticdynamics. From the potential and flux landscape and the associated dynamical de-composition equations, which are based on the stochastic dynamics, we constructa set of non-equilibrium thermodynamic equations quantifying the relations of thenon-equilibrium entropy, entropy flow, entropy production, and other thermody-namic quantities. On both the dynamic level and thermodynamic level, we findthat the probability flux plays a central role. We will give detailed explanations,discussions and illustrations with specific examples in chapter 3.

1.3 Scope and StructureThe systems considered in this dissertation are spatially homogeneous and

inhomogeneous systems, described by Langevin and Fokker-Plank dynamics (in-cluding their deterministic limit), on a one or multiple (including infinite) dimen-sional state space. We do not consider Fokker-Planck equations derived from mas-ter equations by truncating the Kramers-Moyal expansion [18,138] which usuallymodels systems with intrinsic fluctuations. We shall consider even state variablesunder time reversal. Also, the non-equilibrium thermodynamics is covered onthe ensemble average level. These restrictions define a manageable scope of an

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extensive subject for this dissertation to deal with and will be lifted in future work.The rest of this dissertation is structured as follows. In chapter 2 we inves-

tigate the global stability and dynamics of non-equilibrium systems in terms ofthe potential and flux landscape theory, which is a consolidation of the work inRef. [50]. In chapter 3 we address the thermodynamics of non-equilibrium sys-tems in the context of the potential and flux landscape theory based on the workof Ref. [139]. The conclusion of this dissertation is given in chapter 4.

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Chapter 2

Global Stability and Dynamics ofNon-Equilibrium Systems

In this chapter we study the global stability and dynamics of spatially homo-geneous and inhomogeneous non-equilibrium systems within the potential andflux landscape framework.1 We first in Sec. 2.1 review and refine our previouslyestablished potential and flux landscape theory for quantifying the global stabil-ity and dynamics of spatially homogeneous systems [9, 12]. Then in Sec. 2.2 weextend the potential and flux landscape theory to spatially inhomogeneous non-equilibrium systems. Our extended theory is then applied to reaction diffusionsystems and, in particular, the Brusselator reaction diffusion model in Sec. 2.3.Finally, we give a summary of this chapter in Sec. 2.4.

2.1 Global Stability and Dynamics of Spatially Ho-mogeneous Non-Equilibrium Systems

First, we introduce the deterministic and stochastic dynamics spatially homo-geneous non-equilibrium systems. Then we present and expand on the potentialand flux landscape framework developed previously for spatially homogeneousnon-equilibrium systems governed by Fokker-Planck equations [9, 10, 13, 14]. Inparticular, we investigate the force decomposition equation in this framework and

1Much of the material in this chapter was originally co-authored with Jin Wang. Reprint-ed with permission from W. Wu and J. Wang, The Journal of Chemical Physics, 139, 121920(2013). Copyright 2013, AIP Publishing LLC. Yet the figures and tables in this chapter are allnew. Sec. 2.1.4 is new. Sec. 2.1.3 and Sec. 2.3 contain new materials.

8

reveal various other meanings of this equation by looking at it from multiple per-spectives. Then we review and discuss a general method for uncovering the Lya-punov functions quantifying the global stability of the deterministic and stochasticspatially homogeneous non-equilibrium dynamical systems described by Fokker-Planck equations [1–12]. Finally, we study a specific dynamical system in detailto illustrate the general theory presented in this section.

2.1.1 Deterministic and Stochastic Dynamics of Spatially Ho-mogeneous Systems

We discuss the deterministic dynamics first and then the stochastic Langevindynamics and the corresponding Fokker-Planck dynamics.

Deterministic Dynamics

The state of a spatially homogeneous system can usually be characterized bya state vector q = (q1, ..., qi, ..., qn), representing the system’s degrees of free-dom under study. The concrete meaning of the state vector and its components issystem-specific. For instance, the state vector of a spatially homogeneous (well-stirred) chemical reaction system may represent the collection of the concentra-tions of all the chemical species involved in the system at each moment; the con-centration of each chemical species is a component of the state vector.

The deterministic dynamics of an autonomous spatially homogeneous systemin general can be described by an ordinary differential equation of the state vector:

d

dtq = F (q), (2.1)

where F (q) is the deterministic driving force of the system. Mathematically, it is avector field on the state space of the system (i.e., the space of q). In general, F (q)may come from the contribution of several different sources. If those sources areidentifiable in the studied system, the driving force can be written in a decomposedform in terms of different contributing sources (labeled below by the index r):

F (q) =∑r

Fr(q), (2.2)

where Fr(q) represents the deterministic driving force from source r. For ex-ample, the deterministic driving force of chemical-reaction systems can be de-composed in terms of different chemical reactions (labeled by r) [52]: F (q) =

9

∑r Fr(q) =

∑r νrwr(q), where Fr(q) = νrwr(q) is the deterministic driving

force from chemical reaction r. Here, νr is the vector whose components are thestoichiometric coefficients of chemical reaction r. And wr(q) is the rate of chem-ical reaction r, dependent on the concentrations of the chemical species involved.

Langevin Dynamics

When stochastic fluctuations are present which change the state of the systemstochastically, the dynamics of the system becomes stochastic. In many cases, thestochastic dynamics of the system is governed by a Langevin equation [15]:

d

dtq = F (q) + ξ(q, t), (2.3)

where ξ(q, t) is the stochastic driving force in addition to the deterministic driv-ing force F (q). In general, ξ(q, t) can be decomposed according to statisticallyindependent fluctuation sources (labeled by s) [15, 140] :

ξ(q, t) =∑s

Gs(q)Γs( t), (2.4)

where Γs( t) are Gaussian white noises with the following statistical property:

< Γs(t) >= 0, < Γs(t)Γs′(t′) >= δss′δ(t− t′). (2.5)

In Eq. (2.4) the vector Gs(q) characterizes the direction and strength of the s-tochastic driving force from source s, while the Gaussian noise Γs(t) character-izes its stochastic nature. Stochastic fluctuations from different sources labeledby different s are statistically independent as shown by the second equation inEq. (2.5). From Eqs. (2.4) and (2.5) we can derive the statistical property of thetotal stochastic driving force from the contributions of all the fluctuation sources:

< ξ(q, t) >= 0, < ξ(q, t)ξ(q, t′) >= 2D(q)δ(t− t′), (2.6)

where D(q) as one half of the fluctuation correlator is the diffusion matrix (thename is justified shortly in the Fokker-Planck dynamics) given by:

D(q) =1

2

∑s

Gs(q)Gs(q). (2.7)

D(q) is a nonnegative-definite symmetric square matrix by construction. It ac-counts for the combined effects of stochastic fluctuations from all the fluctuationsources as indicated by its expression in Eq. (2.7).

10

Fokker-Planck Dynamics

The Lagevin equation in Eq. (2.3) traces a single stochastic trajectory in thestate space qt for a given initial state, which represents the stochastic dynamicalevolution of the state of the system with time. It is useful to know how the proba-bility distribution Pt(q) for the random state vector qt evolves with time instead ofindividual stochastic trajectories. The dynamical evolution of the probability dis-tribution corresponding to the Langevin dynamics in Eq. (2.3) (interpreted as anIto stochastic differential equation) is governed by a partial differential equationknown as the Fokker-Planck equation [15, 16]:

∂

∂tPt(q) = −∇ ·

(F (q)Pt(q)

)+∇ · ∇ · (D(q)Pt(q)) . (2.8)

Here ∇ = ∂/∂q is the n-dimensional vector differential operator in the state s-pace. The notation ∇ · ∇ · (D(q)Pt(q)) means two successive operations of ∇·on D(q)Pt(q), which in its component form reads

∑ab ∂a∂b(Dab(q)Pt(q)), with

the short notation ∂a ≡ ∂/∂qa. F (q) in Eq. (2.8) is called the drift vector inthe Fokker-Planck equation and is given by the deterministic driving force in theLangevin equation. D(q) in Eq. (2.8) is the diffusion matrix in the Fokker-Planckequation and is given by Eq. (2.7) accounting for the effects of stochastic fluctua-tions in the Langevin equation.

The Fokker-Planck equation in Eq. (2.8) can be identified as a continuity e-quation representing probability conservation:

∂

∂tPt(q) = −∇ · Jt(q). (2.9)

The (transient) probability flux is given by

Jt(q) = F ′(q)Pt(q)−D(q) · ∇Pt(q), (2.10)

where F ′(q) = F (q) − ∇ · D(q) is the effective drift vector (or effective driv-ing force), which is the original drift vector F (q) modified by a diffusion-induceddrift vector −∇ · D(q). The Fokker-Planck equation in the form of a continuityequation allows for an interpretation of the Fokker-Planck dynamics as a proba-bility transport dynamics in the state space. Probability is transported in the statespace through the probability flux which, according to Eq. (2.10), comes from thecontribution of a drift process in the state space characterized by the drift velocityfield F ′(q) and a diffusion process in the state space characterized by the diffusionmatrix D(q). This also justifies the appropriateness of the names, drift vector anddiffusion matrix, used in the Fokker-Planck equation.

11

2.1.2 Potential and Flux Landscape Theory for Spatially Ho-mogeneous Non-Equilibrium Systems

We consider the steady state Ps(q) of the Fokker-Planck equation which satis-fies ∂Ps(q)/∂t = 0. Then Eqs. (2.9) and (2.10) become

∇ · Js(q) = 0, (2.11)

Js(q) = F ′(q)Ps(q)−D(q) · ∇Ps(q). (2.12)

Equation (2.11) states the steady state probability flux Js is divergence-free every-where in the state space. Thus it is a solenoidal curl vector field like the magneticfield. Its field line has no beginning or end; it either extends to infinity or forms aclosed loop, since the field has no sinks or sources due to the divergence-free con-dition. The deterministic driving force F (q) governing the deterministic dynam-ics and the diffusion matrix D(q) encoding the effects of stochastic fluctuationstogether through Eqs. (2.11) and (2.12) determine the steady state probability dis-tribution Ps(q) and the steady state probability flux Js(q). Ps(q) characterizes theglobal stochastic steady state of the system, while Js(q) governs the global steadyprobability transport dynamics of the system in the state space. Although Ps doesnot change with time, there is still probability transport going on in the state spacewhen Js = 0, which does not change Ps as long as Js is divergence-free.

The detailed balance condition in the steady state as the equilibrium condi-tion of the system characterizing microscopic reversibility is represented by zerosteady state probability flux Js = 0 [15]. (This is for even state variables q undertime reversal as is what we consider in this dissertation. Also, there are subtletiesand extensions regarding this condition which will be discussed in Secs. 3.1.2 and3.2.2 in the next chapter.) This means there is no probability transport dynamics inthe state space. The global stochastic steady state and steady probability transportdynamics (vanish) in this case are characterized by Ps alone. Equation (2.12) inthis case becomes F ′(q)Ps(q) −D(q) · ∇Ps(q) = 0. Dividing Ps on both sidesof the equation and rearranging it lead to the potential condition [15]:

F ′(q) = −D(q) · ∇U(q), (2.13)

where U ≡ − lnPs (up to an additive constant) is the (dimensionless) equilibriumpotential landscape. It plays a double role. On the one hand U connects to thesteady state probability distribution via Ps = e−U . Therefore U is the ‘weight’of the probability distribution. Its topography reflects the shape of the steady

12

state probability distribution. We can say the potential landscape U characterizes(as Ps does) the global stochastic steady state and steady probability transportdynamics (vanish) for equilibrium systems. On the other hand, it connects to thedeterministic driving force F governing the deterministic dynamics of the systemand the diffusion matrix D governing the stochastic fluctuation dynamics throughEq. (2.13). For equilibrium systems with detailed balance the effective drivingforce F ′ has the form of the gradient of the potential landscape U with respect tothe diffusion matrix D. F and D are connected to each other by the potential Uthrough Eq. (2.13), which places a constraint on them. When the diffusion matrixis invertible, the potential condition can be explicitly expressed as a constraintequation on F and D :

∇∧[D−1(q) · F ′(q)

]= 0, (2.14)

where the operator ∇∧ is the generalization of the curl operator ∇× in 3 di-mensions to n dimensions, defined as [∇ ∧ A]ab = ∂aAb − ∂bAa. For stochasticsystems, the constraint on F and D due to the detailed balance condition meansthe deterministic dynamics of the system governed by F and the stochastic fluctu-ation dynamics represented by D have to cooperate with each other in a specificway to balance each other out and produce zero probability flux, thus maintain-ing reversibility of the system, although each of them individually may have anon-zero contribution to the probability flux.

In general, when the deterministic dynamics and the stochastic fluctuation dy-namics of the system are not cooperating with each other in the specific way thatproduce the detailed balance condition characterized by vanishing steady stateprobability flux, the steady state probability flux Js can be non-zero. The non-zeroflux Js indicates microscopic irreversibility that breaks detailed balance and drivesthe system away from equilibrium. Dividing Ps(q) on both sides of Eq. (2.12) andrearranging it gives the ‘force decomposition equation’ [9, 10, 13, 14]:

F ′(q) = −D(q) · ∇U(q) + Vs(q), (2.15)

where U(q) ≡ − lnPs(q) (up to an additive constant) is the generalized dimen-sionless non-equilibrium potential landscape and Vs(q) ≡ Js/Ps is the steady stateprobability flux velocity. They are related to each other by

Vs(q) · ∇U(q) = ∇ · Vs(q), (2.16)

which is obtained from Eq. (2.11), ∇ · Js = ∇ · (e−U Vs) = 0. The steady stateprobability flux velocity Vs is related to Js through Vs = Js/Ps. In general Vs is

13

not divergence-free; thus it is generally not a solenoidal vector field. Accordingto Eq. (2.16) the necessary and sufficient condition for Vs to be divergence-free is∇ · Vs(q) = Vs(q) · ∇U(q) = 0, that is, Vs is perpendicular to the gradient of thepotential landscape U everywhere in the state space. When this condition is notsatisfied, Vs is not a solenoidal vector field while Js still is. Yet since they are relat-ed to each other by Js = PsVs, they are parallel to each other everywhere (exceptfor Js = 0 or Vs = 0 where the concept of parallel fails). Therefore, their fieldlines have the same curling shapes, though their field-line densities, convention-ally representing their respective field magnitudes, may differ at different pointsin the state space due to the factor Ps(q). We summarize the (non-)equilibriumdynamics in relation to flux and force in Table 2.1.

Detailed Balance Detailed Balance Breaking

Js Js = 0 Js = 0, ∇ · Js = 0

Vs Vs = 0 Vs = 0, ∇ · Vs = Vs · ∇U

F ′ F ′ = −D · ∇U F ′ = −D · ∇U + Vs

Table 2.1: (Non-)equilibrium dynamics of spatially homogeneous systems2

For non-equilibrium systems, the effective driving force can be decomposedinto two terms according to Eq. (2.15). One term is the gradient (with respect tothe diffusion matrix D) of the potential landscape U (related to the steady stateprobability distribution Ps = e−U ) which characterizes the global stochastic stateof the system. The other term is the curling steady state probability flux velocityVs (related to the steady state probability flux Js = PsVs ) which represents non-equilibrium effects that break detailed balance. This is analogous to the motion ofan electron moving in a gradient electric field and a curl magnetic field. Alterna-tively, one may move the term ∇ ·D within the effective driving force F ′ to theright side of the equation and make a statement about the decomposition of the de-terministic driving force F into three terms. In Sec. 2.1.3 we shall see the potentiallandscape U in the small fluctuation limit is a Lyapunov function of the determin-istic system, quantifying the global stability of the deterministic system. And thepotential landscape U and the flux velocity Vs together, in the small fluctuation

2As has been noted previously, the relation between detailed balance (breaking) and probabilityflux presented here in this chapter applies to even state variables and there are also subtleties andextensions to be discussed in chapter 3.

14

limit, through the force decomposition equation, decide the deterministic driv-ing force that governs the deterministic dynamics of the non-equilibrium system.Therefore, for non-equilibrium spatially homogeneous systems, while the globalstability is quantified by the underlying potential landscape U closely related tothe steady state probability distribution, quantification of the global dynamics ofthe non-equilibrium systems requires both the potential landscape U and the fluxvelocity Vs, in contrast to equilibrium systems where U alone is enough.

The way Eq. (2.15) is written down and the above statements made in relationto this equation is based on the perspective of seeing F (or F ′ ) as the force in thisequation, while seeing D, U and Vs as that which compose the force F (or F ′ ).This perspective is natural when one is working in the small fluctuation limit andconnecting the stochastic dynamics to the deterministic dynamics of the system.However, it is not the only perspective to look at this equation. Equation (2.15) onits own is simply a statement of the relation between the four important quantities:F (or F ′ ), D, U , and Vs. In this sense we can call it the Constitutive Equation ofthe system. Grouping different quantities together allows one to look at Eq. (2.15)from different perspectives. From a certain perspective, it is natural to group F(or F ′ ) and D together as they are both directly from the Langevin dynamics andgroup U and Vs together as they are both quantities directly related to the charac-terization of the Fokker-Planck dynamics. The pair (U , Vs) in a way serves as abridge between the pair (F , D) and the pair (Ps, Js). On the one hand, (U , Vs)connects to (Ps, Js) through Ps = e−U and Js = PsVs, which characterize theglobal stochastic steady state and steady probability transport dynamics of non-equilibrium systems in the Fokker-Planck equation. On the other hand, (U , Vs)connects through Eq. (2.15) to (F , D), which are the deterministic driving forcethat governs the deterministic dynamics of the system and the diffusion matrix thatgoverns the stochastic fluctuation dynamics of the system, both coming from theLangevin equation. Therefore, through the pair U and Vs, a bridge is built betweenthe Fokker-Planck dynamics governing the evolution of probability distribution-s and the Langevin dynamics governing the evolution of stochastic trajectories.So from such a perspective, Eq. (2.15) is seen as the statement of the relation be-tween two pairs of quantities: the pair (F , D) and the pair (U , Vs). From anotherperspective by grouping F , D and U together, Eq. (2.15) can be seen as a state-ment about the composition of Vs. It states that there two parts contributing to thesteady-state probability flux velocity that determines the steady-state probabilitytransport dynamics. One part is the drift process in the state space contributingthe drift velocityF ′(q). The other part is the diffusion process in the state spacecontributing D(q) · ∇U(q), which specifies how the gradient of the potential U

15

and the diffusion matrix D together determine the diffusion contribution to theprobability flux velocity. Drift and diffusion together produce the total probabilityflux velocity Vs that drives the probability transport dynamics in the state space. Inaddition, in the context of non-equilibrium thermodynamics, Eq. (2.15) is closelyrelated to the entropy flow decomposition equation detailed in chapter 3. So it canalso be interpreted from that perspective. There could also be other possible per-spectives. Therefore, the meaning of Eq. (2.15) is multi-dimensional, dependingon the perspective one looks at it and the context in which one studies it.

We can also derive a force decomposition equation in terms of transient quan-tities, from the definition of the transient probability flux in Eq. (2.10):

F ′(q) = −D(q) · ∇S(q, t) + Vt(q), (2.17)

where S(q, t) ≡ − lnPt(q) and Vt(q) ≡ Jt(q)/Pt(q). Subtracting Eq. (2.15) fromEq. (2.17) leads to the following equation:

Vt(q) = −D(q) · ∇ ln

(Pt(q)

Ps(q)

)+ Vs(q). (2.18)

The meanings of two equations, Eq. (2.17) and Eq. (2.18), will be further investi-gated in chapter 3 and form part of an extension of the potential and flux landscapeframework to be presented in Sec. 3.1.2. Here we only mention that Eq. (2.18)plays an important role in the proof of the Lyapunov property that quantifies theglobal stability of the stochastic system with finite fluctuations.

2.1.3 Lyapunov Function Quantifying the Global Stability ofSpatially Homogeneous Non-Equilibrium Systems

We review and discuss a general method for uncovering Lyapunov function-s quantifying the global stability of spatially homogeneous systems modeled byFokker-Planck equations [1–12]. We first give a brief review of the general Lya-punov function theory relevant to the current discussion.3 Then we discuss theLyapunov function for deterministic systems and finally for stochastic systems.

Lyapunov Function of Dynamical Systems

We first consider autonomous dynamical systems governed by an ordinary d-ifferential equation (ODE) ˙q = F (q) on an n-dimensional Euclidean state space.

3This material is new and does not exist in Ref. [50].

16

What is particularly interesting is the long-term behavior of a dynamical systemand the stability thereof. The simplest long-term behavior is a fixed point. Thestability of a fixed point can be studied by Lyapunov functions [141]. Withoutloss of generality, assume the fixed point is at the origin q = 0, i.e., F (0) = 0.A Lyapunov function is a continuous scalar function V(q) defined on a region Ωcontaining the fixed point with two additional properties. One is that V(q) is pos-itive definite, i.e., V(q) ≥ 0 with equality if and only if q = 0. The other is that itstime derivative is negative semidefinite, i.e., V(q) ≤ 0 for all q ∈ Ω\0. Whensuch a Lyapunov function exists, the fixed point q = 0 is (Lyapunov) stable, whichmeans a trajectory near the fixed point will indefinitely stay close to it. If the timederivative of V(q) is negative definite (stronger than negative semidefininte), i.e.,V(q) < 0 for all q ∈ Ω\0, then the fixed point q = 0 is asymptotically stable,which means a trajectory near the fixed point will converge to it eventually. Fur-ther, if the Lyapunov function V(q) is defined globally on the entire state spaceΩ = Rn, radially unbounded (i.e., V(q) → ∞ as ||q|| → ∞), and its time deriva-tive is globally negative definite (i.e., V(q) < 0 for all q ∈ Rn\0), then the fixedpoint q = 0 is globally asymptotically stable, which means all the trajectories willconverge to the fixed point eventually. We summarize these results in Table 2.2.

V(q) V(q)

stable V(q) ≥ 0 (“=” iff q = 0) V(q) ≤ 0 ∀q ∈ Ω\0asymptotically

stableV(q) ≥ 0 (“=” iff q = 0) V(q) < 0 ∀q ∈ Ω\0

globallyasymptotically

stable

V(q) → ∞ (||q|| → ∞),V(q) ≥ 0 (“=” iff q = 0)

V(q) < 0 ∀q ∈ Rn\0

Table 2.2: Lyapunov function and stability of a fixed point4

Fixed points are not the only type of long-term behavior a dynamical systemcan exhibit. Other long-term behaviors include limit cycles and strange attractors.For these more general behaviors, the LaSalle’s invariant set theorem generalizesthe above Lyapunov function theory [141]. An invariant set Ξ is a set that evolvesto itself under the dynamics, that is, q(0) ∈ Ξ implies q(t) ∈ Ξ for all t ≥ 0.5 The

4The notation “iff” in the table is short for “if and only if”.5In some literatures this is defined as a positively invariant set.

17

LaSalle’s invariant set theorem is stated as follows [141].LaSalle’s Theorem: Let Ω ⊂ Rn be a compact invariant set with respect to

the dynamics ˙q = F (q). Let V(q) be a continuously differentiable scalar functiondefined on Ω such that V(q) ≤ 0 in Ω. Let Λ be the set of all points in Ω whereV(q) = 0. Let Θ be the largest invariant set in Λ. Then every trajectory starting inΩ approaches Θ as t → +∞.

In the above theorem the function V(q) is not required to be positive definiteas required for the Lyapunov function of a fixed point. It may still be called aLyapunov function in a generalized sense. The three sets in the theorem have therelation Θ ⊆ Λ ⊆ Ω, where Θ and Ω are both invariant sets while Λ = q |V(q) =0 is not necessarily one. Also note that the set Ωc = q |V(q) ≤ c with V(q) ≤ 0in Ωc is an invariant set. Thus if we can find a function V(q) with V(q) ≤ 0 definedon some domain not necessarily invariant, we can restrict that domain to Ωc forsome number c so that V(q) is defined on an invariant set. If Ωc is also compact(equivalent to bounded and closed for Euclidean space) then Ω in the theorem canbe taken as Ωc. In particular, if V(q) is radially unbounded (with V(q) ≤ 0), thenΩc for any c can serve as Ω in the theorem.

In addition to finite dimensional dynamical systems governed by an ODE,there are also dynamical systems described by partial differential equations ormore general functional differential equations on an infinite dimensional statespace. When the state space of an infinite dimensional dynamical system is aHilbert space (a linear space with an inner product that is complete) or more gener-ally a Banach space (a complete normed linear space) which generalize Euclideanspace, the above stability results for finite dimensional dynamical systems in termsof Lyapunov function (that for a fixed point and the LaSalle’s theorem) have beenextended in a pretty straightforward way to these infinite dimensional systems butwith some subtleties involved [142].6 We avoid these technical difficulties for in-finite dimensional dynamical systems in this chapter and work on a more formallevel when dealing with these systems (more details given in Sec. 2.2.1).

Lyapunov Function of the Deterministic Dynamical System

It turns out that the leading order of the potential landscape of the stochasticsystem in the small fluctuation limit gives a Lyapunov function of the correspond-ing deterministic system [81,82]. To investigate the system’s behavior under small

6A subtle difference between an infinite dimensional linear space (e.g., Hilbert space or Banachspace) and a finite dimensional Euclidean space is that a subset of Euclidean space is compact ifand only if it is closed and bounded, while this is not true for an infinite dimensional linear space.

18

fluctuation conditions, we write the diffusion matrix as D(q) = DD(q), wherethe state-independent scale parameter D characterizes the magnitude of the diffu-sion matrix or the fluctuation strength and D(q) is the re-scaled diffusion matrix.The corresponding steady-state Fokker-Planck equation then becomes

−∇ ·(F (q)Ps(q)

)+D∇ · ∇ ·

(D(q)Ps(q)

)= 0. (2.19)

When D is small, the steady state probability distribution has the following asymp-totic form [81, 82]:

Ps(q, D) =1

Z(D)exp [−U(q, D)] =

1

Z(D)exp

[− 1

D

∞∑k=0

DkU (k)(q)

], (2.20)

where Z(D) =∫dq exp [−U(q, D)] is the normalization factor. Inserting E-

q. (2.20) into Eq. (2.19) and keeping only the leading order of D leads to theFokker-Planck Hamilton-Jacobi equation [81, 82]:

HFP (∇U (0), q) = F (q) · ∇U (0)(q) +∇U (0)(q) · D(q) · ∇U (0)(q) = 0, (2.21)

where

U (0)(q) = limD→0

DU(q, D) = limD→0

−D ln[Ps(q, D)Z(D)] (2.22)

is the leading-order potential landscape called the intrinsic potential (also termedthe non-equilibrium potential). Equation (2.21) is of the same form as the Hamilton-Jacobi equation with zero energy in mechanical systems. The correspondingHamiltonian of the system reads:

HFP (p, q) = p · F (q) + p · D(q) · p, (2.23)

where p = ∇U (0) is the canonical momentum conjugate to q. The deterministicdynamics ˙q = F (q) forms a subdynamics (p = 0) of the Hamilton’s canonicalequations determined by the Hamiltonian HFP in Eq. (2.23) [81, 82].

There does not always exist a global single-valued continuously-differentiablefunction U (0)(q) as a solution of the Fokker-Planck Hamilton-Jacobi equation. Fornon-equilibrium systems U (0)(q) may lose its differentiability under certain con-ditions though it is still continuous [81,82]. When the continuously differentiablesolution U (0) of Eq. (2.21) exists, it gives a Lyapunov function of the correspond-ing deterministic system, which is shown in the following. Due to the relation

19

Ps = e−U and Eq. (2.22), U (0) is bounded from below if Ps is normalizable. Also,if natural boundary conditions are used for Ps, then U (0) is radially unbounded.Its time derivative is calculated as follows:

d

dtU (0)(q) =

d

dtq·∇U (0)(q) = F (q)·∇U (0)(q) = −∇U (0)(q)·D(q)·∇U (0)(q) ≤ 0,

where we have used the deterministic equation and the nonnegative-definite prop-erty of the scaled diffusion matrix D(q). Therefore, according to the discussionof the LaSalle’s theorem, the intrinsic potential U (0) is a Lyapunov function ofthe deterministic dynamical system. It can thus be used to study the global (alsolocal) stability of the deterministic system. If D(q) is strictly positive-definite,then U (0)(q) will cease decreasing only when ∇U (0)(q) = 0. Therefore the setΛ = q |U (0)(q) = 0 is simply the set of the extremum points of U (0)(q). Theasymptotic behaviors of the deterministic system (fixed points, limit cycles, or s-trange attractors) are thus contained in the set of the extremum points of U (0)(q)according to the LaSalle’s theorem. In particular, attractors correspond to mini-ma of U (0)(q). Local minima correspond to locally stable attractors, while globalminima correspond to globally stable attractors [81, 82].

Furthermore, the Fokker-Planck Hamilton-Jacobi equation in Eq. (2.21) is e-quivalent to the following two equations [81, 82]:

F (q) = −D(q) · ∇U (0)(q) + V (0)s (q), (2.24)

V (0)s (q) · ∇U (0)(q) = 0. (2.25)

In fact, we can always write down Eq. (2.24) where V(0)s is a vector field to be

determined. Then we plug it into Eq. (2.21) and we can derive Eq. (2.25). Weremark here that Eq. (2.24) is simply the small fluctuation limit of the generalforce decomposition equation in Eq. (2.15). Comparing Eq. (2.24) with Eq. (2.15),we can see the diffusion-induced force ∇ ·D in Eq. (2.15) vanishes in the smallfluctuation limit in Eq. (2.24) so that F ′ becomes F . The potential U in Eq. (2.15)is only left with the lowest order U (0) in Eq. (2.24). V

(0)s (q) in Eq. (2.24) is able

to be identified as the zero-order steady state probability flux velocity in the smallfluctuation limit. It is named as the intrinsic steady state probability flux velocity(or intrinsic flux velocity for short). Equation (2.25) is a result of Eq. (2.16) in thesmall fluctuation limit. It states that the intrinsic flux velocity V

(0)s is perpendicular

to the gradient of the intrinsic potential U (0). This is because the lowest order ofthe divergence term ∇· Vs is one less than the term Vs ·∇U (notice there is a D−1

in U ). This property is not generally true for systems with finite fluctuations.

20

Therefore, in the small fluctuation limit, the potential landscape U and thesteady state probability flux velocity Vs that serve as a bridge between the Fokker-Planck dynamics (governing the global probability transport dynamics) and theLangevin dynamics (governing the single stochastic trajectory dynamics) give aglobal description of the deterministic system in terms of U (0) and V

(0)s . The in-

trinsic potential landscape U (0), related to the stochastic steady state probabilitydistribution in the small fluctuation limit through P

(0)s ∼ exp[−U (0)/D], is the

Lyapunov function of the deterministic non-equilibrium system that quantifies itsglobal stability. The intrinsic potential landscape U (0) together with the intrinsiccurling probability flux velocity V

(0)s , related to the intrinsic steady-state prob-

ability flux by J(0)s = P

(0)s V

(0)s , determine through Eq. (2.24) the driving force

F (q) which governs the deterministic dynamics of the non-equilibrium system.While the topography of the intrinsic potential landscape guides the dynamicsdown the gradient of the landscape, the intrinsic flux velocity drives the system toflow around in a curling way. In particular, since the asymptotic dynamics of thedeterministic system takes place in the set of the extreme points of U (0) where thegradient term vanishes ∇U (0) = 0, the driving force of the asymptotic dynamicsis thus purely the intrinsic flux velocity V

(0)s as seen from Eq. (2.24). It is V (0) that

is responsible for the time-dependent asymptotic dynamics of, for instance, limitcycles and strange attractors. For equilibrium systems, the intrinsic flux veloci-ty V

(0)s vanishes. The entire dynamics of the system is then determined only by

the gradient of the intrinsic potential landscape without the curling flux velocitycontribution. The asymptotic dynamics of equilibrium systems is therefore static(fixed points) since it lacks the driving force V

(0)s .

Lyapunov Function of the Stochastic Dynamical System

To investigate the Lyapunov function of spatially homogeneous dynamicalsystems with finite fluctuations governed by Fokker-Planck equations, we intro-duce the concept of relative entropy (or Kullback-Leibler divergence) well knownin information theory [16, 115, 118, 143].7 The relative entropy of the stochasticspatially homogeneous system with respect to the stationary probability distribu-

7The relative entropy will also be discussed in chapter 3 in the context of non-equilibriumthermodynamics, where it plays an integral role in the formulation of a set of non-equilibriumthermodynamic equations. In this chapter, however, we are not concerned with its thermodynamicmeanings. We just remind the reader here to not confuse it with the regular entropy.

21

tion is defined as follows:

A[Pt(q)] =

∫dq Pt(q)

(ln

Pt(q)

Ps(q)

), (2.26)

where Pt(q) is the solution of the time-dependent Fokker-Planck equation (E-q. (2.8)) and Ps(q) is the solution of the corresponding stationary Fokker-Planckequation (Eqs. (2.11) and (2.12)). We assume Pt(q) and Ps(q) are both positivefunctions normalized to one. The relative entropy quantifies the deviation of thetransient probability distribution Pt from the stationary probability distribution Ps.It turns out to be a Lyapunov function(al) of the stochastic dynamical system.

We note that the stochastic dynamics here is governed by a linear partial differ-ential equation (the Fokker-Planck equation) rather than an ordinary differentialequation. It can be regarded as an infinite dimensional dynamical system. The(stochastic) state of the system is described by a probability distribution functionP (q). The state space is thus an infinite dimensional function space, which can beappropriately defined by specifying P (q) to satisfy certain conditions (e.g., beingpositive, normalized, smooth enough, with certain boundary conditions imposed,with a metric or norm defined etc.). We assume the function state space forms aHilbert or Banach space so that the Lyapunov function theory can still be applied.In this case, the Lyapunov function becomes a functional since it is a function ofthe stochastic state P (q) which in itself is a function. The stationary probabilitydistribution Ps(q) satisfies the stationary Fokker-Planck equation. It is the analogof a fixed point of an ODE. We study its stability by investigating the Lyapunovproperty of the relative entropy defined in Eq. (2.26). We refer readers to Table2.2 for the Lyapunov function theory in relation to a fixed point of an ODE.

To prove the Lyapunov property of A[Pt(q)], we first show it is positive defi-nite and then prove its time derivative is negative semidefinite [1, 2, 7, 12, 16, 100,115, 118, 140, 144].

A[Pt(q)] = −∫

dq Pt(q) lnPs(q)

Pt(q)

≥ −∫

dq Pt(q)

(Ps(q)

Pt(q)− 1

)=

∫dq Pt(q)−

∫dq Ps(q) = 0,

where we have used the inequality lnx ≤ x − 1 for x > 0 and that Ps(q) andPt(q) are both normalized to one. The equality holds if and only if x = 1, i.e.,

22

Pt(q) = Ps(q). Therefore, we have proven A[Pt(q)] is positive definite. Then wecalculate its time derivative.d

dtA[Pt(q)] =

∫dq

∂Pt(q)

∂tln

Pt(q)

Ps(q)+

∫dq

∂Pt(q)

∂t

= −∫

dq(∇ · Jt(q)

)(ln

Pt(q)

Ps(q)

)=

∫dq Jt(q) · ∇

(ln

Pt(q)

Ps(q)

)=

∫dq Pt(q)Vt(q) · ∇

(ln

Pt(q)

Ps(q)

)=

∫dq Pt(q)

[−D(q) · ∇

(ln

Pt(q)

Ps(q)

)+ Vs(q)

]· ∇(ln

Pt(q)

Ps(q)

)= −

∫dq Pt(q)∇

(ln

Pt(q)

Ps(q)

)·D(q) · ∇

(ln

Pt(q)

Ps(q)

)+

∫dq Pt(q)Vs(q) · ∇

(ln

Pt(q)

Ps(q)

),

where we have used Eq. (2.18) and Gauss’s theorem. We assume boundary termsvanish under appropriate boundary conditions. (This forms part of the definitionof the function state space.) The first term in the last equation is non-positive:

−∫

dq Pt(q)∇(ln

Pt(q)

Ps(q)

)·D(q) · ∇

(ln

Pt(q)

Ps(q)

)≤ 0, (2.27)

since D(q) is positive semidefinite and Pt(q) is positive. The second term vanishesbecause ∫

dq Pt(q)Vs(q) · ∇(ln

Pt(q)

Ps(q)

)=

∫dq Ps(q)Vs(q) · ∇

(Pt(q)

Ps(q)

)=

∫dq Js(q) · ∇

(Pt(q)

Ps(q)

)= −

∫dq(∇ · Js(q)

)(Pt(q)

Ps(q)

)= 0,

where we have used ∇ · Js = 0 and Gauss’ theorem. Again we assume certainboundary conditions ensure the boundary terms vanish. Thus we have proven

23

the time derivative of A[Pt(q)] is negative semidefinite (non-positive). Thereforethe relative entropy A[Pt(q)] is a Lyapunov functional of the stochastic spatiallyhomogeneous dynamical system governed by the Fokker-Planck equation. The s-tationary distribution Ps(q) is stable, which means a probability distribution P (q)near Ps(q) will stay indefinitely close to it under the Fokker-Planck dynamics. Ifthe diffusion matrix D(q) is positive definite, which is a stronger condition thanpositive semidefinite, then the equality in dA[Pt(q)]/dt ≤ 0 holds if and only ifPt(q) = Ps(q) according to Eq. (2.27). This means the time derivative of A[Pt(q)]is negative definite. Therefore, in this case, Ps(q) is asymptotically stable. In otherwords, a probability distribution P (q) near Ps(q) will eventually evolve into the s-tationary distribution Ps. Further, P (q) in the definition of A[P (q)] in Eq. (2.26) isarbitrary in an appropriately defined function state space. This suggests that Ps(q)is globally asymptotically stable,8 which would mean that starting from an arbi-trary initial condition P (q, t0) in the state space, the system will eventually evolveinto the stationary probability distribution Ps(q) following the Fokker-Planck dy-namics. This demonstrates how the relative entropy A[Pt(q)] can be used to studythe stability of stochastic dynamical systems with finite fluctuations.

2.1.4 An Illustrative ExampleTo illustrate the general theory presented in previous sections, we study a t-

wo dimensional autonomous dynamical system, which is a particular case of thenormal form of the Hopf bifurcation [141]. Although the asymptotic behaviorof the deterministic dynamics of this system is known, the study of this system(including the stochastic dynamics) using the potential and flux landscape theorypresented below is new.


The state of the system is specified by the state vector (x, y)ᵀ. We assume thedeterministic dynamics of the system follows the following set of ODEs.

x = (1− x2 − y2)x− yy = (1− x2 − y2)y + x

. (2.28)

8For a finite dimensional dynamical system described by an ODE, showing a fixed point tobe globally asymptotically stable requires the condition that the Lyapunov function is radiallyunbounded. For an infinite dimensional dynamical system as is in this case, this condition maybecome subtle and we leave it open.

24

In other words, the deterministic driving force is given by

F (x, y) =

((1− x2 − y2)x− y(1− x2 − y2)y + x

). (2.29)

The stream lines of the vector field F (x, y) are plotted in Fig. 2.1. It shows quiteclearly the dynamical behavior of the system. As seen from Fig. 2.1, the driving

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Figure 2.1: Streamlines of the deterministic driving force

force flows out from the origin and approaches a circle around the origin from theinside. The driving force outside the circle also flows towards and approaches thecircle from the outside. This suggests the system has an unstable fixed point atthe origin and a stable limit cycle around the origin. More rigorously, this systemhas only one fixed point at the origin (x, y) = (0, 0), which is the only solution tothe set of algebraic equations obtained by setting the right side of Eq. (2.28) as 0.Linear stability analysis shows this fixed point is unstable. In addition to the fixedpoint, one can also prove there is a stable limit cycle on the unit circle:

x(t) = cos ty(t) = sin t

. (2.30)

This solution can be verified directly by plugging it into Eq. (2.28). Later onwe shall construct a Lyapunov function using the theory presented in previous

25

sections to identify and study the stability of the fixed point at the origin and thelimit cycle on the unit circle for this particular system.

Langevin Dynamics

We assume the system is influenced by external fluctuations such that the dy-namics of the system can be modeled by the following Langevin equation:

x = (1− x2 − y2)x− y + ξ1(t)y = (1− x2 − y2)y + x+ ξ2(t)

, (2.31)

where ξi(t) (i = 1, 2) are Gauss white noises with the statistical property:

< ξi(t) >= 0, < ξi(t)ξj(t′) >= 2Dδijδ(t− t′), (2.32)

for i, j = 1, 2. This means the two fluctuating forces ξ1(t) and ξ2(t) are statisti-cally independent. Also, the fluctuation strength characterized by D is the samefor both fluctuating forces and is independent of the state of the system. These as-sumptions are reasonable for external fluctuations under certain conditions. Thediffusion matrix in this case is thus given by D(x, y) = DI , where I is 2×2 iden-tity matrix. The solution of the Langevin equation for a specific initial conditionis a stochastic trajectory in the state space.

Fokker-Planck Dynamics

The Fokker-Planck equation governing the evolution of probability distribu-tions which corresponds to the Langevin equation in Eq. (2.31) reads

∂

∂tPt(x, y) = − ∂

∂x

([(1− x2 − y2)x− y]Pt(x, y)

)− ∂

∂y

([(1− x2 − y2)y + x]Pt(x, y)

)(2.33)

+D

(∂2

∂x2+

∂2

∂y2

)Pt(x, y).

The steady state distribution Ps(x, y) satisfies the stationary Fokker-Planck equa-tion which is obtained by setting the right side of Eq. (2.33) as zero. When thedeterministic driving force is a nonlinear function, in general, the Fokker-Planckequation cannot be solved exactly, even for the steady state. However, for thisparticular dynamical system we are studying, there is a rotational symmetry in the

26

system which allows us to obtain the exact form of the steady state distribution ofEq. (2.33) [5, 81]. It is easy to verify directly that the probability distribution

Ps(x, y) =1

Z(D)exp

− 1

4D

(x2 + y2 − 1

)2 (2.34)

is the steady state of Eq. (2.33). Z(D) is the normalization factor given by

Z(D) = π32

√D

[1 + erf

(1

2√D

)],

where erf(·) is the error function defined as erf(z) = 2√π

∫ z

0e−s2ds. The steady

state probability distribution Ps(x, y) in Eq. (2.34) for D = 1 is plotted in Fig. 2.2.Its shape is like a hat. The origin sinks in indicating a lower probability, while thecircle around the origin bulges up indicating a higher probability.

Figure 2.2: Steady state probability distribution for D = 1

Potential and Flux Landscape

With the steady state distribution Ps(x, y) obtained in Eq. (2.34), we now havethe expression of the potential landscape

U(x, y) = − lnPs(x, y) =1

4D

(x2 + y2 − 1

)2+ lnZ(D). (2.35)

27

Note that the potential landscape is defined up to a constant independent of thestate of the system. The term lnZ(D) only depends on D and does not dependon x and y. Thus it has no essential influence on the potential landscape andcan be dropped. Fig. 2.3 shows the shape of the potential landscape. It is like aMexican hat. Contrary to the shape of the hat in Fig. 2.2, now the center bulges upwhile the circle around the center sinks in. This is in agreement with the relationU(x, y) = − lnPs(x, y) and the shape of the steady state distribution shown inFig. 2.2.

Figure 2.3: Potential landscape for D = 1

Calculation of the gradient of U(x, y) gives

∇U(x, y) =1

D

(x2 + y2 − 1

)( xy

). (2.36)

What is also convenient is to consider the negative gradient of U(x, y), i.e., −∇U .For D = 1 its streamlines are plotted in Fig. 2.4. As can be seen in Fig. 2.4, −∇Upoints in the radial direction. Inside the unit circle, it points outward away fromthe origin and towards the unit circle. Outside the unit circle, it points inward andtowards the unit circle. Both regions are showing the behavior of pulling the stateof the system towards the unit circle.

Note that since D = DI , we have ∇·D = 0. Thus the effective driving forceis simply the deterministic driving force in this case: F ′ = F −∇ ·D = F . The

28

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Figure 2.4: Streamlines of the negative gradient of potential landscape for D = 1

force decomposition equation in Eq. (2.15), F ′ = −D ·∇U + Vs, is then reducedto F = −D∇U + Vs. Plugging in the expressions of F in Eq. (2.29) and ∇U inEq. (2.36), we then derive the expression of the steady state flux velocity

Vs(x, y) = F (x, y) +D∇U(x, y) =

(−yx

). (2.37)

The fact that the flux velocity Vs(x, y) is not identically zero indicates that thesystem is in non-equilibrium without detailed balance. This can be verified forconcrete systems that can be modeled by the dynamical system we are studying.Further, one can prove directly that in this particular case the flux velocity Vs isdivergence-free (a curl field) and it is perpendicular to the gradient of the potentiallandscape, that is, ∇ · Vs = Vs · ∇U = 0. (See Table 2.1.) The streamlines ofthe flux velocity are plotted in Fig. 2.5. As we can see, the flux velocity points inthe transverse direction, perpendicular to the radial direction (the direction of thegradient of the potential landscape). This agrees with the fact that Vs · ∇U = 0.Also, the streamlines of Vs form closed circles. This is in agreement with the factthat in this particular case Vs is a curl field satisfying ∇ · Vs = 0.

The force decomposition equation in this case F = −D∇U + Vs decomposesthe driving force of the system into two perpendicular parts. One part is propor-tional to the negative gradient of the potential landscape, which drives the system

29

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Figure 2.5: Streamlines of the flux velocity

in the radial direction. The other part is the flux velocity, which drives the systemin the transverse direction. These two parts together determine the dynamics ofthe system.

Global Stability and Dynamics

We calculate the intrinsic potential landscape and verify its Lyapunov property.According to the expression of in the potential landscape in Eq. (2.35), we canobtain the expression of the intrinsic potential landscape by its definition:

U (0)(x, y) = limD→0

DU(x, y) =1

4

(x2 + y2 − 1

)2. (2.38)

Note that even if one keeps the term lnZ(D) in Eq. (2.35), limD→0D lnZ(D) = 0means it does not contribute to the final result. Also notice that in this particularcase U (0) can be obtained by simply setting D = 1 in the expression of U inEq. (2.35) and removing the additive constant that has no essential effects. Thismeans U and U (0) are proportional to each other in this case. But we should pointout this is a special situation; it does not have to be so in general. The shape of theintrinsic potential landscape looks like a Mexican hat as shown in Fig. 2.6. Thesummit of the mountain at the center corresponds to the origin of the state space,

30

Figure 2.6: The intrinsic potential landscape as the Lyapunov function

while the valley around the mountain corresponds to the unit circle on the statespace. The behavior of the negative gradient of U (0) is also depicted in Fig. 2.4.

We have shown in the general case that U (0) satisfies a Fokker-Planck Hamilton-Jacobi equation (Eq. (2.21)). For this particular system, this equation becomes

F · ∇U (0) +∇U (0) · ∇U (0) = 0, (2.39)

since D = D/D = I in this case. Using the expressions of F in Eq. (2.29) andU (0) in Eq. (2.38), we easily verify that this is indeed true.

Next we verify the Lyapunov property of U (0). The expression of U (0) inEq. (2.38) shows that it is non-negative, i.e., U (0)(x, y) ≥ 0. We calculate its timederivative as follows:

U (0) = x∂xU(0) + y∂yU

(0)

=[(1− x2 − y2)x− y

](x2 + y2 − 1)x

+[(1− x2 − y2)y + x

](x2 + y2 − 1)y

= −(x2 + y2 − 1)2(x2 + y2)

≤ 0. (2.40)

31

Thus we have proven directly that U (0) = (x2 + y2 − 1)2/4 is a Lyapunov func-

tion of the dynamical system in Eq. (2.28). Its extremum states, according to E-q. (2.40), are given by the set Λ = (x, y)|(0, 0), x2+ y2 = 1, which is the originand the unit circle. These correspond to the asymptotic behaviors of the systemin the long term. It may be intuitive to see that the origin as a single point corre-sponds to a fixed point. But it may not be so obvious why the unit circle shouldcorrespond to a limit cycle instead of a continuous set of fixed points which is notimpossible. This can be further understood by the following investigation.

The Fokker-Planck Hamilton-Jacobi equation in Eq. (2.39) is equivalent to thefollowing two equations:

F = −∇U (0) + V (0)s , (2.41)

V (0)s · ∇U (0) = 0. (2.42)

Using the expressions of F in Eq. (2.29) and U (0) in Eq. (2.38), one can calculateV

(0)s from Eq. (2.41) and find that in this particular case V

(0)s = Vs = (−y, x)ᵀ.

That is, the intrinsic flux velocity (in the small fluctuation limit) is equal to theflux velocity without taking the small fluctuation limit in this particular case. ThusFig. 2.5 for flux velocity also applies to the intrinsic flux velocity. Also, in additionto Eq. (2.42) which states that the intrinsic flux velocity is perpendicular to thegradient of the intrinsic potential landscape, we also have ∇·V (0)

s = 0 which statesthat the intrinsic flux velocity is also a curl field in this particular case. However,these are special properties for this particular system. Equation (2.41) is the forcedecomposition equation in the small fluctuation limit. Now the driving force ofthe system is decomposed into two parts. One part is the gradient of the intrinsicpotential landscape and the other part is the intrinsic flux velocity perpendicularto the first part. The gradient part of the driving force guides the system down theintrinsic potential landscape, while the intrinsic flux velocity drives the system inthe transverse direction on the level set of the intrinsic potential landscape. SeeFig. 2.4, Fig. 2.5 and Fig. 2.6 for perspectives. (The field lines of the gradientof the intrinsic potential landscape are along the radial direction. The field linesof the intrinsic flux velocity form circles around the origin.) When the systemreaches its asymptotic sets where ∇U (0) = 0, the intrinsic flux velocity V

(0)s (if

nonzero) continues to drive the asymptotic dynamics. In this particular case, whenthe system is at the origin, we have V

(0)s (0, 0) = (0, 0)ᵀ. Thus the intrinsic flux

velocity vanishes at the origin and it will not drive the system away from theorigin. When the system is on the unit circle with x2 + y2 = 1, the intrinsic fluxvelocity V

(0)s = (−y, x)ᵀ continues to drive the system along the unit circle. This

32

is why the unit circle forms a limit cycle rather than a continuous set of fixedpoints. It is the intrinsic flux velocity V

(0)s that connects the points on the unit

circle to form a limit cycle.Further, the stability of the fixed point at the origin and the limit cycle on the

unit circle can be studied by investigating the property of the extremum statesof the intrinsic potential landscape U (0). If we introduce the radius variable r =√

x2 + y2, then U (0)(r) = (r2−1)2/4. It is easy to prove that r = 0 is a maximumpoint of U (0) while r = 1 is a minimum point of U (0). Thus the fixed point at theorigin is unstable, while the limit cycle on the unit circle is stable. This can alsobe understood from the shape of U (0) in Fig. 2.6. When the system is at the origincorresponding to the summit of the mountain, a small perturbation will drive thesystem away from the summit down the hill. This shows the origin is an unstablefixed point. When the system deviates from the unit circle corresponding to thevalley around the mountain, it will tend to go down the hill and return to the valley.This shows that the limit cycle is stable.

For the stochastic system described by the Fokker-Planck equation in Eq. (2.33),the relative entropy A[Pt] =

∫dxPt ln(Pt/Ps) is a Lyapunov functional of the

Fokker-Planck equation, showing that the probability distribution Pt eventuallyconverge to the stationary distribution Ps following the Fokker-Planck dynamic-s, since in this particular case D = DI is positive definite. However, since wedo not have the exact transient solution Pt(x, y) of this Fokker-Planck equation,we cannot verify it explicitly. In Sec. 3.2.5 of chapter 3, we will study Ornstein-Uhlenbeck processes which can be solved exactly for both the steady state Ps andthe transient state Pt. That allows one to verify the Lyapunov property of therelative entropy A[Pt] explicitly for these systems.

2.2 Global Stability and Dynamics of Spatially In-homogeneous Non-Equilibrium Systems

We extend results in Sec. 2.1 for spatially homogeneous non-equilibrium sys-tems to general spatially inhomogeneous non-equilibrium systems. In Sec. 2.2.1we briefly show how to make systematic extensions from spatially homogeneoussystems to spatially inhomogeneous systems. In Sec. 2.2.2 we focus on the dy-namical equations for spatially inhomogeneous systems. In Sec. 2.2.3 we extendthe potential and flux landscape framework to general spatially inhomogeneousnon-equilibrium systems described by functional Fokker-Planck equations. In

33

Sec. 2.2.4 we generalize the method of uncovering Lyapunov functions to de-terministic and stochastic spatially inhomogeneous non-equilibrium systems de-scribed by functional Fokker-Planck equations.

2.2.1 The Method of Formal ExtensionWe give a short introduction to the formal extension method. For a more elabo-

rate presentation, we refer to the supplementary material of Ref. [50]. This methodexploits the connection of formal notations between spatially homogeneous andinhomogeneous systems. It facilitates systematic formal extensions from spatiallyhomogeneous systems to spatially inhomogeneous systems, without confrontingdirectly the mathematical difficulties involved in dealing with spatially inhomoge-neous systems with infinite degrees of freedom. A more mathematically rigoroustreatment of spatially inhomogeneous systems based on stochastic dynamics onHilbert space will be presented in Sec. 3.3 of chapter 3. In this chapter we sim-ply assume that the state space of spatially inhomogeneous systems we considerforms a Hilbert or Banach space.

For a spatially homogeneous system the state of the system with n degrees offreedom at any given moment can be specified by a single n-dimensional statevector q = q1, ..., qa, ..., qn. For a spatially inhomogeneous system (or spatiallyextended system, field) we assume the state of the system at any given moment canbe specified by a complete set of local quantities, such as local concentrations orlocal densities, that can be regarded as continuous functions of the physical space,represented as a vector-valued field ϕ(x) = ϕ1(x), ..., ϕa(x), ..., ϕn(x). Thusthe system has infinite degrees of freedom, labeled by the discrete vector indexa and the continuous space index x together. Formally, the extension of the stateof the system from a spatially homogeneous system to a spatially inhomogeneoussystem is achieved through extending the discrete index a of q to a discrete indexa and a continuous space index x of ϕ(x). To facilitate the extension initially, wecan introduce a discrete implicit space index λ and a discrete explicit space indexxλ. Then the formal extension procedure from q to ϕ(x) is represented by thefollowing flow chart.

qcomponentize−−−−−−−−−−−−

vectorizeqa

extend index−−−−−−−−−−−−reduce index

ϕ(aλ)

explicitize−−−−−−−−−−implicitize

ϕa(xλ)continuize−−−−−−−−−−discretize

ϕa(x)vectorize−−−−−−−−−−−−

componentizeϕ(x)

(2.43)Notice the index λ is written as if it is a vector index as a is, while xλ is writtenas if it is a continuous space function argument x. This procedure does not only

34

apply to extension of states but also applies to all state indexes, such as the indexof the driving force and the diffusion matrix. Furthermore, a state function for aspatially homogeneous system O(q) will become a functional O[ϕ] for a spatiallyinhomogeneous system, where [ϕ] is a short notation indicating dependence onthe function ϕ(x). Sum over the discrete index will become integral for the con-tinuous index. Partial derivatives over the state variables will become functionalderivatives. Integrals over the state variables will become functional integrals.Some of the most used extensions are summarized in the following.

(q)extension−−−−−−−−−−reduction

[ϕ]

F (q)extension−−−−−−−−−−reduction

F (x)[ϕ]

D(q)extension−−−−−−−−−−reduction

D(x, x ′)[ϕ]∑λ

continuize−−−−−−−−−−discretize

∫dx

∆V

δλλ′continuize−−−−−−−−−−discretize

∆V δ(x− x ′)

∂

∂ϕa(xλ)


∆Vδ

δϕa(x)∏λ

∫dϕ(xλ)


∫D[ϕ] (2.44)

2.2.2 Deterministic and Stochastic Dynamics of Spatially In-homogeneous Systems

We use the results summarized in Eqs. (2.43) and (2.44) to formally extendsome of the most relevant and essential dynamical equations of spatially homo-geneous systems to spatially inhomogeneous systems. A more mathematicallyrigorous treatment will be given in Sec. 3.3.2.


The formal extension of the deterministic driving force to spatially inhomo-geneous systems is the deterministic driving force field functional F (x)[ϕ] (seeEq. (2.44)). The deterministic dynamical equation for spatially inhomogeneous

35

systems (i.e., the deterministic field equation) as an extension of Eq. (2.1) is there-fore [51]:

∂

∂tϕ(x, t) = F (x)[ϕ]. (2.45)

In many cases F (x)[ϕ] can be represented by (nonlinear) partial differential oper-ators or integral operators acting on the field ϕ(x). For reaction diffusion systems,the deterministic driving force functional has the form F (x)[ϕ] = f(ϕ(x)) +

D · ∇2ϕ(x), which is determined only by the local values of the field ϕ(x). Ingeneral, the driving force functional could have non-local forms, for instance,F (x)[ϕ] =

∫dx ′G(x, x ′) · ϕ(x ′).

Functional Langevin Equation

When stochastic fluctuations (intrinsic or extrinsic) are present, a stochasticdescription is required. We assume the stochastic spatially inhomogeneous sys-tems we are concerned with can be modeled by the following functional Langevinequation [15, 45–47, 51]:

∂

∂tϕ(x, t) = F (x)[ϕ] + ξ(x, t)[ϕ], (2.46)

where ξ(x, t)[ϕ] is the stochastic driving force field functional accounting for theeffect of stochastic fluctuations that change the state of the system. This equationis a formal extension of Eq. (2.3). ξ(x, t)[ϕ] in general can be decomposed in termsof different sources generating statistically independent stochastic fluctuations.We label these stochastic fluctuation sources by a discrete index s and a continuousindex y together. The discrete index s does not have to be the state space vectorindex and the continuous index y does not have to be the space index. Then as aformal extension of Eq. (2.4) we have

ξ(x, t)[ϕ] =∑s

∫dy G (sy)(x)[ϕ] Γs(y, t), (2.47)

where Γs(y, t) are Gaussian noises with the following statistical property:

< Γs(y, t) >= 0, < Γs(y, t)Γs′(y′, t) >= δss′δ(y − y ′)δ(t− t′). (2.48)

The statistical property of the stochastic driving force can be obtained by E-qs. (2.47) and (2.48):

< ξ(x, t)[ϕ] >= 0, < ξ(x, t)[ϕ]ξ(x ′, t′)[ϕ] >= 2D(x, x ′)[ϕ]δ(t− t′), (2.49)

36

where D(x, x ′)[ϕ] is the diffusion matrix functional accounting for the combinedeffect of stochastic fluctuations from all the independent sources labeled by (sy)and is given by

D(x, x ′)[ϕ] =1

2

∑s

∫dy G (sy)(x)[ϕ]G (sy)(x

′)[ϕ]. (2.50)

D(x, x ′)[ϕ] has the nonnegative-definite property by construction. For any realvector-valued integrable functions f(x), the following inequality holds∫∫

dxdx ′ f(x) ·D(x, x ′)[ϕ] · f(x ′) ≥ 0. (2.51)

Functional Fokker-Planck Equation

Once the functional Langevin equation for the spatially inhomogeneous sys-tems is written down, we can investigate the evolution of individual stochastictrajectories in the field configuration state space. Since the evolution of the stateof the system is stochastic, what is useful to know is how the probability distribu-tion of the state of the system evolves with time. In other words, what would be thecorresponding Fokker-Planck equation governing the evolution of the probabilitydistribution for spatially inhomogeneous dynamical systems?

From the form of the Fokker-Planck equation for spatially homogeneous sys-tems given by Eq. (2.8) and the method of formal extension as summarized inEqs. (2.43) and (2.44), we could expect the Fokker-Planck equation for spatiallyinhomogeneous systems would become some form of functional Fokker-PlanckEquation involving functionals and functional derivatives. In the following wegive a formal derivation of the functional Fokker-Planck equation using the for-mal extension method. First we write Eq. (2.8) in its component form:

∂

∂tPt(q) = −

∑a

∂a (Fa(q)Pt(q)) +∑ab

∂a∂b (Dab(q)Pt(q)) . (2.52)

Then we extend the input of the state of the spatially homogeneous system (q) tothe functional dependence form of [ϕ] of spatially inhomogeneous systems:

∂

∂tPt[ϕ] = −

∑a

∂a (Fa[ϕ]Pt[ϕ]) +∑ab

∂a∂b (Dab[ϕ]Pt[ϕ]) .

Next we extend the space index a and b to the double discrete index (aλ) and (bλ′)and further explicitize the space index xλ and xλ′ . The extension of the derivative

37

∂a =∂

∂qato double index with the explicit space index xλ is

∂

∂ϕa(xλ). So now

we have

∂

∂tPt[ϕ] = −

∑aλ

∂

∂ϕa(xλ)(Fa(xλ)[ϕ]Pt[ϕ])

+∑aλbλ′

∂

∂ϕa(xλ)

∂

∂ϕb(xλ′)(Dab(xλ, xλ′)[ϕ]Pt[ϕ]) .

Then we use the rules of continuization for the sum of the discrete space index λand for the partial derivatives given in the Eq. (2.44):∑

λ


∫dx

∆V,

∂

∂ϕa(xλ)


∆Vδ

δϕa(x).

Combining them together gives∑λ

∂

∂ϕa(xλ)


∫dx

δ

δϕa(x).

We also replace Fa(xλ) and Dab(xλ, xλ′)[ϕ] with their continuous forms Fa(x)and Dab(x, x

′)[ϕ]. Doing all these together we obtain the following functionalFokker-Planck equation in its component form:

∂

∂tPt[ϕ] = −

∑a

∫dx

δ

δϕa(x)(Fa(x)[ϕ]Pt[ϕ])

+∑ab

∫∫dxdx ′ δ

δϕa(x)

δ

δϕb(x ′)(Dab(x, x

′)[ϕ]Pt[ϕ]) .(2.53)

By using the vector and matrix form and the short notation for the functionalderivative, we have the following functional Fokker-Planck equation in its vector-matrix form:

∂

∂tPt[ϕ] = −

∫dx δϕ(x) ·

(F (x)[ϕ]Pt[ϕ]

)+

∫∫dxdx ′δϕ(x) · δϕ(x ′) · (D(x, x ′)[ϕ]Pt[ϕ]) . (2.54)

Thus we have obtained the general form of the functional Fokker-Planck equationgoverning the evolution of the probability distribution functional for general spa-tially inhomogeneous systems through applying the formal extension method. A

38

special form of the general functional Fokker-Planck equation given in Eq. (2.54)has been derived before for reaction diffusion systems through the procedure ofdiscretizing space and taking the continuum limit [15]. Here we have given a for-mal derivation, equivalent to discretizing space and taking the continuum limit, ofthe more general form of the equation. Functional Fokker-Planck equations havebeen utilized, for instance, in studying biological systems [145]. Such functionalFokker-Planck equations can also arise from functional master equations for spa-tially inhomogeneous systems under certain conditions [15, 51]. In that case thestochastic fluctuations are intrinsic and D(x, x ′)[ϕ] has specific forms.

The functional Fokker-Planck equation in Eq. (2.54) can be interpreted as acontinuity equation in the field configuration state space representing probabilityconservation:

∂

∂tPt[ϕ] = −

∫dx δϕ(x) · Jt(x)[ϕ], (2.55)

where the probability flux field functional is given by

Jt(x)[ϕ] = F ′(x)[ϕ]Pt[ϕ]−∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′)Pt[ϕ], (2.56)

where we have introduced the effective driving force field functional

F ′(x)[ϕ] = F (x)[ϕ]−∫

dx ′δϕ(x ′) ·D(x, x ′)[ϕ]. (2.57)

The functional Fokker-Planck Equation in the form of a continuity equation indi-cates the dynamical evolution of the system governed by the functional Fokker-Planck equation is a probability transport dynamics in the field configuration statespace. Since the field configuration state space is an infinite dimensional functionspace, it is much harder to ‘visualize’ it compared to spatially homogeneous sys-tems (especially systems with low degrees of freedom). Yet some analogies arestill useful. Probability distribution in the field configuration state space is redis-tributed through probability transport process that is determined by the probabilityflux field. Equation (2.56) indicates that there are two sources contributing to theprobability flux field. One source is through ‘drifting’ where the effective drivingforce field functional F ′(x)[ϕ] represents a steady drift velocity field in the fieldconfiguration state space, instructing how probability is transported through drift-ing. The other source is through ‘diffusing’ where the diffusion matrix functionalin the field configuration state space instructs how probability diffuses from high

39

density regions to low density regions. Thus the probability distribution function-al generates a probability flux field functional in the field configuration state spacethrough the drift effect and the diffusion effect, which are instructed, respectively,by the effective driving force field functional and the diffusion matrix function-al. The generated probability flux field functional transports probability in thefield configuration state space and thus redistributes the probability distributionfunctional. The redistributed probability distribution functional again generatesa probability flux field functional through drift and diffusion effects. That againredistributes the probability distribution. Thus the probability transport dynamicsin the field configuration state space is embodied in the feedback loop between theprobability distribution functional and the probability flux field functional, whichis instructed by the deterministic driving force field functional and the diffusionmatrix functional. The probability flux field functional is the driving force of theprobability transport dynamics in the field configuration state space. The proba-bility transport dynamics in the state space for both spatially homogeneous andinhomogeneous systems can be represented by the following graph, where ‘PD’represents ‘Probability Distribution’ and ‘PF’ represents ‘Probability Flux’.

..PD .PF.diffuse

.drift

.redistribute

Figure 2.7: Probability transport dynamics

2.2.3 Potential and Flux Field Landscape Theory for SpatiallyInhomogeneous Non-Equilibrium Systems

We extend the potential and flux landscape theory for spatially homogeneoussystems presented in Sec. 2.1.2 to spatially inhomogeneous systems. We also dis-cuss some aspects specific to spatially inhomogeneous systems that was not re-vealed in spatially homogeneous systems.

For steady state probability distribution ∂Ps[ϕ]/∂t = 0, the functional Fokker-

40

Planck equation in the form of Eqs. (2.55) and (2.56) become∫dx δϕ(x) · Js(x)[ϕ] = 0, (2.58)

Js(x)[ϕ] = F ′(x)[ϕ]Ps[ϕ]−∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′)Ps[ϕ]. (2.59)

Equation (2.58) is a global constraint which requires the integral over the wholephysical space to be 0 rather than requiring the integrand itself to be 0. The steadystate probability flux field functional Js(x)[ϕ] has vanishing functional divergenceeverywhere in the field configuration state space. By analogy with spatially ho-mogeneous systems, we may call it a ‘solenoidal’ or curl vector field in the fieldconfiguration state space, as it has no sinks or sources and thus having to rotatearound. F (x)[ϕ] governing the deterministic dynamics and D(x, x ′)[ϕ] account-ing for the stochastic fluctuation dynamics together through Eq. (2.59) determinethe steady state probability distribution functional Ps[ϕ] and the curl steady stateprobability flux field functional Js(x)[ϕ]. Ps[ϕ] characterizes the global stochasticsteady state of spatially inhomogeneous systems while Js(x)[ϕ] characterizes theglobal steady probability transport dynamics in the field configuration state space.

Detailed Balance and Local Equilibrium Condition

The detailed balance condition as the equilibrium condition characterizing mi-croscopic reversibility is represented by vanishing steady state probability flux:

Js(x)[ϕ] = 0. (2.60)

(This is also for even state variables. Its further extension is given in chapter 3.)This condition means there is no probability transport dynamics happening in thefield configuration state space. The global stochastic steady state and dynamicsof the system is characterized by Ps[ϕ] alone without Js(x)[ϕ]. For spatially in-homogeneous systems Eq. (2.60) requires the probability flux field Js(x)[ϕ] to bezero in every physical space point (or spatial cell considering finite resolutions)for every state the system is in. Thus the detailed balance condition for spatial-ly inhomogeneous systems has a local nature (balanced on every physical spacepoint or in every spatial cell) which characterizes the local equilibrium conditionof the system. The detailed balance condition as the local equilibrium condition

41

for spatially inhomogeneous systems poses a very strong constraint on the sys-tem. Setting the left side of Eq. (2.59) as 0, dividing Ps[ϕ] on both sides of theequation and rearranging it lead to the ‘potential condition’ for spatially inhomo-geneous systems, which is an extension of the potential condition for spatiallyhomogeneous systems given in Eq. (2.13):

F ′(x)[ϕ] = −∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′)U [ϕ], (2.61)

where U [ϕ] ≡ − lnPs[ϕ] is the equilibrium potential field functional. Hence, forspatially inhomogeneous systems when the detailed balance condition as the lo-cal equilibrium condition is satisfied, the effective driving force field functionalhas the form of the functional gradient of the the potential field landscape with re-spect to the diffusion matrix functional D(x, x ′)[ϕ]. U [ϕ] serves as a link betweenF (x)[ϕ] (determining the deterministic dynamics) and D(x, x ′)[ϕ] (characteriz-ing the stochastic fluctuation dynamics), which places a constraint on these two.When the diffusion matrix functional D(x, x ′)[ϕ] is non-singular ( x and x ′ areregarded as two continuous matrix indexes together with two discrete matrix in-dexes within D ), Eq. (2.61) can be written explicitly as a constraint equation onF ′(x)[ϕ] and D(x, x ′)[ϕ]:

δϕb(x ′)

[∫dx ′′D−1

ac (x, x′′)[ϕ]Fc(x

′′)[ϕ]

]= δϕa(x)

[∫dx ′′D−1

bc (x′, x ′′)[ϕ]Fc(x

′′)[ϕ]

], (2.62)

where the inverse of D(x, x ′)[ϕ] is defined by the following equation:∑b

∫dx ′Dab(x, x

′)D−1bc (x

′, x ′′) = δacδ(x− x ′′). (2.63)

Equation (2.62) is the extension of Eq. (2.14) to spatially inhomogeneous systems.

Detailed Balance Breaking and Non-Equilibrium Condition

In general, the detailed balance as the local equilibrium condition may notbe satisfied, in which case the system can have non-zero probability flux fieldJs(x)[ϕ] = 0, which characterizes detailed balance breaking and indicates that thesystem is not in local equilibrium. Js(x)[ϕ] = 0 means it is not 0 all over the phys-ical space for every state the system is in. Yet it can still be 0 in some regions in the

42

physical space. So there could be a distinction between partial non-equilibriumand complete non-equilibrium situations in the physical space for spatially inho-mogeneous systems. When the probability flux field is non-zero in some regionsin the physical space while still 0 in some other regions, some parts of the systemare in non-equilibrium while others are still in equilibrium locally. In this casewe can say the system is in partial non-equilibrium (or partial equilibrium) in thephysical space. When the probability flux field is non-zero globally at every spacepoint, we can say the system is in complete non-equilibrium in the physical space,since all the parts of the system are in non-equilibrium characterized by non-zeroprobability flux. This is a new feature manifested in spatially inhomogeneoussystems that was not revealed in spatially homogeneous systems.

Dividing Ps[ϕ] on both sides of Eq. (2.59) and rearranging the equation givesthe following functional force decomposition equation, which is an extension ofEq. (2.15) to spatially inhomogeneous systems:

F ′(x)[ϕ] = −∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′)U [ϕ] + Vs(x)[ϕ], (2.64)

where U [ϕ] = − lnPs[ϕ] is the generalized non-equilibrium potential field andVs(x)[ϕ] = Js(x)[ϕ]/Ps[ϕ] is the steady state probability flux velocity field. Theyare related by ∫

dx Vs(x)[ϕ] · δϕ(x)U [ϕ] =

∫dx δϕ(x) · Vs(x)[ϕ], (2.65)

which is a result of Eq. (2.58) expressed in terms of U [ϕ] and Vs(x)[ϕ]. Thus innon-equilibrium spatially inhomogeneous systems with detailed balance breaking,the effective driving force field can be decomposed into two terms. One term isthe functional gradient (with respect to the diffusion matrix functional) of thepotential field U , which characterizes the global stochastic state of the system.The other term is the curling probability flux velocity field which represents theeffect of detailed balance breaking that drives the system away from equilibrium.

In Sec. 2.2.4 we will see that in the small fluctuation limit the zero-order poten-tial field landscape becomes a Lyapunov functional of the deterministic spatiallyinhomogeneous system quantifying the global stability of the system. In the smallfluctuation limit the functional force decomposition equation states how the zero-order potential field landscape U [ϕ] and the probability flux velocity field func-tional Vs(x)[ϕ] together determine the deterministic driving force field functionalof the system (governing the deterministic dynamics of the spatially inhomoge-neous system). Therefore, for non-equilibrium spatially inhomogeneous systems,

43

while the global stability is quantified by the underlying potential field landscapeU [ϕ], quantification of the underlying global dynamics of the non-equilibrium s-patially inhomogeneous systems requires both the potential field landscape U [ϕ]

and the probability flux velocity field Vs(x)[ϕ], in contrast to equilibrium systemswhere U [ϕ] alone is sufficient. As discussed in Sec. 2.1.2 for spatially homoge-neous systems, the force decomposition equation is only one perspective of manyto look at that equation. Other perspectives discussed there may also apply heresimilarly and we shall not repeat again. We summarize the equilibrium and non-equilibrium dynamics of spatially inhomogeneous systems in Table 2.3.

Detailed Balance Detailed Balance Breaking

Js(x)[ϕ] Js(x)[ϕ] = 0 Js(x)[ϕ] = 0,∫dx δϕ(x) · Js(x)[ϕ] = 0

Vs(x)[ϕ] Vs(x)[ϕ] = 0Vs(x)[ϕ] = 0,

∫dx Vs(x)[ϕ] · δϕ(x)U [ϕ] =∫

dx δϕ(x) · Vs(x)[ϕ]

F ′(x)[ϕ]

F ′(x)[ϕ] =−∫dx ′D(x, x ′)[ϕ]·δϕ(x ′)U [ϕ]

F ′(x)[ϕ] =

−∫dx ′D(x, x ′)[ϕ]·δϕ(x ′)U [ϕ]+Vs(x)[ϕ]

Table 2.3: (non-)equilibrium dynamics of spatially inhomogeneous systems9

Similar to spatially homogeneous systems, a time-dependent force decom-position equation can also be derived from the definition of the time-dependentprobability flux field in Eq. (2.56):

F ′(x)[ϕ] = −∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′)S[ϕ] + Vt(x)[ϕ], (2.66)

where S[ϕ] ≡ − lnPt[ϕ] and Vt(x)[ϕ] ≡ Jt(x)[ϕ]/Pt[ϕ]. From Eq. (2.64) andEq. (2.66) we also have the following equation:

Vt(x)[ϕ] = −∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′) ln

(Pt[ϕ]

Ps[ϕ]

)+ Vs(x)[ϕ], (2.67)

which will used in the study of the Lyapunov function(al) of spatially inhomoge-neous stochastic dynamics systems.

9These relations are further generalized in chapter 3.

44

2.2.4 Lyapunov Functional Quantifying the Global Stability ofSpatially Inhomogeneous Non-Equilibrium Systems

We extend the methods that are used to uncover Lyapunov functions of spa-tially homogeneous systems discussed in Sec. 2.1.3 to spatially inhomogeneoussystems to quantify the global stability of the system. Similar to spatially homo-geneous systems, the intrinsic potential field landscape of the spatially inhomo-geneous system in the small fluctuation limit is a Lyapunov functional of the de-terministic system quantifying its global stability. The relative entropy functionalof the spatially inhomogeneous system is a Lyapunov functional of the stochasticsystem with finite fluctuations which can be used to explore its global stability.

Lyapunov Functional of Deterministic Spatially Inhomogeneous Systems

To be able to adjust the strength of fluctuation, which usually applies to sys-tems with extrinsic fluctuations, we write D(x, x ′)[ϕ] = DD(x, x ′)[ϕ]. Then thecorresponding stationary functional Fokker-Planck equation becomes

−∫

dx δϕ(x) ·(F (x)[ϕ]Ps[ϕ]

)+ D

∫∫dxdx ′δϕ(x) · δϕ(x ′) ·

(D(x, x ′)[ϕ]Ps[ϕ]

)= 0. (2.68)

When D is small, assume the steady state distribution has the asymptotic form:

Ps(D)[ϕ] =exp −U(D)[ϕ]

Z(D)=

1

Z(D)exp

[− 1

D

∞∑k=0

DkU (k)[ϕ]

]. (2.69)

Plugging it into Eq. (2.68) and keeping only the leading order of D, we derive thefunctional Fokker-Planck Hamilton-Jacobi equation:

HFFP [ δϕU(0), ϕ ]

=

∫dx F (x)[ϕ] · δϕ(x)U

(0)[ϕ]

+

∫∫dxdx ′

(δϕ(x)U

(0)[ϕ])· D(x, x ′)[ϕ] ·

(δϕ(x ′)U

(0)[ϕ])

= 0, (2.70)

45

where U (0)[ϕ] = limD→0

DU(D)[ϕ] = limD→0

[−D ln (Ps(D)[ϕ]Z(D))] is the zero-order potential field landscape of spatially inhomogeneous systems. We call itthe intrinsic potential field landscape. Note that δϕU (0) is a short notation forδϕ(x)U

(0)[ϕ]. Equation (2.70) has the form of a Hamilton-Jacobi equation for fieldswith zero energy. The corresponding Field Hamiltonian of the system is

HFFP [ π, ϕ ] =

∫dx F (x)[ϕ] · π(x) +

∫∫dxdx ′π(x) · D(x, x ′)[ϕ] · π(x ′),

(2.71)where π(x) = δϕ(x)U

(0)[ϕ] is the canonical momentum field conjugate to ϕ(x).As the solution of the functional Fokker-Planck Hamilton-Jacobi equation,

U (0)[ϕ] can be proven to be the Lyapunov functional of the corresponding de-terministic system governed by the deterministic field equation ∂ϕ(x, t)/∂t =

F (x)[ϕ]. The time derivative of U (0)[ϕ] is non-negative as shown below:

d

dtU (0)[ϕ] =

∫dx

∂ϕ(x, t)

∂t· δϕ(x)U

(0)[ϕ]

=

∫dx F (x)[ϕ] · δϕ(x)U

(0)[ϕ]

= −∫∫

dxdx ′(δϕ(x)U

(0)[ϕ])· D(x, x ′)[ϕ] ·

(δϕ(x ′)U

(0)[ϕ])

≤ 0.

Within the proof we have used the chain rule of taking derivatives involving func-tionals, the deterministic field equation, and the nonnegative-definite property ofthe diffusion matrix functional given by Eq. (2.51). If we require D(x, x ′)[ϕ]

and thus D(x, x ′)[ϕ] to be positive definite, then the equality holds if and onlyif δϕ(x)U

(0)[ϕ] = 0. Therefore the intrinsic potential field landscape U (0)[ϕ] isa Lyapunov functional of the spatially inhomogeneous deterministic dynamicalsystems. It decreases monotonically in time while the corresponding determin-istic system reaches its asymptotic behaviors, which are contained within the setof the extremum states of U (0)[ϕ], i.e., Λ = ϕ(x)|δϕ(x)U (0)[ϕ] = 0. Thereforewe can use the intrinsic potential field landscape to quantify the global stability ofspatially inhomogeneous deterministic dynamical systems.

Similar to spatially homogeneous systems, Eq. (2.70) is equivalent to the fol-lowing two equations:

F (x)[ϕ] = −∫

dx ′D(x, x ′)[ϕ] ·(δϕ(x ′)U

(0)[ϕ])+ V (0)

s (x)[ϕ], (2.72)

46

∫dx V (0)

s (x)[ϕ] · δϕ(x)U(0)[ϕ] = 0. (2.73)

The first equation is the force decomposition equation in the small fluctuationlimit (see Eq. (2.64)), where V

(0)s (x)[ϕ] is the lowest order of Vs(x)[ϕ] in the s-

mall fluctuation limit (the intrinsic probability flux velocity field functional). Thesecond equation is the orthogonality condition stating that V (0)

s (x)[ϕ] is perpen-dicular to the functional gradient of U (0)[ϕ] in the field configuration state space.According to Eq. (2.72), when the system reaches its asymptotic behaviors whereδϕ(x ′)U

(0)[ϕ] = 0, the driving force field of the asymptotic dynamic is V (0)s (x)[ϕ].

The intrinsic potential field landscape U (0)[ϕ] and the intrinsic probabilityflux velocity field functional V (0)

s (x)[ϕ], as the small fluctuation limit of U [ϕ]

and Vs(x)[ϕ], provides a global description in the field configuration state spaceof the deterministic spatially inhomogeneous non-equilibrium system. U (0)[ϕ]is the Lyapunov functional of the deterministic spatially inhomogeneous non-equilibrium system characterizing its global stability. U (0)[ϕ] and V

(0)s (x)[ϕ] to-

gether through Eq. (2.72) determine F (x)[ϕ] that governs the deterministic dy-namics of the system. The intrinsic potential field landscape guides the dynamicsdown the direction of its functional gradient, while the intrinsic probability fluxvelocity field drives the system in a curling way. For equilibrium spatially inhomo-geneous systems, the intrinsic probability flux velocity field functional V (0)

s (x)[ϕ]vanishes. In this case, the dynamics of the system is determined by the functionalgradient of the intrinsic potential field landscape alone. This has been shown to betrue for the well stirred spatially homogeneous systems. Here we see a generalizedduality law in terms of potential field landscape and flux velocity field functionalfor spatially inhomogeneous non-equilibrium systems. In this way, we realize theforce field decomposition in the field configuration state space that determines thenon-equilibrium dynamics of spatially inhomogeneous systems.

Lyapunov Functional of Stochastic Spatially Inhomogeneous Systems

Now we turn to the Lyapunov functional of stochastic spatially inhomoge-neous systems with finite fluctuations. Similar to spatially homogeneous systems,the relative entropy for the functional Fokker-Planck equation is defined as:

A[Pt[ϕ]] =

∫D[ϕ]Pt[ϕ]

(ln

Pt[ϕ]

Ps[ϕ]

). (2.74)

47

We assume Pt[ϕ] and Ps[ϕ] are both positive and normalized to the same constant.We show that A[Pt[ϕ]] has the Lyapunov property. However, we must mention thatthe following ‘proof’ only has a formal nature. (The space of P [ϕ] is a space offunctionals; the property of such a space is unclear.) We first prove that A[Pt[ϕ]] isnon-negative and then prove its time derivative is non-positive. The formal proofis similar to that for spatially homogeneous systems, just in functional languages.

A[Pt[ϕ]] = −∫

D[ϕ]Pt[ϕ]

(ln

Ps[ϕ]

Pt[ϕ]

)≥ −

∫D[ϕ]Pt[ϕ]

(Ps[ϕ]

Pt[ϕ]− 1

)=

∫D[ϕ]Pt[ϕ]−

∫D[ϕ]Ps[ϕ] = 0,

where we have used lnx ≤ x− 1 for x > 0 and Ps[ϕ] and that Pt[ϕ] are both nor-malized to the same constant. Then we calculate the time derivative of A[Pt[ϕ]].

d

dtA[Pt[ϕ]] =

d

dt

∫D[ϕ] Pt[ϕ]

(ln

Pt[ϕ]

Ps[ϕ]

)=

∫D[ϕ]

∂Pt[ϕ]

∂t

(ln

Pt[ϕ]

Ps[ϕ]

)+

∫D[ϕ]

∂Pt[ϕ]

∂t

=

∫D[ϕ]

(−∫

dx δϕ(x) · Jt(x)[ϕ])(

lnPt[ϕ]

Ps[ϕ]

)=

∫D[ϕ]

∫dx Jt(x)[ϕ] ·

[δϕ(x)

(ln

Pt[ϕ]

Ps[ϕ]

)]=

∫D[ϕ]

∫dx Pt[ϕ]Vt(x)[ϕ] ·

[δϕ(x)

(ln

Pt[ϕ]

Ps[ϕ]

)]=

∫D[ϕ]

∫dx Pt[ϕ]

[−∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′) ln

(Pt[ϕ]

Ps[ϕ]

)+Vs(x)[ϕ]

]·[δϕ(x)

(ln

Pt[ϕ]

Ps[ϕ]

)]= −

∫D[ϕ] Pt[ϕ]

∫∫dxdx ′

[δϕ(x)

(ln

Pt[ϕ]

Ps[ϕ]

)]·D(x, x ′)[ϕ] ·

[δϕ(x ′) ln

(Pt[ϕ]

Ps[ϕ]

)]+

∫D[ϕ]

∫dx Pt[ϕ]Vs(x)[ϕ] ·

[δϕ(x)

(ln

Pt[ϕ]

Ps[ϕ]

)],

48

where we have plugged in Eq. (2.67). The first term in the last equation is non-positive, because the diffusion matrix functional within the integrand has thenonnegative-definite property represented by Eq. (2.51) and Pt[ϕ] is positive. Thesecond term in the last line vanishes because∫

D[ϕ]

∫dx Pt[ϕ]Vs(x)[ϕ] · δϕ(x)

(ln

Pt[ϕ]

Ps[ϕ]

)=

∫D[ϕ]

∫dx Pt[ϕ]Vs(x)[ϕ] ·

Ps[ϕ]

Pt[ϕ]δϕ(x)

(Pt[ϕ]

Ps[ϕ]

)=

∫D[ϕ]

∫dx Js(x)[ϕ] · δϕ(x)

(Pt[ϕ]

Ps[ϕ]

)= −

∫D[ϕ]

∫dx(δϕ(x) · Js(x)[ϕ]

)(Pt[ϕ]

Ps[ϕ]

)= 0.

Therefore, dA[Pt[ϕ]]/dt ≤ 0. If the diffusion matrix functional is positive def-inite, the equality holds only when Pt[ϕ] = Ps[ϕ]. In the above formal proofwe have used the continuity equation of the functional Fokker-Planck equationand the Gauss’ theorem in the field configuration state space several times. Weassume certain boundary conditions, if applicable, are imposed so that boundaryterms within applying Gauss’ theorem vanish. Thus the relative entropy A[Pt[ϕ]]has the Lyapunov property of decreasing monotonically in time, while the time-dependent probability distribution functional approaches the steady state proba-bility distribution functional showing its asymptotic stability. The relative entropyquantifies the global stability of stochastic spatially inhomogeneous systems.

2.3 Reaction Diffusion SystemsWe apply the general framework developed in Sec. 2.2 to reaction diffusion

systems [15, 51–53, 146–148] and in particular the Brusselator reaction diffusionmodel [1, 93].10

2.3.1 Dynamics of Reaction Diffusion SystemsWe first consider the deterministic dynamics of reaction diffusion sytems and

then the stochastic dynamics.

10The material of the Brusselator model presented here is new and does not exist in Ref. [50].

49


The macroscopic state of multi-species chemical reaction diffusion systemscan be characterized by the set of local concentrations of each chemical speciesover the physical space ϕ(x) = ϕ1(x), ..., ϕm(x), ...ϕn(x). The macroscopicequation governing the deterministic dynamics of reaction diffusion systems isthe reaction diffusion equation [15, 51, 52]:

∂

∂tϕ(x, t) = f(ϕ(x, t)) +D · ∇2ϕ(x, t), (2.75)

where D is a diagonal matrix with the diagonal element Dm (m = 1, 2, ..., n)as the diffusion constant of chemical species m. The RHS of the equation is thedeterministic driving force of the system. The first term is the chemical reactionforce, coming from the contribution of local chemical reactions. The second termis the the diffusion force from local diffusion processes.

The chemical reaction force has the following decomposed form in terms ofdifferent chemical reactions labeled by the index r:

f(ϕ(x, t)) =∑r

νrwr(ϕ(x, t)). (2.76)

The vector νr, with its m-th component being the stoichiometric coefficient νmr,characterizes the number of molecules involved within chemical reaction r andthus which direction the state of the system changes in the state space due tochemical reaction r. wr(ϕ(x, t)) is the rate of chemical reaction r which charac-terizes how fast the state of the system changes due to chemical reaction r. Itsexpression can be inferred by the law of mass action under certain conditions. νrand wr(ϕ(x, t)) together characterize chemical reaction r.

The diffusion force can also be written alternatively in its decomposed formin terms of different diffusion processes of different chemical species labeled bythe index m [51]:

D · ∇2ϕ(x, t) =∑m

emDm∇2ϕm(x, t), (2.77)

where em = (0, ..., 1, ...0)ᵀ with value 1 at the mth component and 0 elsewhereis the m-th base vector in the concentration space. The vector em characterizeswhich direction the state of the system changes due to the diffusion of speciesm. Dm∇2ϕm(x, t) characterizes how fast the state of the system changes due

50

to the diffusion of species m. Combining Eq. (2.76) and Eq. (2.77) we have thefollowing decomposed form of the deterministic reaction diffusion equation interms of elementary state transitions [51]:

∂

∂tϕ(x, t) =

∑r

νrwr(ϕ(x, t)) +∑m

emDm∇2ϕm(x, t). (2.78)

To illustrate the above ideas, we consider the Brusselator model with oscilla-tory behaviors of chemical reactions [1,93]. It consists of four chemical reactions:

Ak1−→ X

2X + Yk2−→ 3X

B +Xk3−→ Y +D

Xk4−→ E

The two chemical species of interest are X and Y . The concentrations of A andB are kept constant, while D and E are removed immediately upon production.kr (r = 1, 2, 3, 4) are the respective reaction rate constants of these four chemicalreactions. The state of the system is characterized by the concentrations of X andY denoted by [X] and [Y ], respectively. When the chemical species are not wellstirred as these chemical reactions take place, local concentrations of X and Y areused to characterize the state of the system. That means [X] and [Y ] become localquantities (concentration fields) dependent on the physical space coordinate x. Weidentify the two components of the state of the system ϕ(x) = ϕ1(x), ϕ2(x) asfollows: ϕ1(x) = [X] and ϕ2(x) = [Y ]. Each reaction is characterized by a pairνr and ωr. For the first reaction, the number of X increases by 1 while the numberof Y does not change if the reaction takes place once. Thus we have ν1 = (1, 0)ᵀ.According to the law of mass action, the reaction rate of the first reaction is givenby ω1 = k1[A]. For the second reaction, the net increase of the number of X is1 (decreased by 2 and increased by 3) while the number of Y decreases by 1 ifthe reaction happens once. That means ν2 = (1,−1)ᵀ. The reaction rate is givenby ω2 = k2[X]2[Y ] according to the law of mass action. Similarly, for the thirdreaction we have ν3 = (−1, 1)ᵀ and ω3 = k3[B][X]. For the fourth reaction wehave ν4 = (−1, 0)ᵀ and ω4 = k4[X]. We summarize these results in Table 2.4.According to Eq. (2.76), the chemical reaction force for the Brusselator model is

f([X], [Y ]) =4∑

r=1

νrωr =

(k1[A] + k2[X]2[Y ]− k3[B][X]− k4[X]

−k2[X]2[Y ] + k3[B][X]

)(2.79)

51

Reaction 1 Reaction 2 Reaction 3 Reaction 4

νr (1, 0)ᵀ (1,−1)ᵀ (−1, 1)ᵀ (−1, 0)ᵀ

ωr k1[A] k2[X]2[Y ] k3[B][X] k4[X]

Table 2.4: Characterization of chemical reactions in the Brusselator model

When the chemical species are not well stirred, there will be diffusion processesacross the physical space. Denote the diffusion constants of X and Y as D1 andD2, respectively. The diffusion force, according to Eq. (2.77), is given by

D · ∇2ϕ(x, t) =2∑

m=1

emDm∇2ϕm(x, t) =

(D1∇2[X]

D2∇2[Y ]

). (2.80)

Combining Eq. (2.79) and Eq. (2.80) together, we have the deterministic reactiondiffusion equation of the Brusselator model as follows:

∂[X]

∂t= k1[A] + k2[X]2[Y ]− k3[B][X]− k4[X] +D1∇2[X]

∂[Y ]

∂t= −k2[X]2[Y ] + k3[B][X] +D2∇2[Y ]

(2.81)

By making the following scale transformations:√k2/k4 [X] → ϕ1

√k2/k4 [Y ] → ϕ2

k4t → t Dm/k4 → Dm√k21k2/k

34 [A] → A (k3/k4) [B] → B

(2.82)

we can rewrite Eq. (2.81) in dimensionless quantities:∂ϕ1

∂t= A− (B + 1)ϕ1 + ϕ 2

1ϕ2 +D1∇2ϕ1

∂ϕ2

∂t= Bϕ1 − ϕ 2

1ϕ2 +D2∇2ϕ2

(2.83)

52

Functional Langevin Dynamics

When there are stochastic fluctuations influencing the dynamics of the sys-tem, the dynamics becomes stochastic. For simplicity, we only consider stochasticfluctuations from external environmental stochastic influences, or extrinsic fluctu-ations. In addition to the deterministic dynamics of reaction diffusion systems inEq. (2.78), there will also be stochastic fluctuations influencing the dynamics ofthe system. We assume the stochastic reaction diffusion system can be describedby the following functional Langevin equation:

∂

∂tϕ(x, t) =

∑r

νrwr(ϕ(x, t)) +∑m

emDm∇2ϕm(x, t) + ξ(x, t)[ϕ], (2.84)

where ξ(x, t)[ϕ] is the stochastic driving force field functional representing spatial-temporal environmental random influences, with the following statistical property:

< ξ(x, t)[ϕ] >= 0, < ξ(x, t)[ϕ]ξ(x ′, t′)[ϕ] >= 2D(x, x ′)[ϕ]δ(t− t′). (2.85)

Equation (2.84) is a special case of the general functional Langevin equation inEq. (2.46), with the deterministic driving force field functional F (x)[ϕ] adaptedto reaction diffusion systems given by the RHS of Eq. (2.78). Its solution is astochastic trajectory in the concentration field configuration state space.

For the Brusselator model with spatial-temporal stochastic fluctuations, thefunctional Langevin equation becomes:

∂ϕ1

∂t= A− (B + 1)ϕ1 + ϕ 2

1ϕ2 +D1∇2ϕ1 + ξ1

∂ϕ2

∂t= Bϕ1 − ϕ 2

1ϕ2 +D2∇2ϕ2 + ξ2

(2.86)

where ξ1 and ξ2 are two random field functionals, written more specifically asξ1(x, t)[ϕ1, ϕ2] and ξ2(x, t)[ϕ1, ϕ2]. They have the following statistical properties:

< ξi(x, t)[ϕ1, ϕ2] >= 0,

< ξi(x, t)[ϕ1, ϕ2]ξj(x′, t′)[ϕ1, ϕ2] >= 2Dij(x, x

′)[ϕ1, ϕ2]δ(t− t′),

where i, j = 1, 2. The specific forms of Dij(x, x′)[ϕ1, ϕ2] (i, j = 1, 2) are depen-

dent on the properties of the random environmental fluctuations. If these fluctua-tions are independent of the state of the system, which is usually the case for many

53

environmental fluctuations approximately, we can drop the functional dependenceof ϕ1 and ϕ2 and write Dij(x, x

′). If these fluctuations are spatially uncorrelated,then Dij(x, x

′) (i, j = 1, 2) have the form Dij(x)δ(x − x ′). If the fluctuationstrengths are also uniform across space, we have Dijδ(x− x ′). Further, if the tworandom fields ξ1 and ξ2 are uncorrelated and have equal fluctuation strength, thediffusion matrix field functional would have the simple form Dδijδ(x − x ′). Weassume this is the case for simplicity.

Functional Fokker-Planck Dynamics

The evolution of the probability distribution functional corresponding to thefunctional Langevin equation for reaction diffusion systems in Eq. (2.84) is gov-erned by the functional Fokker-Planck equation for reaction diffusion systems:

∂

∂tPt[ϕ] = −

∫dx δϕ(x) ·

([∑r

νrwr(ϕ(x)) +∑m

emDm∇2ϕm(x)

]Pt[ϕ]

)+

∫∫dxdx ′ δϕ(x) · δϕ(x ′) · (D(x, x ′)[ϕ]Pt[ϕ]) , (2.87)

This equation is a special case of the general functional Fokker-Planck equationin Eq. (2.54), where the deterministic driving force field functional F (x)[ϕ] isadapted to reaction diffusion systems.

For the Brusselator model with spatial-temporal stochastic fluctuations char-acterized by the diffusion matrix field functional Dδijδ(x − x ′), the functionalFokker-Planck equation corresponding to its functional Langevin equation in E-q. (2.86) is then given by:

∂

∂tPt[ϕ] = −

∫dx

δ

δϕ1(x)

([A− (B + 1)ϕ1 + ϕ 2

1ϕ2 +D1∇2ϕ1

]Pt[ϕ]

)−∫

dxδ

δϕ2(x)

([Bϕ1 − ϕ 2

1ϕ2 +D2∇2ϕ2

]Pt[ϕ]

)+D

∫dx

[(δ

δϕ1(x)

)2

+

(δ

δϕ2(x)

)2]Pt[ϕ]. (2.88)

Unfortunately, we do not know how to solve this functional differential equationexactly, not even for the steady state.

54

2.3.2 Potential and Flux Field Landscape for Reaction Diffu-sion Systems

We apply the functional force decomposition equation developed for generalspatially inhomogeneous systems in Eq. (2.64) to reaction diffusion systems.[∑

r

νrwr(ϕ(x)) +∑m

emDm∇2ϕm(x)

]−∫

dx ′δϕ(x ′) ·D(x, x ′)[ϕ]

= −∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′)U [ϕ] + Vs(x)[ϕ]. (2.89)

In other words, the effective driving force field for non-equilibrium reaction dif-fusion dynamics (the LHS of the above equation) can be decomposed into twoparts. One part is the functional gradient of the potential field landscape U [ϕ],with respect to the diffusion matrix field D(x, x ′)[ϕ]. The other part is the steadystate probability velocity field Vs(x)[ϕ], which breaks detailed balance and indi-cates the reaction diffusion system is away from local equilibrium. (Note that theterm

∫dx ′δϕ(x ′) ·D(x, x ′)[ϕ] is the force field from inhomogeneous diffusion in

the field configuration state space, which vanishes if stochastic fluctuations areindependent of the state of the system.)

The detailed balance condition as the local equilibrium condition or the poten-tial condition is given by the absence of Vs(x)[ϕ] in Eq. (2.89):[∑

r

νrwr(ϕ(x)) +∑m

emDm∇2ϕm(x)

]−∫

dx ′δϕ(x ′) ·D(x, x ′)[ϕ]

= −∫

dx ′D(x, x ′)[ϕ] · δϕ(x ′)U [ϕ]. (2.90)

This is a special form of the general detailed balance condition or potential con-dition for spatially inhomogeneous systems in Eq. (2.61) adapted to reaction dif-fusion systems. It says the effective driving force field for equilibrium reactiondiffusion systems has the form of a functional gradient of the potential field land-scape (with respect to D(x, x ′)[ϕ]). When this condition is satisfied, the reactiondiffusion system will be in detailed balance in all the spatial points, i.e., in lo-cal equilibrium. When it is not satisfied, non-vanishing Vs(x)[ϕ] will then breakdetailed balance and indicates non-equilibrium conditions of the system.

For the Brusselator reaction diffusion model, the functional force decom-position equation as a special case of Eq. (2.89). Noticing that the diffusion

55

matrix field functional is assumed to have the form Dδijδ(x − x ′), the term∫dx ′δϕ(x ′) · D(x, x ′)[ϕ] vanishes and the spatial integral in the functional gra-

dient term can be carried out. We finally reach the following functional forcedecomposition equation for the Brusselator reaction diffusion model:

A− (B + 1)ϕ1 + ϕ 21ϕ2 +D1∇2ϕ1 = −Dδϕ1U [ϕ] + V s

1 [ϕ]

Bϕ1 − ϕ 21ϕ2 +D2∇2ϕ2 = −Dδϕ2U [ϕ] + V s

2 [ϕ](2.91)

The notation δϕm (m = 1, 2) is short for the functional derivative δ/δϕm(x)(m = 1, 2). The non-zero flux velocity field functionals V s

1 [ϕ] and V s2 [ϕ] indicate

detailed balance breaking and thus non-equilibrium condition for this particularsystem. If the system satisfies the detailed balance condition, they vanish fromthese two equations. We would then have the following potential condition for theBrusselator reaction diffusion model, indicating the system is in equilibrium:

A− (B + 1)ϕ1 + ϕ 21ϕ2 +D1∇2ϕ1 = −Dδϕ1U [ϕ]

Bϕ1 − ϕ 21ϕ2 +D2∇2ϕ2 = −Dδϕ2U [ϕ]

. (2.92)

We investigate whether this potential condition can be satisfied. Note that theabove potential condition implies δϕ1(x)δϕ2(x ′)U [ϕ] = δϕ2(x ′)δϕ1(x)U [ϕ]. Thismeans in Eq. (2.92) the functional derivative δϕ2 of the left side of the first equationand the functional derivative δϕ1 of the left side of the second equation should beequal to each other. We thus have ϕ 2

1 (x)δ(x− x ′) = [B−2ϕ1(x)ϕ2(x)]δ(x− x ′),which further implies ϕ 2

1 (x) = B − 2ϕ1(x)ϕ2(x). This should hold for all al-lowed ϕ1(x) and ϕ2(x). But it is not possible as B is a a parameter independentof the state of the system. This means for stochastic fluctuations characterized bythe diffusion matrix field Dδijδ(x− x ′), the stochastic Brusselator reaction diffu-sion model cannot satisfy the potential condition (detailed balance) in Eq. (2.92),i.e., the system cannot exist in equilibrium. Thus the steady state probability fluxvelocity field functional V s

m[ϕ] (m = 1, 2) must be non-zero, breaking detailedbalance and indicating the system is in non-equilibrium. This seems to agree withthe fact that the Brusselator system has non-equilibrium chemical oscillations.

2.3.3 Lyapunov Functional Quantifying the Global Stability ofReaction Diffusion Systems

We investigate the Lyapunov functions that quantify the global stability ofreaction diffusion systems. The intrinsic potential field landscape U (0)[ϕ] (see E-q. (2.69)) is a Lyapunov functional of the deterministic reaction diffusion system.

56

According to the discussions in Sec. 2.2 for general spatially inhomogeneous sys-tems, U (0)[ϕ] satisfies a functional Fokker-Planck Hamilton-Jacobi equation (E-q. (2.70)). Adapting this equation to reaction diffusion systems gives:

HFFP [δϕU(0), ϕ]

=

∫dx

(∑r

νrwr(ϕ(x)) +∑m

emDm∇2ϕm(x)

)· δϕ(x)U

(0)[ϕ]

+

∫∫dxdx ′

(δϕ(x)U

(0)[ϕ])· D(x, x ′)[ϕ] ·

(δϕ(x ′)U

(0)[ϕ])

= 0, (2.93)

where D(x, x ′)[ϕ] is the rescaled diffusion matrix field functional. The Lyapunovproperty of U (0)[ϕ] for reaction diffusion systems can also be proven by adaptingthe general proof to reaction diffusion systems:

d

dtU (0)[ϕ] =

∫dx

∂ϕ(x, t)

∂t· δϕ(x)U

(0)[ϕ]

=

∫dx

(∑r

νrwr(ϕ(x)) +∑m

emDm∇2ϕm(x)

)· δϕ(x)U

(0)[ϕ]

= −∫∫

dxdx ′(δϕ(x)U

(0)[ϕ])· D(x, x ′)[ϕ] ·

(δϕ(x ′)U

(0)[ϕ])

≤ 0.

Therefore, U (0)[ϕ] quantifies the global stability of the deterministic reaction dif-fusion system. The asymptotic dynamics of the deterministic reaction diffusionsystem in the long term takes place in the region of the extremum states of U (0)[ϕ]in the concentration field configuration state space.

The functional Fokker-Planck Hamilton-Jacobi equation for reaction diffusionsystems in Eq. (2.93) is equivalent to the following two equations:∑

r

νrwr(ϕ(x)) +∑m

emDm∇2ϕm(x)

= −∫

dx ′D(x, x ′)[ϕ] ·(δϕ(x ′)U

(0)[ϕ])+ V (0)

s (x)[ϕ], (2.94)

∫dx V (0)

s (x)[ϕ] · δϕ(x)U(0)[ϕ] = 0. (2.95)

57

According to Eq. (2.94), for non-equilibrium reaction diffusion systems withoutdetailed balance, the intrinsic potential field landscape U (0)[ϕ] and the intrinsicflux velocity field V

(0)s (x)[ϕ] together determine the driving force field and thus

the dynamics of reaction diffusion systems. Also, the asymptotic dynamics ofreaction diffusion systems, where δϕU

(0)[ϕ] = 0, is driven purely by the intrin-sic flux velocity field V

(0)s (x)[ϕ]. For equilibrium reaction diffusion systems with

detailed balance, V (0)s (x)[ϕ] = 0. The dynamics of the equilibrium reaction d-

iffusion system is then only determined by the intrinsic potential field functionalU (0)[ϕ]. And the asymptotic dynamics is static due to the lack of asymptotic driv-ing force field V

(0)s (x)[ϕ].

For reaction diffusion system with finite stochastic fluctuations described bythe functional Fokker-Planck equation (Eq. (2.87)), the relative entropy functionalA[Pt[ϕ]] =

∫D[ϕ]Pt[ϕ] ln (Pt[ϕ]/Ps[ϕ]) has the Lyapunov property of decreasing

monotonically in time. It can be used to study the global stability of the stochasticreaction diffusion system.

Then we further apply the above results of general reaction diffusion systemsto the Brusselator reaction diffusion model. The intrinsic potential field landscapeU (0)[ϕ] of this particular system satisfies the following functional Fokker-PlanckHamilton-Jacobi equation:

HFFP [δϕU(0), ϕ]

=

∫dx(A− (B + 1)ϕ1 + ϕ 2

1ϕ2 +D1∇2ϕ1

)δϕ1U

(0)[ϕ]

+

∫dx(Bϕ1 − ϕ 2

1ϕ2 +D2∇2ϕ2

)δϕ2U

(0)[ϕ]

+

∫dx(δϕ1U

(0)[ϕ])2

+(δϕ2U

(0)[ϕ])2

= 0. (2.96)

Unfortunately, we do not know whether this equation can be solved exactly ei-ther. What we do know is that if we can find a solution U (0)[ϕ] of this equation,then U (0)[ϕ] is a Lyapunov functional of the deterministic Brusselator reactiondiffusion model decreasing monotonically dU (0)[ϕ]/dt ≤ 0. And the asymptot-ic behaviors of the system is contained within the set of the extremum states ofU (0)[ϕ] defined as Λ = ϕ|δϕU (0)[ϕ] = 0.

Applying Eq. (2.94) to the Brusselator reaction diffusion model, we have its

58

functional force decomposition equation in the small fluctuation limit:A− (B + 1)ϕ1 + ϕ 2

1ϕ2 +D1∇2ϕ1 = −δϕ1U(0)[ϕ] + V

(0) s1 [ϕ]

Bϕ1 − ϕ 21ϕ2 +D2∇2ϕ2 = −δϕ2U

(0)[ϕ] + V(0) s2 [ϕ]

(2.97)

This means the driving force field of the deterministic Brusselator reaction diffu-sion model can be decomposed into two parts. One part is the functional gradientof the intrinsic potential field landscape U (0)[ϕ] and the other part is the intrinsicflux velocity field V

(0)s [ϕ]. U (0)[ϕ] as the Lyapunov functional of the deterministic

system guides the system down its functional gradient, while V(0)s [ϕ] drives the

system in the transverse direction, perpendicular to the direction of the function-al gradient of U (0)[ϕ]. This last statement comes from Eq. (2.95), which for theBrusselator reaction diffusion model becomes∫

dx V(0) s1 [ϕ]δϕ1U

(0)[ϕ] + V(0) s2 [ϕ]δϕ2U

(0)[ϕ] = 0. (2.98)

When the system reaches its asymptotic behavior in the long term, the functionalgradient of U (0)[ϕ] vanishes. Thus the asymptotic dynamics is purely driven bythe transverse component V (0)

s [ϕ].For the stochastic Brusselator reaction diffusion model governed by the func-

tional Fokker-Planck equation in Eq. (2.88), we can use the relative entropy func-tional of the system A[Pt[ϕ]] =

∫D[ϕ]Pt[ϕ] ln (Pt[ϕ]/Ps[ϕ]) to investigate the

stochastic system’s global stability since it has the Lyapunov property of decreas-ing monotonically in time. Since we did not (do not know how to) solve the func-tional Fokker-Planck equation for the Brusselator reaction diffusion model, wedo not have the explicit expressions of Pt[ϕ] and Ps[ϕ] to calculate A[Pt[ϕ]] andthus verify its Lyapunov property directly. In chapter 3, we study the stochasticneuronal model which can be solved exactly with analytical expressions obtained.The relative entropy A(t) is calculated explicitly for a specific initial condition.And we can verify explicitly that A(t) decreases with time, which is also illustrat-ed in Fig. 3.1.

2.4 SummaryIn this chapter, we established a potential and flux field landscape theory

to quantify the global stability and dynamics of spatially inhomogeneous non-equilibrium field systems. We extended our potential and flux landscape theory

59

for spatially homogeneous systems to spatially inhomogeneous non-equilibriumfield systems described by functional Fokker-Planck equations. Within this ex-tended potential and flux field landscape framework, we found that for equilibri-um spatially inhomogeneous systems in detailed balance, the potential field alonedetermines both the global stability and dynamics of the system. The topogra-phy of the potential field landscape in terms of the basin of attraction and barrierheights can quantify the global stability of the system. The functional gradien-t of the potential field landscape alone gives the effective driving force field ofthe system. However, for non-equilibrium spatially inhomogeneous systems withdetailed balance broken, although the topography of the potential field landscapecharacterizes the global stability of the system, the dynamics of the system cannotbe determined by the potential field landscape alone. Both the functional gradientof the potential field and the non-zero curl probability flux are required to deter-mine the non-equilibrium dynamics of the spatially inhomogeneous system. Thismimics the dynamics of a charge particle in both an electric field and a magneticfield. The non-zero flux characterizes the non-equilibrium nature of the systemand implies the system is an open system exchanging matter, energy and informa-tion with the environment.

In the small fluctuation limit, the intrinsic potential field as the leading orderof the potential field, closely related to the steady state probability distribution,is a Lyapunov functional of the deterministic spatially inhomogeneous systemthat characterizes the global stability of the deterministic system. In the smallfluctuation limit, the driving force governing the deterministic dynamics of thespatially inhomogeneous system is determined by both the functional gradient ofthe intrinsic potential field and the curling intrinsic probability flux velocity fornon-equilibrium systems with detailed balance broken. The relative entropy func-tional of the stochastic spatially inhomogeneous non-equilibrium system is foundits Lyapunov functional, quantifying the global stability of the stochastic spatiallyinhomogeneous non-equilibrium system with finite fluctuations. We also appliedour general extended framework for spatially inhomogeneous systems to the morespecific reaction diffusion systems with extrinsic fluctuations and illustrated thetheory further with the Brusselator reaction diffusion model.

Our potential and flux field landscape theory offers an alternative generalapproach to other field-theoretic techniques. It is also an extension of the non-equilibrium potential approach by including the indispensable curl flux field, suit-able for studying the global stability and dynamics of spatially inhomogeneousnon-equilibrium field systems. In the next chapter we shall investigate the non-equilibrium thermodynamics in the potential and flux landscape framework.

60

Chapter 3

Non-Equilibrium Thermodynamicsof Stochastic Systems

In this chapter, we explore the non-equilibrium thermodynamics of spatial-ly homogenous and inhomogeneous non-equilibrium systems.1 The key featurein our approach is to use the potential-flux landscape framework as a bridge toconstruct the non-equilibrium thermodynamics from the underlying stochastic dy-namics. This chapater is structured as follows. In Sec. 3.1 we construct the non-equilibrium thermodynamics for spatially homogeneous systems with one statetransition mechanism. In Sec. 3.2 we generalize the non-equilibrium thermody-namic framework to spatially homogeneous systems with multiple state transitionmechanisms. In Sec. 3.3 we formulate the non-equilibrium thermodynamic frame-work for spatially inhomogeneous systems. In Sec. 3.4 we give a summary of thischapter.

3.1 Non-Equilibrium Thermodynamics for Spatial-ly Homogeneous Stochastic Systems with One S-tate Transition Mechanism

We consider spatially homogeneous systems described by Langevin and Fokker-Planck equations with one state transition mechanism. By state transition mecha-

1Most of the material in this chapter (including all the appendices at the end of this dissertation)was originally co-authored with Jin Wang. Reprinted with permission from W. Wu and J. Wang,The Journal of Chemical Physics, 141, 105104 (2014). Copyright 2014, AIP Publishing LLC.

61

nism we mean mechanisms that are responsible for transitions of the state of thesystem, which can take on different forms in different contexts. For a system ofinteracting particles coupled to several heat reservoirs with distinct temperatures,each heat reservoir represents a different state transition mechanism. For chemi-cal reaction systems, each chemical reaction channel is a different state transitionmechanism. The non-equilibrium conditions of a system is usually sustained bycoupling the system to multiple reservoirs, thus involving multiple (state) transi-tion mechanisms. Therefore the study of systems with one transition mechanismmay seem to be of very limited application. Yet we shall show in Sec. 3.2 undercertain conditions the results for one transition mechanism also apply to multipletransition mechanisms that are equivalent to one effective transition mechanism.

We first introduce the Langevin and Fokker-Planck stochastic dynamics. Thenwe present the extended potential-flux landscape framework. The stochastic dy-namics is then placed into the general non-equilibrium thermodynamic context.The more specific non-equilibrium isothermal process is then studied, with thefirst and second laws of thermodynamics uncovered together with a set of non-equilibrium thermodynamic equations. After a summary and discussion, we con-clude this section with an extension of the results to systems with one generaltransition mechanism.

3.1.1 Stochastic DynamicsWe consider spatially homogeneous systems governed by the following Langevin

equation [15, 16, 50, 140]:

dq = F (q, t)dt+∑s

Gs(q, t)dWs(t). (3.1)

Here q is the state vector representing the dynamical variables of the system.Ws(t) (s = 1, 2, ...) are statistically independent standard Wiener processes. Theindex s labels fluctuation sources, which all come from one transition mechanism(e.g., one heat reservoir). For generality, time-dependence of the deterministic andstochastic driving forces F (q, t) and Gs(q, t) are considered, which accounts forchanging external conditions that can be represented by a set of time-dependentexternal control parameters λ [114, 129, 130, 133]. The solution of Eq. (3.1)traces a stochastic trajectory.

When Eq. (3.1) is interpreted as an Ito stochastic differential equation, the evo-lution of the probability distribution is governed by the following Fokker-Planck

62

equation [15, 16]:

∂

∂tPt(q, t) = −∇ ·

(F (q, t)Pt(q, t)−∇ · (D(q, t)Pt(q, t))

), (3.2)

where the drift vector F (q, t) is given by the deterministic driving force and the d-iffusion matrix D(q, t) =

∑s Gs(q, t)Gs(q, t)/2 characterizes the stochastic fluc-

tuating forces. D(q, t) is nonnegative definite symmetric by construction, but werequire it to be positive definite symmetric. Equation (3.2) is to be solved with agiven initial distribution P (q, t0) in a given region of the state space (call it theaccessible state space) with proper boundary conditions imposed (e.g., reflective,periodic or natural boundary conditions) [16]. Pt(q, t) as the solution of Eq. (3.2)may be called the transient probability distribution, with the subscript t indicating‘transient’. We assume certain conditions are satisfied so that Pt(q, t) is alwayspositive and normalized to 1 in the accessible state space [15, 16]. Equation (3.2)has the form of a continuity equation ∂tPt +∇ · Jt = 0, representing probabilityconservation. The transient probability flux Jt is identified as:

Jt(q, t) = F ′(q, t)Pt(q, t)−D(q, t) · ∇Pt(q, t), (3.3)

where F ′(q, t) = F (q, t) − ∇ · D(q, t) is the effective drift vector (or effectivedriving force).

At each instant of time, with the time t in F (q, t) and D(q, t) fixed (equivalent-ly, with the external control parameters λ fixed), we define an instantaneous sta-tionary distribution Ps(q, t) by setting the right side of Eq. (3.2) as zero [114,129]:

∇ ·(F (q, t)Ps(q, t)−∇ · (D(q, t)Ps(q, t))

)= 0. (3.4)

Mathematically, this is a family of stationary Fokker-Planck equations parameter-ized by t (or λ). Each one of them is just the usual stationary Fokker-Planckequation; the existence and uniqueness of its solution is guaranteed by certainconditions on the drift vector, diffusion matrix and boundary conditions [16]. Weshall thus assume that at each instant of time Ps(q, t) is unique, positive and nor-malized. Equation (3.4) also has the form ∇ · Js = 0, with the instantaneousstationary probability flux given by:

Js(q, t) = F ′(q, t)Ps(q, t)−D(q, t) · ∇Ps(q, t). (3.5)

∇ · Js = 0 means Js is divergence-free. There is no sink for the flux to go intoor source for it to come out from; it has to rotate and thus has a curl nature (asolenoidal vector field).

63

The time variable t in Ps(q, t) and Js(q, t) solely comes from the time depen-dence of F (q, t) and D(q, t) that represents changing external conditions. Al-though Ps(q, t) is time-dependent when external conditions change, it is generallynot a solution of the transient (dynamical) Fokker-Planck equation (Eq. (3.2)).In particular, Ps(q, t) does not depend on the initial distribution P (q, t0), whilethe transient distribution Pt(q, t) does [15, 16, 129]. Therefore, Ps(q, t) shouldnot be confused with Pt(q, t). When the external conditions do not change withtime, which means F (q) and D(q) are time-independent, the transient distributionPt(q, t) will relax to the supposedly unique stationary distribution Ps(q) in thelong time limit, and the relaxation process can be characterized by a relaxationtime scale (determined by the eigenvalues of the Fokker-Planck operator) [16].When external conditions change with time, meaning F (q, t) and D(q, t) aretime-dependent, the instantaneous stationary distribution Ps(q, t) plays the role ofthe reference distribution at each moment which the transient distribution Pt(q, t)would try to relax to at that moment. If the external conditions change on a timescale much slower than the relaxation time scale (relatively speaking, relaxationhappens very fast), the system would approximately stay in the instantaneous sta-tionary distribution at each moment and thus Pt(q, t) ≈ Ps(q, t) all the time. In thecontext of thermodynamics this describes quasi-static processes (not necessarilyreversible) [100]. In a comparable situation in quantum mechanics, this definesthe so-called adiabatic processes, where the word adiabatic has no direct relationwith heat exchange, though.

3.1.2 Generalized Potential-Flux Landscape FrameworkThe potential-flux landscape framework presented in the following is an ex-

tension of the previous framework on non-equilibrium steady state processes [9,10, 12–14, 50] and that presented in Sec. 2.1.2 of chapter 2, by accommodatingtransient processes and time-dependent external conditions. It will facilitate theestablishment of the non-equilibrium thermodynamic formalism later on.

The definition of the stationary probability flux in Eq. (3.5) can be reformulat-ed as a stationary dynamical decomposition equation (also called force decompo-sition equation in previous work [9, 10, 13, 14]) and chapter 2:

F ′(q, t) = −D(q, t) · ∇U(q, t) + Vs(q, t), (3.6)

where U(q, t) = − lnPs(q, t) is the stationary potential landscape and Vs(q, t) =

Js(q, t)/Ps(q, t) is the stationary flux velocity. We shall simply refer to flux ve-locity as flux when its meaning is clear from the notation and context. The ‘time’

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t in the above equation is not of a dynamical nature; it can be replaced by externalcontrol parameters λ. In the stationary dynamical decomposition equation, theeffective driving force is decomposed into a gradient-like part of the stationarypotential landscape and a curl-like part of the stationary flux velocity. The wordgradient-like refers to the fact that there is a diffusion matrix D before the gra-dient, while curl-like means Js = PsVs is divergence-free although Vs itself isgenerally not. For even state variables, the detailed balance condition character-izing equilibrium steady state is indicated by vanishing stationary flux, Vs = 0, orequivalently, Js = 0. Equation (3.6) is then reduced to F ′ = −D · ∇U , which isthe potential condition characterizing detailed balance [15, 16]. Thus the dynam-ics of the equilibrium steady state is determined solely by the gradient-like partof the stationary potential landscape, without the contribution from the curl-likepart of the stationary flux. Non-zero stationary flux Vs or Js indicates detailedbalance breaking in the steady state, where the dynamics of the non-equilibriumsteady state is determined by both the gradient-like part of the stationary poten-tial landscape and the curl-like part of the stationary flux. In this sense, thesetwo parts represent, respectively, the detailed balance preserving and detailed bal-ance breaking parts of the steady state. In chapter 2 we have elaborated on theapplication of the stationary dynamical decomposition equation in the study ofthe system’s global stability and dynamics; we shall not repeat this subject herefurther again.

The definition of the transient flux in Eq. (3.3) can be reformulated as a tran-sient dynamical decomposition equation:

F ′(q, t) = −D(q, t) · ∇S(q, t) + Vt(q, t), (3.7)

where S(q, t) = − lnPt(q, t) may be termed the transient potential landscape andVt(q, t) = Jt(q, t)/Pt(q, t) the transient flux velocity. In Eq. (3.7), the effectivedriving force is decomposed into a gradient-like part of the transient potentiallandscape and a part that is the transient flux velocity Vt, which is no longer curl-like since the transient flux Jt(= PtVt) is generally not divergence-free. However,Vt and Jt still have significant physical meanings, which will be clarified later.

From the stationary and transient dynamical decomposition equations in E-qs. (3.6) and (3.7), we can derive a ‘relative’ dynamical constraint equation:

Vr(q, t) = −D(q, t) · ∇A(q, t), (3.8)

where A(q, t) = U(q, t) − S(q, t) = ln[Pt(q, t)/Ps(q, t)], as the difference be-tween the stationary and transient potential landscapes, can be termed the relative

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potential landscape and Vr(q, t) = Vt(q, t) − Vs(q, t), as the difference betweenthe transient and stationary flux velocities, may be called the relative flux veloc-ity. Equation (3.8) expresses a constraint between A and Vr. Since D(q, t) ispositive definite, Eq. (3.8) implies that the necessary and sufficient condition forVr = 0 (i.e., Vt = Vs) is A = 0 (i.e., Pt = Ps). In other words, Pt = Ps isequivalent to Vt = Vs. When the transient distribution reaches the (instantaneous)stationary distribution, the transient flux also coincides with the (instantaneous)stationary flux. Deviation of the transient distribution from the stationary distri-bution indicates deviation of the transient flux from the stationary flux and viceversa. Therefore, the relative potential landscape A and the relative flux veloci-ty Vr directly characterize the non-stationary condition, namely deviation of thetransient state from the instantaneous steady state. What is inherent in such a de-viation is the tendency to reduce the deviation through a relaxation process, anirreversible non-equilibrium process, until or unless there is no more deviation.Thus non-zero Vr (or A) is also an indicator of the irreversible non-equilibriumrelaxation process inherent in the non-stationary condition.

There are two typical situations where deviation of the transient state from thesteady state can be created, that is, by displacing either the transient state or thesteady state. When external conditions are constant, the steady state is fixed. Ifthe system is prepared in a state different from the steady state, a non-stationarycondition is created and the induced relaxation process towards the fixed steadystate follows. We can call it the state-preparation induced relaxation. When ex-ternal conditions change with time, so is the (instantaneous) steady state. Evenif the system is initially prepared in a steady state, the change of external condi-tions, if fast enough (compared to the relaxation time scale), can displace the sys-tem out of the instantaneous steady state. This way a non-equilibrium relaxationprocess is induced by changing external conditions. We can call it the external-driving induced relaxation. If the external conditions change constantly and sofast that relaxation cannot keep up, the system will be constantly displaced outof the instantaneous steady state. (A systematic investigation on relevant issuesin chemical reaction systems based on non-Markovian dynamics can be found inRef. [34].) In the opposite scenario, which is the quasi-static limit, the steady s-tate changes so slowly due to slow external driving that it cannot shake off the fastrelaxation process; the system will be approximately sticking to the instantaneoussteady state all the time and no deviation is created. In general, deviation of thetransient state from the steady state may result from both of the following factors:(1) initial preparation of the transient state; (2) external driving of the steady state.Yet we also remark that external driving does not always drive the steady state

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away from the transient state; it can also drive them closer to each other if there isalready a deviation. In other words, external driving can work both ways of creat-ing and increasing a deviation or reducing and eliminating an existent deviation.The relative flux velocity Vr (or the relative potential landscape A) characterizesthe ‘relative’ deviation of the transient state from the steady state and indicatesthe non-equilibrium relaxation process that always tries to reduce and eliminatethe deviation; whether that attempt is successful or not depends on which direc-tion external driving works towards and their relative ‘speed’. We note that thequalifier ‘induced’ in the term ‘induced relaxation’ is a matter of perspective; ifdeviation from the steady state is given, then the relaxation process attempting toreduce that deviation may as well be labeled as ‘spontaneous’ as it is driven by aninherent tendency.

Then we come back to the meaning of the transient flux Vt and Jt. In fact, van-ishing transient flux Vt = 0 (Jt = 0) implies the detailed balance condition in thesteady state Vs = 0 (Js = 0). This is because Vt = 0 (Jt = 0) implies ∇ · Jt = 0,which is the same equation as the instantaneous stationary Fokker-Planck equa-tion in Eq. (3.4), just with Ps replaced by Pt. By the assumption (justified underreasonable conditions) that the instantaneous stationary solution is unique at eachmoment, we have Pt = Ps and thus Vs = Vt = 0 (also Js = 0). The reversestatement, however, is not true. Vs = 0 (Js = 0) does not imply Vt = 0 (Jt = 0).Even if the steady state obeys detailed balance Vs = 0 (Js = 0), the system canstill be in the process of relaxing from the transient state to the steady state (withdetailed balance); thus Vt = Vs + Vr = Vr = 0 according to the discussion underEq. (3.8). The condition Vt = 0 (Jt = 0) is equivalent to the following two con-ditions both satisfied: (1) detailed balance in the steady state: Vs = 0 (Js = 0);(2) stationarity: Vr = 0 (A = 0). In other words, there are two basic ways togenerate a non-zero transient flux Vt or Jt. One way is detailed balance breakingin the steady state, indicated by non-zero stationary flux Vs (or Js), which charac-terizes the irreversible non-equilibrium nature of the steady state. The other wayis non-stationarity, indicated by non-zero relative flux Vr (or A), which charac-terizes the irreversible non-equilibrium nature of the relaxation process from thetransient state to the (instantaneous) steady state. These are the two fundamentalaspects of non-equilibrium processes [115]; together they are captured in the tran-sient flux Vt (or Jt). (Since the relaxation process contains two different facets,state preparation and external driving, we may as well say there are three basicaspects of non-equilibrium processes [129]). Therefore, Vt (or Jt) characterizesthe combined non-equilibrium effects from both the detailed balance breaking inthe steady state characterized by Vs (or Js) and the relaxation process induced by

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deviation from the steady state characterized by Vr (or A).Hence, we reinterpret the following equation:

Vt(q, t) = Vs(q, t) + Vr(q, t) (3.9)

as a flux decomposition equation. It represents a fundamental decomposition ofthe total non-equilibrium irreversible process characterized by Vt, into the station-ary (steady-state) non-equilibrium irreversible process characterized by Vs and therelaxational non-equilibrium irreversible process characterized by Vr. (Note thatthe subscript t in Vt can represent both ‘transient’ and ‘total’, the subscript s in Vs

can represent both ‘stationary’ and ‘steady-state’, and the subscript r in Vr can rep-resent both ‘relative’ and ‘relaxation(al)’.) The equilibrium condition is character-ized by the vanishing of all the three fluxes Vt = Vs = Vr = 0 (with Vs = Vr = 0implied by Vt = 0). Non-zero fluxes Vt, Vs and Vr represent the essential charac-teristics of non-equilibrium processes. Thus by digging out the physical meaningof the fluxes as characterizing different aspects of non-equilibrium processes, wehave elevated Eq. (3.9) from merely an equation defining Vr into an equation thatrepresents a fundamental decomposition of non-equilibrium processes.

The flux decomposition equation in Eq. (3.9) can also be regarded as a form ofdynamical decomposition equation as the stationary dynamical decomposition e-quation in Eq. (3.6) and the transient dynamical decomposition in Eq. (3.7). Strict-ly speaking, Eq. (3.8) is a dynamical constraint equation as there is no decompo-sition. Yet collectively we shall simply refer to Eqs. (3.6)-(3.9) as the dynamicaldecomposition equations. The quantities involved in all these four equations areall dynamical quantities that can be defined and constructed directly from the s-tochastic dynamical equation. Yet there is more. In addition to the dynamicalaspects, these equations can also be understood in the context of non-equilibriumthermodynamics, where they will take on thermodynamic meanings with theirthermodynamic aspects revealed. The dynamical decomposition equations serveas a bridge connecting stochastic dynamics with non-equilibrium thermodynam-ics, within which the fluxes Vt, Vs and Vr play an essential role in characterizingthe non-equilibrium nature of the (thermo)dynamics of the system. Therefore, itcan be expected that these equations and in particular the fluxes will also manifestthemselves on the non-equilibrium thermodynamic level.

Before discussing the non-equilibrium thermodynamics, we remark that thestationary potential landscape U = − lnPs and the transient potential landscapeS = − lnPt are only defined up to a common additive constant (corresponding toa common multiplicative constant of probability distributions), which leaves un-changed the definition of the relative potential landscape A = U −S = ln(Pt/Ps)

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as well as the dynamical decomposition equations in Eqs. (3.6)-(3.9). In otherwords, only their difference (i.e., relative values) rather than absolute values isphysically meaningful; the common additive constant means a freedom to choosethe reference point to measure them. This issue becomes important when it comesto spatially inhomogeneous systems.

3.1.3 Non-Equilibrium Thermodynamic ContextNow we place the stochastic dynamics into the non-equilibrium thermody-

namic context. We discuss three related description levels, namely the microscop-ic level, the macroscopic level and the ensemble level.

We assume the microscopic state of the system is specified by a microstatevector q. The space of the microstate q is called the microstate space. The mi-crostate q evolves according to a stochastic dynamics described by the Langevinequation (Eq. (3.1)), tracing a stochastic trajectory on the microstate space. In gen-eral, the microscopic stochastic dynamics is regulated by macroscopic externalconditions (e.g., mechanical constraints, the temperatures of the heat reservoirsthe system is coupled to), which enter the microscopic dynamical equation in theform of external control parameters [129,130]. Such dynamics may emerge as aneffective internal description of open systems, which can be derived from the larg-er dynamics of the system and the environment by eliminating the environment’sdegrees of freedom [149]. (For a concrete example based on master equations andinvestigations on the consistency of the internal dynamics and the larger dynamic-s, see Ref. [150].) We note that the word ‘microscopic’ used here is based on thepremise that the stochastic dynamics is applicable, which may require the descrip-tion of the system to be on a certain level of coarse-graining. Thus the descriptionmay actually be on a ‘mesoscopic’ level in terms of its physical scale. We shall,however, proceed with the word ‘microscopic’, with its implication in mind.

We define the macroscopic external conditions that the system is subject toas a macrostate, which is assumed to be specified by a set of macrostate vari-ables, denoted by λ ≡ ..., λi, ..., regulating the microscopic internal dynam-ics in the form of external control parameters. The space of the macrostate λis referred to as the macrostate space. The macrostate defined this way, howev-er, does not have to represent the properties of the system itself; it may reflectrelations between the system and the environment. In particular, it can repre-sent non-equilibrium constraints imposed on the system by the environment orsustained non-equilibrium conditions through constant system-environment in-teraction and exchange of matter and energy [1, 100, 105, 108, 150]. Therefore,

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in addition to equilibrium macrostates, λ may also represent non-equilibriummacroscopic steady states [1, 100]. An extension of the equilibrium state space toaccommodate non-equilibrium steady states has been proposed in a phenomeno-logical thermodynamic framework [100]. The idea of incorporating matter andenergy fluxes as state variables has also been suggested in ‘extended irreversiblethermodynamics’ [102]. Generally, equilibrium macrostates form a subspace ofthe extended space of macroscopic steady states, with those parameters character-izing non-equilibrium macroscopic conditions vanishing. We also consider time-dependent external conditions which can be represented by λ(t). To simplifynotation, we often just write λ. The time dependence in F (q, t), Gs(q, t) andthat which originates from them can now be replaced by λ, taking on the follow-ing form: F (q, λ), Gs(q, λ), D(q, λ), Ps(q, λ), Js(q, λ), U(q, λ)and Vs(q, λ).

An ensemble of systems compatible with the macrostate λ (i.e., all under thesame macroscopic conditions specified by λ), existing in different microstateswith certain probabilities, can be described by an ensemble probability distribu-tion P (q, λ). The Fokker-Planck equation (Eq. (3.2)) is interpreted as the dy-namical equation governing the evolution of an ensemble of systems, initiallyprepared in a particular ensemble distribution, each of which is independently e-volving according to the Langevin equation (Eq. (3.1)). We denote the transientensemble distribution by Pt(q, λ; t). (The time variable t behind the semicolonrepresents the ‘total’ time variable, defined by the transient Fokker-Planck equa-tion (Eq. (3.2)), with ∂tP (q, λ; t) understood as the derivative with respect tothe total time variable.) The stationary ensemble distribution Ps(q, λ) is thedistribution that an ensemble of systems, regardless of their initial preparation,eventually settle in when the macrostate λ is fixed. However, there could al-so be a non-zero stationary probability flux Js(q, λ). Therefore, correspondingto each macroscopic steady state λ, there is a stationary pair Ps(q, λ) andJs(q, λ) of the ensemble, which is the microscopic statistical characterizationof the macrostate λ. (Equivalently, λ can be characterized by the station-ary potential landscape U(q, λ) and the stationary flux velocity Vs(q, λ) s-ince U = − lnPs and Vs = Js/Ps [50]). Equilibrium macrostates λeq can beidentified with those ensembles satisfying the detailed balance condition, charac-terized by Js(q, λeq) = 0 (or Vs(q, λeq) = 0). Thus equilibrium macrostatescan be described micro-statistically by the stationary equilibrium ensemble dis-tribution Ps(q, λeq) alone. Non-equilibrium macroscopic steady states λneqcan be identified with those ensembles with detailed balance broken, indicated

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by Js(q, λneq) = 0 (or Vs(q, λneq) = 0). Therefore, they must be describedmicro-statistically by the pair Ps(q, λneq) and Js(q, λneq) together. The sta-tionary probability flux on the microstate space connects different microstates intoa coherent global state, representing the non-equilibrium macroscopic steady stateon the micro-statistical level. The pair P (q, λ) and J(q, λ) together may bemore appropriately called an ensemble ‘state’ [88], though P (q, λ) alone is alsoreferred to as that. (A relevant fact is that given the dynamical equation, the prob-ability flux and probability distribution are related via, e.g., Eqs. (3.3) and (3.5).)When the macrostate λ changes with time (external driving), Ps, Js, U and Vs

also change accordingly, since they all directly depend on λ. Thus externaldriving drives the steady state, where ‘steady state’ means both the macroscopicsteady state and the ensemble steady state and these two meanings are consistent.For a discussion on the micro-macro correspondence of non-equilibrium states interms of master equations, see Ref. [2].

Three types of state have been introduced, namely the microstate q, the macrostateλ and the ensemble state P (q, λ) (and J(q, λ)). Accordingly, there arethree types of state functions. A function that depends on the microstate q, whichmay also depend on the macrostate λ, since the microstate and its dynamics aresubject to macroscopic conditions, is called a microstate function. A function thatdoes not depend on the microstate q, but only depends on the macrostate λ, iscalled a macrostate function. We use calligraphy letters (e.g., S , A, U) to denotemacrostate functions. A macrostate function can usually be expressed as the en-semble average of a microstate function; if so, it is also an ensemble state functionas it is a function(al) of the ensemble distribution. To specify such a macrostatefunction, we need to specify both the microstate function and the ensemble dis-tribution in taking the ensemble average. In particular, whether the transient orthe stationary ensemble is used need to be indicated, in contrast with equilibriumstatistical mechanics where only the equilibrium ensemble is of concern.

3.1.4 State Functions of Non-Equilibrium Isothermal Process-es

In the following we focus on systems in an environment with a constant tem-perature T , which at the same time may still be subject to other non-equilibriumconditions, such as non-equilibrium steady states sustained by chemical potentialdifference or nonconservative forces [105,129,150], non-equilibrium transient re-laxation processes due to time-dependent external driving [34, 129, 130] or initial

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state preparation [109, 115]. The environment in this situation serves as both aheat reservoir providing a constant temperature and a source of non-equilibriumconditions through system-environment interaction and exchange of matter andenergy. Such non-equilibrium processes taking place at a constant temperatureare referred to as non-equilibrium isothermal processes.

For some systems modeled by Langevin and Fokker-Planck dynamics, the dif-fusion matrix D characterizing the fluctuating forces is, under certain conditions,proportional to the temperature of the heat reservoir (environment), as a result ofthe fluctuation-dissipation theorem [15, 16, 151]. We shall assume that the sys-tems we consider also satisfy this condition D ∝ T . It means the fluctuationstrength (the magnitude of D) is proportional to T . Intuitively, higher tempera-ture corresponds to stronger fluctuations. Thus the temperature plays the role ofa fluctuation strength parameter. In general, D ∝ T breaks down when the tem-perature becomes too low as quantum effects dominate. We define the re-scaledtemperature-independent diffusion matrix D(q, λ′) = T−1D(q, λ), whereλ′ represents λ with T excluded.

Then we consider the state functions for non-equilibrium isothermal process-es. For such processes three non-equilibrium (macro)state functions, namely thenon-equilibrium entropy, internal energy and free energy have been introducedformally for master equations as well as for Fokker-Planck equations [11,12,114,115]. Here we investigate these quantities in connection with the potential-fluxlandscape framework. We also discuss some subtleties involved in their defini-tions, especially regarding the indeterminacy of the microscopic non-equilibriumenergy function and its connection to macroscopic external conditions, and wepropose possible solutions.

Non-Equilibrium Entropy

The non-equilibrium entropy of the system, more specifically, the non-equilibriummacroscopic transient entropy, is defined by applying the Gibbs entropy formulato the transient ensemble distribution (Boltzmann constant kB is set to 1 through-out this article) [11, 12, 109, 112, 118, 120, 125]:

S(λ; t) = ⟨S(q, λ; t)⟩t = −∫

Pt(q, λ; t) lnPt(q, λ; t)dq, (3.10)

where ⟨·⟩t represents average over the transient ensemble Pt. The microstate func-tion S(q, λ; t) = − lnPt(q, λ; t) is the transient potential landscape intro-duced in Eq. (3.7). Now its thermodynamic meaning is the microscopic transient

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entropy; its average over the transient ensemble gives the macroscopic transiententropy. We note that S = − lnPt is also referred to as the ‘stochastic entropy’if it is defined on a stochastic trajectory [128–130]. As mentioned before, S isdefined up to an additive constant, so is its ensemble average S = ⟨S⟩t, whichmeans we are only concerned with entropy differences.

If the transient distribution Pt in Eq. (3.10) is replaced by the stationary distri-bution Ps, the resulting entropy Ss = ⟨− lnPs⟩s may be called the (non-equilibriummacroscopic) stationary entropy. In the equilibrium limit Pt = Ps = Pe, E-q. (3.10) recovers the equilibrium entropy Se = ⟨− lnPe⟩e. Thus Eq. (3.10) seemsto be a natural extension of the equilibrium entropy to non-equilibrium systems.Yet there are also other definitions of the non-equilibrium entropy in the literaturebased on the Gibbs entropy postulate [101], which is more relevant to the conceptof relative entropy [115, 118, 143] that will also be discussed later.

Non-Equilibrium Internal Energy and Cross Entropy

We know that for equilibrium systems in contact with a heat reservoir, withthe microscopic energy E given a priori (e.g., the Hamiltonian of the conser-vative system), the equilibrium distribution is the canonical ensemble distribu-tion Pe = Z−1e−E/T = e(Ae−E)/T , where Z is the partition function and Ae =−T lnZ is the equilibrium free energy. The equilibrium internal energy is givenby Ue = ⟨E⟩e, where ⟨·⟩e represents the ensemble average over Pe. For isother-mal non-equilibrium systems without detailed balance (e.g., a dissipative system),a microscopic energy function connected to the non-equilibrium stationary distri-bution in a similar fashion as the equilibrium canonical ensemble is not given apriori. Yet one may still ask whether it is possible to identify or construct sucha microscopic energy function, if the non-equilibrium stationary distribution issolved [9, 10, 13, 14, 82, 85, 115, 152].

For general Fokker-Planck dynamics describing non-equilibrium isothermalprocesses, there is one condition D ∝ T we can employ. In the study of smallfluctuation problems of the Fokker-Planck equation, the stationary distributioncan be expanded into an asymptotic series in terms of the fluctuation strengthparameter using the WKB method [12, 50, 82, 153]. Given that D ∝ T meansthe temperature T plays the role of a fluctuation strength parameter, the stationary

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distribution can be assumed to have the following asymptotic form for small T :

Ps(q, λ) =exp

[− 1

TU(q, λ)

]Z(λ)

=

exp

[− 1

T

+∞∑k=0

T kU (k)(q, λ′)]

Z(λ), (3.11)

where Z(λ) =∫exp

[−U(q, λ)/T

]dq is the normalization factor; λ′ is

λ with T excluded; U(q, λ) =∑+∞

k=0 TkU (k)(q, λ′) is an asymptotic series

for small T . The equation for U (k)(q, λ′) (k = 0, 1, ...) can be obtained by plug-ging the above ansatz into the stationary Fokker-Planck equation (Eq. (3.4)) andmatch each order of T . Formally, Eq. (3.11) is an extension of the equilibriumcanonical ensemble distribution to non-equilibrium isothermal processes, whereZ(λ) is the analog of the partition function and U(q, λ) serves as the mi-croscopic non-equilibrium energy. However, there are some subtleties, which wediscuss below.

First, in general, U(q, λ) may depend on the temperature T (although thisis not always the case), in contrast with equilibrium systems whose microscopicenergy (e.g., the system’s Hamiltonian) is generically independent of tempera-ture. This indicates that U(q, λ) (if well defined) is an effective microscopicenergy, which may (partially) have a macroscopic origin. Its dependence on the(non-equilibrium) external conditions λ suggests that it may also have includedthe system-environment interaction energy. For perspectives on the issues aroundincluding or not the interaction energy in the microscopic energy function andthus also the internal energy, see Ref. [154]. In our case, since λ can also de-scribe non-equilibrium steady states sustained by constant system-environmentinteraction, it does not seem to be so unreasonable that interaction energy needto be accounted for, in the description of an effective internal dynamics that onlyinvolves the system’s degrees of freedom, in order to have a consistent thermody-namic description.

Second, U(q, λ) and Z(λ) are not uniquely determined by Ps(q, λ).U(q, λ) is only determined up to an additive macrostate function C(λ). Ac-cordingly, Z(λ) is determined up to a multiplicative macrostate function in theform of exp(−C(λ)/T ). This indeterminacy can be seen in another form. Weintroduce the non-equilibrium stationary free energy (the name will be justifiedlater), motivated by the relation of the partition function with equilibrium freeenergy [151]:

As(λ) = −T lnZ(λ). (3.12)

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Note that As is a macrostate function, independent of q. Then Eq. (3.11) can bewritten as:

Ps(q, λ) = exp

[1

T

(As(λ)− U(q, λ)

)]. (3.13)

Further, we haveU(q, λ)− As(λ) = TU(q, λ), (3.14)

where the dimensionless microstate function U(q, λ) = − lnPs(q, λ) is thestationary potential landscape introduced in Eq. (3.6); in the context of non-equilibrium thermodynamics, it is the microscopic stationary entropy (comparewith the microscopic transient entropy S = − lnPt). Equation (3.14) shows thatthe stationary distribution can only determine the difference U(q, λ)−As(λ)rather than each of them individually. There is a common additive macrostatefunction C(λ) in their expressions undetermined. If U(q, λ) and As(λ)have thermodynamic meanings individually (as an extension of their equilibri-um counterparts we expect so), the macrostate function C(λ) not fixed by thestochastic dynamical equation (which determines the stationary distribution) in-dicates that the microscopic internal dynamics alone is not sufficient to give acomplete description of the system’s thermodynamics [121,130,135]. Hence, ad-ditional information is needed to fix (one of) U(q, λ) and As(λ), up to anadditive constant independent of λ.

One way is to fix As(λ). We know from equilibrium thermodynamics thatif the equations of state for the equilibrium free energy Ae(λ) are known, i.e.,Λi

e(λ) = ∂Ae(λ)/∂λi (i = 0, 1, ...) are given, where λi’s are the natural vari-ables of the equilibrium free energy and Λi

e is the variable conjugate to λi, thenthe free energy can be found by integration Ae(λ) =

∫ ∑i Λ

ie(λ)dλi, deter-

mined up to an additive integration constant. This suggests that the ‘equations ofstate’ for the non-equilibrium stationary free energy, namely

Λis(λ) = ∂As(λ)/∂λi, (3.15)

(with i = 0, 1, ... and λ0 ≡ T ) can serve as the additional information requiredto supplement the thermodynamic description of the system apart from the mi-croscopic internal dynamics. With Λi

s(λ) given, As(λ) is determined up toan additive constant by integration: As(λ) =

∫ ∑i Λ

is(λ)dλi. Accordingly,

U(q, λ) is also determined up to an additive constant by Eq. (3.13) or (3.14). Inthe discussion of the second law of thermodynamics in terms of free energy, wewill also propose an operational definition of As consistent with this approach.

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The other way is to fix U(q, λ). Since U(q, λ) as a function of q hasalready been partially determined by the microscopic internal dynamics, the free-dom that is left to be fixed is a macrostate function C(λ). Thus we considerdetermining the ensemble average of U . We define the non-equilibrium station-ary internal energy Us(λ) = ⟨U(q, λ)⟩s, where ⟨·⟩s means ensemble averageover Ps. The undetermined function C(λ) in U is then transferred to Us. Thisshows the non-equilibrium stationary internal energy Us cannot be derived fromthe microscopic internal dynamics; it need to be supplemented to complete thethermodynamic description of the system. If Us is given, then U is determined.Accordingly, As is also determined. In the discussion of the first law of ther-modynamics, we will propose an operational definition of the non-equilibriumstationary internal energy Us.

The equivalence of these two approaches, determining As(λ) or Us(λ),can be seen more directly from Eq. (3.14), by taking the ensemble average over Ps,which gives Us = As + TSs, with the stationary entropy Ss = ⟨U⟩s = ⟨− lnPs⟩sdetermined by the microscopic internal dynamics. Thus with either As(λ) orUs(λ) given, U(q, λ) is then determined. This also validates our statemen-t that U(q, λ) as an effective microscopic energy partially has a macroscop-ic origin; it is determined by both the microscopic internal dynamics and themacroscopic external conditions. In fact, there exists a family of non-equilibriumsystems, all compatible with the same microscopic internal dynamics regulatedby macroscopic external conditions λ, which are not thermodynamically e-quivalent, as they are characterized by different stationary free energy functionsAs(λ), or equivalently stationary internal energy functions Us(λ). This a-grees with the perspective that the subsystem dynamics is not sufficient to handlethe full first law of thermodynamics [121, 130, 135].

With the issues around the microscopic non-equilibrium energy U addressed,we define the non-equilibrium internal energy U as the average of U over thetransient ensemble, i.e., U = ⟨U⟩t. In the steady state Pt = Ps, it coincides withthe non-equilibrium stationary internal energy Us = ⟨U⟩s already introduced. Inthe equilibrium state Pt = Ps = Pe, the microscopic non-equilibrium energy Ureduces to the equilibrium microscopic energy of the system Ue; then U and Us

recover the equilibrium internal energy Ue = ⟨Ue⟩e. According to Eq. (3.14), thenon-equilibrium internal energy is also given by:

U(λ; t) = ⟨U(q, λ)⟩t = TU(λ; t) + As(λ), (3.16)

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where

U(λ; t) = ⟨U(q, λ)⟩t = −∫

Pt(q, λ; t) lnPs(q, λ)dq. (3.17)

The dimensionless macrostate function U = ⟨− lnPs⟩t is the average of the mi-croscopic stationary entropy − lnPs over the transient ensemble distribution Pt.Thus it does not have the exact form of the Gibbs entropy formula. It is neitherthe transient entropy St = ⟨− lnPt⟩t, nor the stationary entropy Ss = ⟨− lnPs⟩s.In the language of information theory [143], U = ⟨− lnPs⟩t is the cross entropy,of the stationary distribution Ps with respect to the transient distribution Pt. Theword ‘cross’ refers to the fact that its definition ⟨− lnPs⟩t involves two differentprobability distributions Ps and Pt. In the steady state Pt = Ps, the transient en-tropy, cross entropy and stationary entropy coincide St = U = Ss = ⟨− lnPs⟩s.Also, U = ⟨U⟩t reduces to Us = ⟨U⟩s in the steady state. Accordingly, Eq. (3.16)reads Us = TSs+ As; equivalently, As = Us−TSs, which is the relation of inter-nal energy, entropy and free energy in non-equilibrium steady states of isothermalprocesses. In the equilibrium state Pt = Ps = Pe, it recovers the equilibriumrelation Ae = Ue − TSe.

The non-equilibrium internal energy U in isothermal processes is related tothe cross entropy U by Eq. (3.16). If in the isothermal process external condi-tions do not change (no external driving), then As(λ) remains constant. Thechange of internal energy U , according to Eq. (3.16), is then proportional to thechange of cross entropy U , i.e., ∆U = T∆U . If external conditions also change,then ∆U = T∆U + ∆As. Internal energy U and cross entropy U have differentphysical meanings; yet their thermodynamic equations turn out to have similarstructures. (That is also why they are denoted with similar symbols.) In non-equilibrium isothermal processes, the non-equilibrium internal energy U is whatwe mainly work with. For more general non-equilibrium processes, even if thereis no obvious way to introduce the non-equilibrium internal energy, the cross en-tropy U = ⟨− lnPs⟩t is still readily defined by the probability distributions Ps andPt.

Non-Equilibrium Free Energy and Relative Entropy

Motivated by the relation of free energy, internal energy and entropy in equi-librium thermodynamics, Ae = Ue − TSe, we introduce the microscopic non-equilibrium free energy

A(q, λ; t) = U(q, λ)− TS(q, λ; t). (3.18)

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The macroscopic non-equilibrium free energy is defined as the ensemble averageof A:

A(λ; t) = ⟨A(q, λ; t)⟩t = U(λ; t)− TS(λ; t). (3.19)

In the equilibrium situation Pt = Ps = Pe, the non-equilibrium free energy A re-covers the equilibrium free energy, as the non-equilibrium internal energy U andnon-equilibrium entropy S reduce to their respective equilibrium counterparts Ue

and Se. Therefore, the non-equilibrium free energy is an extension of the equilib-rium free energy to non-equilibrium isothermal processes. From Eqs. (3.14) and(3.18), we also have

A(q, λ; t) = TA(q, λ; t) + As(λ), (3.20)

where

A(q, λ; t) = U(q, λ; t)−S(q, λ; t) = ln[Pt(q, λ; t)/Ps(q, λ)]. (3.21)

The microstate function A = U − S = ln(Pt/Ps) was introduced in Eq. (3.8)as the relative potential landscape. In the context of non-equilibrium thermody-namics, it is, according to Eq. (3.21), the microscopic relative entropy, betweenthe microscopic stationary entropy U and the microscopic transient entropy S.Taking the transient ensemble average of Eqs. (3.20) and (3.21), we have

A(λ; t) = TA(λ; t) + As(λ), (3.22)

A(λ; t) = U(λ; t)− S(λ; t)

=

∫Pt(q, λ; t) ln[Pt(q, λ; t)/Ps(q, λ)]dq. (3.23)

The macrostate function A = ⟨A⟩t = ⟨ln(Pt/Ps)⟩t in Eq. (3.23) is known as rel-ative entropy (Kullback-Leibler divergence) in information theory [16, 115, 118,143]. It quantifies the deviation of the (transient) distribution Pt from the (station-ary) distribution Ps. According to Eq. (3.23), when Pt = Ps, the relative entropyA vanishes. Thus Eq. (3.22) shows A = As when Pt = Ps, justifying the name‘non-equilibrium stationary free energy’ given to As as it was introduced in E-q. (A.1). Equation (3.22) relates the non-equilibrium free energy in isothermalprocesses to relative entropy [115, 118]. (We remark that this is not the same asthe postulate in Ref. [101], which defines the non-equilibrium entropy in terms ofrelative entropy. The implication of these different postulates need further inves-tigation.) The change of these quantities in isothermal processes are related by

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∆A = T∆A+∆As. If external conditions do not change, we have ∆A = T∆A.For more general non-equilibrium processes, the relative entropy is still readilydefined by A = ⟨ln(Pt/Ps)⟩t, even if the non-equilibrium free energy A is noteasy to introduce.

The non-equilibrium internal energy U and non-equilibrium free energy A inisothermal processes are related, respectively, to the cross entropy U and relativeentropy A, as shown in Eqs. (3.16) and (3.22) as well as their microscopic coun-terparts in Eqs. (3.14) and (3.20). These relations are summarized below:

U = TU + As, U = TU + As, (3.24)

A = TA+ As, A = TA+ As. (3.25)

Note that macrostate functions are denoted by calligraphy letters. The definitionsof the quantities on the left side of these equations are given by U , U = ⟨U⟩t,A = U − TS, A = ⟨A⟩t = U − TS; those on the right side are given byU = − lnPs, U = ⟨U⟩t = ⟨− lnPs⟩t, A = U − S = ln(Pt/Ps), A = ⟨A⟩t =U −S = ⟨ln(Pt/Ps)⟩t. Due to the same mathematical structure in Eqs. (3.24) and(3.25), these two sets of quantities, U , U , A, A and U , U , A, A, satisfy similarequations as their counterparts do. In particular, the microstate functions U , S andA also satisfy a set of dynamical decomposition equations, corresponding to thosefor the potential landscapes (microscopic entropies) U , S and A in Eqs. (3.6)-(3.8). They can be written in the following alternative form that is more convenientfor application in non-equilibrium thermodynamics:

∇U = D−1 · Vs − D−1 · F ′

∇S = D−1 · Vt −D−1 · F ′

∇A = −D−1 · Vr

(3.26)

D = T−1D is the rescaled diffusion matrix independent of T . These three e-quations in Eq. (3.26) are, respectively, the alternative forms of the stationary,transient and relative dynamical decomposition equations. We demonstrate in thefollowing that from the three non-equilibrium state functions U , S and A and thethree dynamical decomposition equations (Eq. (3.26)) together with the flux de-composition equation Vt = Vs + Vr (Eq. (3.9)), an entire set of thermodynam-ic equations, representing the thermodynamic laws governing non-equilibriumisothermal processes, can be constructed, together with the expressions of thethermodynamic quantities involved.

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3.1.5 Thermodynamic Laws of Non-Equilibrium Isothermal Pro-cesses

We study the first law of thermodynamics in terms of the non-equilibriuminternal energy and then the second law of thermodynamics in terms of the non-equilibrium entropy as well as the non-equilibrium free energy. We shall calculatethe rate of change of these thermodynamic functions. It is convenient to firstconsider more general cases. To simplify notation, ⟨·⟩ represents average overthe transient ensemble. Let O be a general microstate function. Its (transient)ensemble average is the macrostate function O = ⟨O⟩. The rate of change of O iscalculated as follows:

dOdt

=d

dt

∫OPt dq

=

∫(∂tO)Pt dq −

∫O(∇ · Jt

)dq

= ⟨∂tO⟩+ ⟨Vt · ∇O⟩, (3.27)

where we have used ∂tPt = −∇ · Jt and integration by parts with vanishingboundary terms under appropriate boundary conditions. By introducing the ad-vective derivative as in fluid dynamics:

DO

Dt=

∂O

∂t+ Vt · ∇O, (3.28)

Eq. (3.27) can also be written as d⟨O⟩/dt = ⟨DO/Dt⟩.

The First Law of Non-Equilibrium Thermodynamics

The rate of change of the non-equilibrium internal energy, according to E-q. (3.27), is given by:

˙U = ⟨∂tU⟩+ ⟨Vt · ∇U⟩. (3.29)

Since U is a function of λ, ⟨∂tU⟩ is generally non-zero if λ change with time(time-dependent external conditions). More explicitly, ⟨∂tU⟩ =

∑i⟨∂λi

U⟩λi.Temperature T ≡ λ0 is excluded here since T is constant in isothermal pro-cesses. Motivated by equilibrium statistical mechanics [151], we introduce the(non-equilibrium) generalized force Λi ≡ ⟨∂λi

U⟩ (i = 1, 2, ...), conjugate to the(non-equilibrium) generalized coordinate λi. Λi and λi form a conjugate pair.

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Λi dλi is the product of the generalized force and the generalized displacemen-t, which represents the thermodynamic work done through changing the externalcondition represented by λi. Therefore,

∑i⟨∂λi

U⟩λi =∑

i Λidλi/dt can be iden-

tified as the total thermodynamic work per unit time (i.e., power) done throughchanging the external conditions specified by λ, a process referred to as exter-nal driving. Thus we define the external-driving power as follows:

Wed = ⟨∂tU⟩ =∑i

⟨∂λiU⟩λi =

∑i

Λiλi, (3.30)

where the subscript ‘ed’ of Wed represents external driving and the sum excludesλ0 = T . The sign convention is that Wed is positive (negative) when the environ-ment is doing work on the system (the system is doing work on the environment)through changing external conditions. For a careful account of the concepts ofwork, especially when interaction energy is involved, see Ref. [154]. Since λcan represent non-equilibrium external conditions, the thermodynamic work donethrough changing λ may also incorporate non-equilibrium effects. Thus Eq.(3.30) is an extension of the equilibrium thermodynamic work to non-equilibriumisothermal processes.

When discussing the equations of state of the stationary free energy, we intro-duced Λi

s = ∂λiAs (i = 0, 1, ...) in Eq. (3.15). Here we introduced the generalized

force Λit ≡ ⟨∂λi

U⟩t (i = 1, 2, ...), with the subscript t spelled out for clarity. Thesetwo sets of quantities are related, due to the following results (proven in AppendixA):

Λis = ∂λi

As = ⟨∂λiU⟩s (i = 1, 2, ...), Λ0

s = ∂T As = ⟨∂T U⟩s − Ss. (3.31)

According to the first equation, Λit coincide with Λi

s in the steady state for i =1, 2, ..., thus justifying the notation used. For i = 0 (λ0 = T ), the second e-quation differs from the equilibrium relation ∂T Ae = −Se by an additional term⟨∂T U⟩s, due to the possible temperature dependence of the effective microscopicnon-equilibrium energy U .

Comparing with the first law of equilibrium thermodynamics, the second ter-m in Eq. (3.29) can be interpreted as the rate of heat transfer. It will becomeclear in the discussion of the second law that this heat is only part of the totalheat transferred between the environment and the system. There is another partcalled the housekeeping heat [100, 108] due to the dissipation in maintaining thenon-equilibrium steady state, which is constantly produced within the system and

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constantly expelled into the environment, leaving the non-equilibrium internal en-ergy unchanged. The part of heat left after removing the housekeeping heat iscalled excess heat [100, 108]. That is what we are dealing with here. Thus wedefine the second term in Eq. (3.29), with a negative sign, as the excess heat flowrate:

Qex = −⟨Vt · ∇U⟩. (3.32)

The adopted sign convention is that heat is positive (negative) when transferredfrom the system to the environment (from the environment to the system), whichseems natural considering that the house-keeping heat can only flow from thesystem to the environment.

Now Eq. (3.29) can be written as a balance equation for the non-equilibriuminternal energy:

˙U = Wed − Qex. (3.33)

This is an extension of the first law of equilibrium thermodynamics to non-equilibriumisothermal processes, as a statement of energy conservation. It states there are twodifferent ways to change the non-equilibrium internal energy. One way is doingwork through changing external conditions. The other way is transferring excessheat, the heat in addition to the housekeeping heat that is generated and expelledconstantly in maintaining non-equilibrium steady states. For equilibrium systemsthe housekeeping heat vanishes and the excess heat and the total heat coincidewith the equilibrium heat; the non-equilibrium internal energy and work also re-duces to their equilibrium counterparts. Then Eq. (3.33) becomes a statement ofthe first law in equilibrium thermodynamics .

The first law of thermodynamics suggests an operational definition of the non-equilibrium stationary internal energy Us, considering how equilibrium internalenergy can be operationally defined by work in adiabatic processes. Consider pro-cesses that are quasi-static (in the sense that λ change so slowly that Pt ≈ Ps allthe time) as well as adiabatic to the excess heat (i.e., Qex = 0 during the process).For perspectives on the constantly generated house-keeping heat in the steady stateas well as some other technical issues involved in such processes, we refer to Re-f. [100]. In such quasi-static adiabatic processes, we can use the thermodynamicwork Wed performed to determine the non-equilibrium stationary internal ener-gy, according to ∆Us = Wed. With Us(λ) operationally defined, the effectivemicroscopic non-equilibrium energy function U(q, λ) can then be constructedfrom the microscopic internal dynamics. Strictly speaking, the first law of non-equilibrium thermodynamics in Eq. (3.33) is not ‘derived’; the determination ofU is dependent on the application of the quasi-static behavior of Eq. (3.33) in the

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first place. Yet the transient behavior of Eq. (3.33) is not arbitrarily assumed, if Uhas already been determined.

The Second Law of Non-Equilibrium Thermodynamics in terms of Entropy

The rate of change of the non-equilibrium entropy is also given by Eq. (3.27).Yet it is not difficult to prove that ⟨∂tS⟩t = 0 with S = − lnPt, due to probabilityconservation. Thus we actually have

S = ⟨Vt · ∇S⟩. (3.34)

Plugging in Eq. (3.26) for ∇S, we have

S = ⟨Vt ·D−1 · Vt⟩ − ⟨Vt ·D−1 · F ′⟩. (3.35)

Thus the rate of change of the non-equilibrium entropy is split into two parts. Thefirst term is always nonnegative and can be identified as the entropy productionrate within the system [2, 11, 12, 112, 118, 120]:

Spd = ⟨Vt ·D−1 · Vt⟩, (3.36)

where the subscript ‘pd’ represents production. The entropy production rate Spd,as a macrostate function, measures the extent of non-equilibrium irreversibilityof the system on the macroscopic level. The transient flux velocity Vt, as a mi-crostate function, indicates the non-equilibrium irreversibility of the system on themicroscopic level. Thus it is not unexpected that they are related to each other inEq. (3.36). Since D−1 is positive definite and the transient ensemble distributionPt is positive in the accessible state space, the necessary and sufficient conditionfor Spd ≡ 0 is Vt ≡ 0, that is, the probability flux Vt (or Jt) vanishes all the timeand everywhere in the accessible (micro)state space. The second term (withoutthe minus sign) in Eq. (3.35) represents the rate of entropy flow from the systemto the environment [2, 11, 12, 112, 118, 120]:

Sfl = ⟨Vt ·D−1 · F ′⟩, (3.37)

where the subscript ‘fl’ represents flow. Therefore Eq. (3.35) can be identified asthe (transient) entropy balance equation:

S = Spd − Sfl. (3.38)

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It states that the entropy of the system is increased when there is entropy pro-duction within the system and decreased when there is entropy flow from thesystem into the environment. This transient entropy balance equation is a directreflection and manifestation of the transient dynamical decomposition equationF ′ = −D · ∇S + Vt, as seen from Eqs. (3.34)-(3.38).

For non-equilibrium isothermal processes, the entropy flow is related to thetotal heat transfer according to dSfl = dQtot/T , where d represents inexactdifferentials. Thus we define the total heat flow rate:

Qtot = T Sfl = ⟨Vt · D−1 · F ′⟩, (3.39)

where we have used Eq. (3.37). The sign convention for heat is as before; heatis positive when flowing from the system to the environment. The non-negativityof the entropy production rate Spd ≥ 0 allows the entropy balance equation (E-q. (3.38)) to be written as an inequality:

dS ≥ −dQtot

T, (3.40)

which has the conventional form of the second law of thermodynamics, except fora sign due to the adopted sign convention of heat already explained.

Comparing Eq. (3.32) and Eq. (3.39), we see that in non-equilibrium isother-mal processes the heat that changes the non-equilibrium internal energy (i.e., theexcess heat) and the heat that contributes to the entropy flow (i.e., the total heat)do not match each other generally. This difference indicates that there is anotherpart of heat (the house-keeping heat) which contributes to the entropy flow yetdoes not change the non-equilibrium internal energy. In other words, the totalheat is composed of the excess heat and the housekeeping heat. Therefore, a s-ingle concept of heat in equilibrium thermodynamics has differentiated into threedistinct yet related concepts of heat in non-equilibrium thermodynamics, namelythe total heat, the excess heat and the housekeeping heat. In terms of the rate ofheat transfer, we have the following heat flow decomposition equation [100,108]:

Qtot = Qhk + Qex. (3.41)

From Eqs. (3.32), (3.39), (3.41) and the dynamical decomposition equation (E-q. (3.26)) for ∇U and ∇A, we derive the expression of the housekeeping heatdissipation rate (‘dissipation’ indicates its dissipative nature):

Qhk = ⟨Vt · D−1 · Vs⟩ = ⟨Vs · D−1 · Vs⟩, (3.42)

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which is always nonnegative according to the last equation. The equality of thetwo expressions of Qhk in Eq. (3.42) is a result of ⟨Vs · ∇A⟩ = 0, proven in Ap-pendix A. The sign property Qhk ≥ 0 means that the housekeeping heat generatedwithin the system is always transferred from the system into the environment andnot in the opposite direction. The excess heat and thus also the total heat can betransferred in both directions.

Since the housekeeping heat is constantly generated within the system due todissipation (an irreversible process) and expelled into the environment in sustain-ing non-equilibrium steady states, we can expect that it does not only contribute tothe entropy flow from the system to the environment, but also the entropy produc-tion within the system. In fact, from the second expression of Qhk in Eq. (3.42),Qhk is intimately related to the stationary flux Vs. The necessary and sufficientcondition for Qhk ≡ 0 is Vs ≡ 0, namely the stationary probability flux vanish-es all the time and everywhere in the accessible state space. Vs as a microstatefunction is a microscopic indicator of detailed balance breaking and thus non-equilibrium irreversibility of the steady state. Hence, Qhk, as a macrostate func-tion, also indicates on the macroscopic level the non-equilibrium irreversibilityof the steady state. This suggests the housekeeping heat contributes to the en-tropy production in the system. This part of entropy production associated withdetailed balance breaking in the steady state has been termed adiabatic entropyproduction [116–118]. (Note that the word ‘adiabatic’ here has no direct relationwith heat transfer. In the thermodynamic context quasi-static entropy productionor steady-state entropy production may be more appropriate, given that there arealso processes adiabatic to heat. In this work we still use the name initially givento it.) In non-equilibrium isothermal processes the adiabatic entropy productionrate is the housekeeping heat dissipation rate divided by temperature, which, usingEq. (3.42), is given by

Sad = T−1Qhk = ⟨Vt ·D−1 · Vs⟩ = ⟨Vs ·D−1 · Vs⟩. (3.43)

The subscript ‘ad’ of Sad represents ‘adiabatic’. Sad is also nonnegative as Qhk is,which agrees with the requirement that entropy production is nonnegative. Similarto Qhk, the necessary and sufficient condition for Sad ≡ 0 is Vs ≡ 0. According tothe relation between entropy flow and heat flow dSfl = dQ/T , the same Sad givenin Eq. (3.43) is also equal, in quantity, to the entropy flow rate associated withthe housekeeping heat transferred from the system to the environment (it can bereferred to as the housekeeping entropy flow rate). Thus we use the same notationSad to represent both the adiabatic entropy production rate and the housekeeping

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entropy flow rate. Their physical meanings, however, are different (one is entropyproduction and the other is entropy flow).

There is also entropy flow associated with the excess heat, called the excessentropy flow [116–118]. Using the expression of Qex in Eq. (3.32), the excessentropy flow rate is given by:

Sex = T−1Qex = −⟨Vt · ∇U⟩, (3.44)

where we have used ∇U = T∇U derived from Eq. (3.24). Therefore, correspond-ing to the heat flow decomposition equation in Eq. (3.41), there is also an entropyflow decomposition equation [116–118]:

Sfl = Sad + Sex. (3.45)

In words, the total entropy flow is composed of the housekeeping entropy flow andthe excess entropy flow, just as the total heat is composed of the housekeeping heatand the excess heat. This non-equilibrium thermodynamic entropy flow decompo-sition equation (as well as the heat flow decomposition equation) is a direct reflec-tion of the stationary dynamical decomposition equation F ′ = −D · ∇U + Vs,as seen from the expressions in Eqs. (3.37), (3.43) and (3.44). Since Sad ≥ 0,Eq. (3.45) can also be written as an inequality:

dSfl ≥dQex

T. (3.46)

Equations (3.45) and (3.46) represent another facet of the second law of non-equilibrium thermodynamics, in addition to what is revealed in (3.38) and (3.40)[116–118].

The difference in the (total) entropy production rate in Eq. (3.36) and the a-diabatic entropy production rate in Eq. (3.43) indicates there is another part ofentropy production, termed nonadiabatic entropy production [116–118]. This isexpressed as an entropy production decomposition equation:

Spd = Sad + Sna, (3.47)

which states the total entropy production is composed of the adiabatic entropyproduction (associated with the non-equilibrium irreversibility in the steady state)and the nonadiabatic entropy production (associated with the non-equilibrium ir-reversibility in the relaxation process from the transient state to the steady state).

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Using the expression of Spd in Eq. (3.36) and that of Sad in Eq. (3.43), we derivethe following equivalent expressions of the nonadiabatic entropy production rate:

Sna = ⟨Vt ·D−1 · Vr⟩ = −⟨Vt · ∇A⟩= ⟨Vr ·D−1 · Vr⟩ = −⟨Vr · ∇A⟩ = ⟨∇A ·D · ∇A⟩, (3.48)

which is always nonnegative. The first expression is a result of Eqs. (3.9), (3.36)and (3.43). The third expression is derived from the first one using the property⟨Vs ·D−1 · Vr⟩ = 0, which is equivalent to ⟨Vs · ∇A⟩ = 0 (already proven in Ap-pendix A) considering the relative dynamical constraint equation (Eq. (3.8)). Theother three expressions can also be derived using these equations. According to E-q. (3.48), the necessary and sufficient condition for Sna ≡ 0 is Vr ≡ 0 (or A ≡ 0).As discussed in the potential-landscape framework, these are the stationary con-ditions, which means the system is staying in the (instantaneous) steady state allthe time. Therefore, non-zero Sna is produced by the irreversible relaxation pro-cess from the transient state to the steady state due to the non-stationary conditioncharacterized by non-zero Vr (or A). For non-equilibrium isothermal processeswe will see later that the nonadiabatic entropy production rate is proportional tothe free energy spontaneous dissipation rate (i.e., dissipative work [131, 132] per

unit time), Sna =˙Asd/T .

The non-equilibrium thermodynamic entropy production decomposition equa-tion (Eq. (3.47)) is a direct reflection of the dynamical decomposition equation ofthe flux: Vt = Vs+ Vr. Within Eq. (3.47) the nonnegative total entropy productionrate Spd is decomposed into two parts that are individually nonnegative, the adia-batic entropy production rate Sad associated with the stationary flux Vs (Eq. (3.43))and the nonadiabatic entropy production rate Sna associated with the relative fluxVr (Eq. (3.48)). The adiabatic entropy production rate Sad, determined by the sta-tionary flux Vs , characterizes the irreversibility of maintaining a non-equilibriumsteady state. The nonadiabatic entropy, determined by the relative flux Vr, char-acterizes the irreversibility of the non-equilibrium spontaneous relaxation fromthe transient state to the steady state. These two sources of non-equilibrium irre-versibility constitute the total entropy production rate, determined by the transient(total) flux Vt, which characterizes the total non-equilibrium irreversibility of thesystem. For the total entropy production to be identically zero, the adiabatic andnonadiabatic entropy productions must vanish individually, which is the equilib-rium condition Pt = Ps = Pe characterized by Vt = Vs = Vr = 0. Since Sad ≥ 0

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and Sna ≥ 0, Eq. (3.47) can also be written as inequalities:

dSpd ≥dAsd

T, dSpd ≥

dQhk

T. (3.49)

Equations (3.47) and (3.49) reveal another face of the second law of non-equilibriumthermodynamics.

Furthermore, plugging the entropy flow decomposition equation (Eq. (3.45))and the entropy production decomposition equation (Eq. (3.47)) into the entropybalance equation (Eq. (3.38)), we can derive another form of the entropy balanceequation [116–118]:

S = Sna − Sex. (3.50)

This is a result of canceling the same quantity Sad with two different meanings inthose two decomposition equations. On the one hand, Sad has the meaning of theadiabatic entropy production rate in the entropy production decomposition equa-tion. On the other hand, it has the meaning of the housekeeping entropy flow ratein the entropy flow decomposition equation. For non-equilibrium isothermal pro-cesses, this is related to the double role the housekeeping heat plays as explainedbefore. The adiabatic entropy production (the housekeeping heat) generated with-in the system is completely transferred out of the system into the environment as apart of the total entropy flow (heat flow). Therefore the adiabatic entropy produc-tion (the housekeeping heat) does not accumulate in the system and thus does notcontribute to the net change of the system entropy (the internal energy). That iswhy the change in the system entropy can be expressed alternatively as a tradeoffbetween the nonadiabatic entropy production and the excess entropy flow givenby Eq. (3.50). Since Sna ≥ 0, Eq. (3.50) can also be written as an inequality:

dS ≥ − dQex

T. (3.51)

According to dQtot = dQhk + dQex and dQhk ≥ 0, we have −dQex/T ≥−dQtot/T . Hence Eq. (3.51) is a stronger statement of the second law of thermo-dynamics than that given by Eq. (3.40) [100,108]. Equations (3.50) and (3.51) rep-resent another dimension of the second law of thermodynamics for non-equilibriumisothermal processes. There is actually more. Another aspect of the second lawwill be revealed by investigating the non-equilibrium free energy.

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The Second Law of Non-Equilibrium Thermodynamics in terms of Free En-ergy

The rate of change of the non-equilibrium free energy is given by

˙A =˙U − T S = (Wed − Qex)− T (Sna − Sex) = Wed − T Sna, (3.52)

where we have used A = U −TS , Eqs. (3.33) and (3.50) as well as Sex = Qex/T .Wed is the external driving power defined in Eq. (3.30). The term T Sna, propor-tional to the nonadiabatic entropy production rate (notice the minus sign beforeit), represents the amount of free energy that is irreversibly lost (i.e., dissipated)per unit time, due to the spontaneous relaxation process. Thus we refer to it as the

free energy spontaneous dissipation rate, denoted by ˙Asd, with the subscript ‘sd’representing ‘spontaneous dissipation’. Using Eq. (3.48), we have:

˙Asd = T Sna = ⟨Vt · D−1 · Vr⟩ = −⟨Vt · ∇A⟩= ⟨Vr · D−1 · Vr⟩ = −⟨Vr · ∇A⟩ = ⟨∇A ·D · ∇A⟩. (3.53)

˙Asd is also nonnegative as with Sna. The necessary and sufficient condition for˙Asd ≡ 0 is also the stationary condition: Vt = Vs (Pt = Ps) characterized by

Vr = 0 (A = 0). Non-zero ˙Asd is a result of the non-stationary condition, underwhich the system is in the non-equilibrium irreversible process of spontaneous-ly relaxing from the transient state to the (instantaneous) steady state. Now E-q. (3.52) becomes the non-equilibrium free energy balance equation:

˙A = Wed −˙Asd. (3.54)

It can also be written as − ˙A = −Wed +˙Asd, interpreted as follows. In the total

decrease of free energy − ˙A, a part of it, −Wed, produces useful work (done on

the environment) and the rest, ˙Asd, is irreversibly dissipated. The irreversiblydissipated free energy can no longer be used (i.e., is not available) to produceuseful work. Thus the spontaneously dissipated free energy Asd is identified asthe (ensemble-averaged) ‘dissipative work’ [131,132], also known as the ‘wastedwork’, ‘lost work’ or ‘lost available work’ in engineering [99, 155]. The free

energy spontaneous dissipation rate ˙Asd is thus identified as the dissipative power(i.e., dissipative work per unit time), also called lost power in engineering [155].

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The notations Wsd and Wsd may be used instead of Asd and ˙Asd. They simply

reflect two different perspectives to look at the same quantity. Since ˙Asd ≥ 0,Eq. (3.54) also means:

− dA ≥ − dWed. (3.55)

This is actually the principle of maximum work extended to non-equilibriumisothermal processes; the amount of free energy decreased is the maximum workthe system can produce in isothermal processes. It is the statement of the secondlaw of thermodynamics in terms of free energy in isothermal processes. ThereforeEqs. (3.54) and (3.55) represent another facet of the second law.

The equation, ˙Asd = T Sna, i.e., Wsd = T Sna, has a close connection with theGouy-Stodola theorem known in engineering for more than a century [99, 155,156]. The difference is that in the Gouy-Stodola theorem Sna is simply the en-tropy production rate, as there was no (need of) distinction of adiabatic entropyproduction and nonadiabatic entropy production, since the steady states are usu-ally equilibrium states in the application of engineering. For living organisms,however, the stable biological structures and functions are sustained by constantinput and output of energy and matter, forming dissipative structures correspond-ing to non-equilibrium steady states, associated with non-zero adiabatic entropyproduction [1]. In engineering, entropy production minimization has been used asan optimization principle for the design of finite-size and finite-time devices [155].In the biological context of living organisms, this optimization principle must berevised. The nonadiabatic entropy production may be minimized. But the adia-batic entropy production associated with maintaining the biological structures andfunctions cannot be minimized without limit, as zero adiabatic entropy produc-tion would also mean disappearance of the dissipative structure and its biologicalfunction, which literally means death for living organisms. Thus a certain levelof adiabatic entropy production must be maintained as the necessary cost to en-sure the required level of stability of the dissipative structure and efficiency of itsbiological function.

The non-equilibrium free energy balance equation, Eq. (3.54), also providesan operational solution to the definition of the non-equilibrium stationary free en-ergy. Consider quasi-static processes in which Pt ≈ Ps all the time. In such

processes ˙Asd = 0 (also Sna = 0). (For perspectives on the constantly generatedhousekeeping heat in such processes, see Ref. [100].) Therefore, we can deter-mine the non-equilibrium stationary free energy with the thermodynamic workperformed in quasi-static processes considering ∆As = Wed. This is consistent

90

with giving the equations of state Λis(λ) and defining As =

∫ ∑i Λ

isdλi, since

the integration is just the thermodynamic work performed in quasi-static processes(for constant T ).

3.1.6 Summary and DiscussionTo summarize, we have the following set of thermodynamic equations for non-

equilibrium isothermal processes (three balance equations, of the non-equilibriuminternal energy, entropy and free energy, and two decomposition equations, of theentropy production and entropy flow):

˙U = Wed − Qex

S = Spd − Sfl

˙A = Wed − ˙Asd

Spd = Sad + Sna

Sfl = Sad + Sex

(3.56)

with the relations A = U − TS, Sfl = Qtot/T , Sad = Qhk/T , Sex = Qex/T ,

Sna =˙Asd/T , Asd ≡ Wsd and the sign properties Sad ≥ 0 (Qhk ≥ 0), Sna ≥ 0

( ˙Asd ≥ 0), Spd ≥ 0. The sign of heat is positive when flowing from the system tothe environment. The sign of work is positive when the environment does workon the system. The equation S = Sna − Sex (Eq. (3.50)) is implied and thus notspelled out. U , S and A are state functions, while the rest are process functions.The expressions of the thermodynamic quantities are also derived.

Within the five equations in Eq. (3.56), the three entropy equations (the en-tropy balance equation, entropy flow and entropy production decomposition e-quations) are a direct reflection of the dynamical decomposition equations onthe thermodynamic level. More specifically, the entropy balance equation S =Spd − Sfl is a manifestation of the transient dynamical decomposition equation∇S = D−1 · Vt − D−1 · F ′ (alternatively, F ′ = −D · ∇S + Vt), where S cor-responds to ∇S, Spd corresponds to D−1 · Vt and Sfl corresponds to D−1 · F ′,as seen from their respective expressions in Eqs. (3.34), (3.36) and (3.37). Theentropy flow equation Sfl = Sad + Sex is a mirror of the stationary dynamical de-composition equation D−1 · F ′ = D−1 · Vs−∇U (or F ′ = −D ·∇U+ Vs), whereSfl corresponds to D−1 · F ′, Sad corresponds to D−1 · Vs and Sex correspondsto −∇U , by examining their respective expressions given in Eqs. (3.37), (3.43)

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and (3.44). The entropy production decomposition equation Spd = Sad + Sna ismapped from the flux decomposition equation D−1 · Vt = D−1 · Vs+D−1 · Vr (orVt = Vs + Vr), where Spd corresponds to D−1 · Vt, Sad corresponds to D−1 · Vs

and Sna corresponds to D−1 · Vr, comparing with their respective expressions inEqs. (3.36), (3.43) and (3.48). To be more accurate, ‘corresponds to’ in these s-tatements is equal to the operation of ⟨Vt · K⟩ =

∫Jt · K dq, where K is the

quantity stated after ‘corresponds to’.In the non-equilibrium thermodynamics pioneered by Onsager [92–94], termed

‘classical irreversible thermodynamics’ [102], it is a basic result of the local equi-librium assumption (where ‘local’ means local in the physical space) that the localentropy production rate (also in the physical space) has a flux-force bilinear form,∑

α Jα · Kα, where Jα and Kα are conjugate thermodynamic fluxes and thermo-dynamic forces, respectively. (They can also be scalars or tensors [102]; herewe consider vectors.) The total entropy production rate is then given by the spa-tial integral of the local entropy production rate over the volume of the system:S∗pd =

∫V

∑α Jα · Kαdx. If there is only one pair of thermodynamic flux and

thermodynamic force, then S∗pd =

∫VJ · Kdx. It has the same mathematical form

as the entropy production rate in Eq. (3.36), Spd =∫Jt · D−1 · Vt dq, with Jt

corresponding to the thermodynamic flux and Kpd ≡ D−1 · Vt corresponding tothe thermodynamic force. This has been noticed before [34, 109, 118].

However, there is a conceptual distinction, which we believe is important, thathas not been given sufficient attention. The thermodynamic fluxes and forces inthe classical irreversible thermodynamics describe transport processes of matterand energy in the physical space. In this context, thermodynamic fluxes representmatter or energy fluxes, which are generated by the corresponding thermodynamicforces in the physical space. For example, temperature gradient is the thermody-namic force generating the heat flow (energy flux). While in our case, q representsa microstate of the system in the microstate space rather than in the physical space.Jt is the probability flux, representing ‘information flow’ in the microstate space.Kpd is the ‘force’ generating information flow in the microstate space. Therefore,Jt and Kpd describe information transport process in the microstate space, ratherthan matter or energy transport process in the physical space. We may still callthem thermodynamic flux and thermodynamic force, but their meanings should beunderstood in the context of information transport in the microstate space. Carealso need to taken in reading the word ‘local’ as it may have two different mean-ings, that is, local in the physical space and local in the microstate space. Inparticular, local in the microstate space does not imply local in the physical space.

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This is most clearly seen in spatially inhomogeneous systems (Sec. 3.3), where aglobal state of the system in the physical space is represented by a local ‘point’ inthe microstate space (field configuration space). This is crucial for understanding,(Sec. 3.3), how a global description of spatially inhomogeneous systems in thephysical space can be disguised as a local description in the microstate space, andhow it is that our treatment is not bound by local equilibrium despite its apparentmathematical similarity with the expressions in classical irreversible thermody-namics that assumes local equilibrium. With the distinction between informationtransport and matter or energy transport stressed, we also mention the possibleconnection between them. The information transport process in the microstatespace may be a result of the matter and energy transport in the physical space.Consider the non-equilibrium steady state which is sustained by constant matterand energy flows generated by constant thermodynamic forces (e.g., temperaturedifference or chemical potential difference). On the micro-statistical level, there isa non-zero stationary probability flux Js on the microstate space, which vanishesas matter and energy flows also vanish when the system is in thermodynamic e-quilibrium. The consistency between these two levels of description suggests thatthe information flux Js is associated with the matter and energy fluxes and thusalso the thermodynamic forces generating them in the physical space. For a casestudy, see Ref. [91].

We have defined Kpd = D−1 · Vt, which we call the thermodynamic forcegenerating the total entropy production, given that Spd =

∫Jt · Kpd dq. Since the

expressions of other thermodynamic quantities in Eq. (3.56) also have the sameform

∫Jt · K dq, we generalize the concept of thermodynamic force (not limited

to entropy production) and also call the K corresponding to other thermodynamicquantities, thermodynamic force. Thus the thermodynamic forces generating theadiabatic entropy production, nonadiabatic entropy production, total entropy flow,excess entropy flow and transient entropy change are, respectively, Kad = D−1 ·Vs, Kna = D−1 · Vr = −∇A, Kfl = D−1 · F ′, Kex = −∇U and KS = ∇S. Forinternal energy, heat flow, free energy and spontaneously dissipated free energy(i.e., dissipative work), we can also define their respective thermodynamic forcesaccordingly; they only differ from those already introduced above by a factor ofT or T−1. We stress again that these thermodynamic forces are to be understoodin the context of information transport in the microstate space; they are thus innature ‘informational’.

These three expressions Kex = −∇U , KS = ∇S and Kna = −∇A deserveparticular attention. Since they express the thermodynamic ‘force’ in a gradient

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form, we can indeed interpret the potential landscapes U , S and A as ‘potentials’,in the same sense that the inverse of temperature can be interpreted as the po-tential of heat flow in classical irreversible thermodynamics [92–94]. Therefore,the stationary potential (landscape) U is the potential of excess entropy flow; thetransient potential (landscape) S is the potential of transient entropy change; therelative potential (landscape) A is the potential of nonadiabatic entropy produc-tion. These potentials in essence are not energy potentials; they are information (orentropy) potentials. This has shed light on the physical meanings of these ‘poten-tials’ introduced in the potential-flux landscape framework in Sec. 3.1.2. Further,if we interpret the diffusion matrix D as a metric tensor on the microstate spaceas in the covariant form of the Fokker-Planck equation [16], then the matrix oper-ation of D or D−1 on a vector simply switches the vector between its covariantform and contravariant form. In this sense we can also refer to D · K or D−1 · Kas the thermodynamic force. Thus we can also say the transient flux Vt is the ther-modynamic force generating total entropy production. Similarly, Vs, Vr and F ′

are, respectively, the thermodynamic forces generating the adiabatic entropy pro-duction, nonadiabatic entropy production and total entropy flow. This has clarifiedthe thermodynamic meanings of these dynamical forces. (Although this is con-venient for conceptual simplification, if non-covariant form of the Fokker-Planckequation is used, the diffusion matrix still need to be tracked.) For some relevantinformation-theoretic studies in thermodynamics, see Ref. [157].

The dynamical decomposition equations in Eqs. (3.6)-(3.9) can now be inter-preted as thermodynamic force decomposition equations, which, through the op-eration ⟨Vt ·K⟩ =

∫Jt ·K dq (also call it the flux-force bilinear form), are mapped

into (three of) the non-equilibrium thermodynamic equations in Eq. (3.56). Thisshows explicitly how, in the potential-flux landscape framework, the decomposi-tion equations on the dynamic level are manifested on the thermodynamic level,thus establishing the connection between the dynamics and thermodynamics ofthe system. In particular, the fluxes Vt, Vs and Vr introduced on the dynamiclevel directly characterize the non-equilibrium aspects of the system. Vs char-acterizing the non-equilibrium irreversible aspect of the steady state is the ther-modynamic force generating adiabatic entropy production, Vr characterizing thenon-equilibrium irreversible aspect of the relaxation process is the thermodynamicforce generating nonadiabatic entropy production, and Vt capturing the combinednon-equilibrium irreversible effect of both the steady state and the transient re-laxation process is the thermodynamic force generating total entropy production.Thermodynamic equilibrium indicated by zero entropy production Spd = Sad =

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Sna = 0 is characterized by the vanishing of all the fluxes: Vt = Vs = Vr = 0. Itis the non-zero fluxes Vt, Vs and Vr that break the reversibility of equilibrium s-tates and create entropy production in non-equilibrium processes. As for the othertwo thermodynamic equations (internal energy and free energy balance equation-s), we mention that their counterparts on the dynamic level are the following twoequations:

DU

Dt=

∂U

∂t+ Vt · ∇U ,

DA

Dt=

∂A

∂t+ Vt · ∇A, (3.57)

where D/Dt is the advective derivative defined in Eq. (3.28). Taking ensembleaverage, changing the advective derivative to the total derivative when pulling itout of the ensemble average ⟨·⟩t, and using ⟨∂tS⟩t = 0, we get the internal energyand free energy balance equations in Eq. (3.56).

3.1.7 Extension to Systems with One General State TransitionMechanism

The non-equilibrium thermodynamic equations in Eq. (3.56) are derived fromnon-equilibrium isothermal processes. We consider a generalization of these e-quations to non-equilibrium processes with one general (effective) transition mech-anism. For such processes we do not introduce temperature into the dynamical e-quation, such as D ∝ T in isothermal processes. Thus we cannot utilize quantitiesbased on the introduction of temperature, including the non-equilibrium internalenergy U and free energy A (see Eqs. (3.16) and (3.22)), to formulate the non-equilibrium thermodynamic equations for such systems. Another perspective isthat temperature introduces an energy scale into the dynamical equation. With-out it, we cannot construct, from the dynamical equation, quantities that have thedimension of energy, such as internal energy, free energy, heat and work.

Fortunately, U and A have their respective dimensionless entropic analogsavailable, namely the cross entropy U and relative entropy A, which are purelyconstructed from Ps and Pt, without referring to temperature (see Eqs. (3.17) andand (3.23)). According to Eqs. (3.24) and (3.25), U = TU+As and A = TA+As.From these relations and the balance equations of U and A in Eq. (3.56), we canderive the balance equations of U and A. For example,

U = T−1(˙U − ˙As

)= T−1

(Wed − Qex −

˙As

)= T−1

(Wed −

˙As

)− T−1Qex. (3.58)

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The second term in the last equation is the excess entropy flow rate Sex = T−1Qex.We define the first term in the last equation as the external driving entropic power,which is the entropic analog of the external driving power Wed:

Sed = T−1(Wed −˙As) = ⟨∂tU⟩ =

∑i

⟨∂λiU⟩λi, (3.59)

where we have used Wed = ⟨∂tU⟩ and U = TU + As. (One difference betweenSed and Wed is that in quasi-static processes Wed = ∆As, which is not necessarilyzero, while Sed = 0 according to the first equation of Eq. (3.59).) Now Eq. (3.58)becomes the cross entropy balance equation:

U = Sed − Sex, (3.60)

which states that the system’s cross entropy is increased (decreased) when the en-vironment is driving the system (the system is driving the environment) throughchanging external conditions and decreased (increased) when excess entropy flowsinto the environment (into the system). Similarly, we can derive the relative en-tropy balance equation:

A = Sed − Sna, (3.61)

which states that the system’s relative entropy is increased (decreased) when theenvironment is driving the system (the system is driving the environment) and al-ways decreased when there is nonadiabatic entropy production within the system.Although Eqs. (3.60) and (3.61) can be derived from the internal energy and freeenergy balance equations in Eq. (3.56), they can also be derived independentlyfrom the definitions U = ⟨U⟩t, A = ⟨A⟩t and the dynamical decomposition equa-tions in Eqs. (3.6)-(3.9). The other three thermodynamic equations in Eq. (3.56)remain unchanged.

Hence, for systems with one general (effective) transition mechanism, we havethe following set of non-equilibrium thermodynamic equations, which are threebalance equations for cross entropy, transient entropy and relative entropy, andtwo decomposition equations for entropy production and entropy flow:

U = Sed − Sex

S = Spd − Sfl

A = Sed − Sna

Spd = Sad + Sna

Sfl = Sad + Sex

(3.62)

96

with the definitions U = ⟨− lnPs⟩, S = ⟨− lnPt⟩, A = ⟨ln(Pt/Ps)⟩ and the signproperties Sad ≥ 0, Sna ≥ 0, Spd ≥ 0. U , S and A are state functions whilethe rest are process functions. For systems described by Langevin or Fokker-Planck dynamics with one transition mechanism, the thermodynamic quantitiesin Eq. (3.62) have explicit expressions, summarized below:

U = ⟨∂tU⟩+⟨Vt · ∇U

⟩, (3.63)

S =⟨Vt · ∇S

⟩, (3.64)

A = ⟨∂tU⟩+⟨Vt · ∇A

⟩, (3.65)

Sed = ⟨∂tU⟩ =∑i

⟨∂λiU⟩λi, (3.66)

Sex = −⟨Vt · ∇U⟩, (3.67)

Spd = ⟨Vt ·D−1 · Vt⟩, (3.68)

Sfl = ⟨Vt ·D−1 · F ′⟩, (3.69)

Sad = ⟨Vt ·D−1 · Vs⟩ = ⟨Vs ·D−1 · Vs⟩, (3.70)

Sna = ⟨Vt ·D−1 · Vr⟩ = −⟨Vt · ∇A⟩= ⟨Vr ·D−1 · Vr⟩ = −⟨Vr · ∇A⟩ = ⟨∇A ·D · ∇A⟩. (3.71)

These results for systems with one general transition mechanism also apply, inparticular, to systems in an isothermal environment. By introducing the (constan-t) temperature of the environment and the quantities defined in relation to tem-perature (internal energy, free energy, heat and work), we recover the results fornon-equilibrium isothermal processes in Eq. (3.56) and the expressions of the ther-modynamic quantities. We leave it as an open question what will happen to theinternal energy and free energy balance equations in Eq. (3.56) if the environmenttemperature changes.

The dimensionless quantities U and A are formally called internal energy andfree energy, respectively, in Ref. [125], as mathematical generalizations of theconcepts of internal energy and free energy in equilibrium thermodynamics. Inthis work we use the entropy language to describe these dimensionless quantities,while reserving the energy language for those quantities that truly have the di-mension of physical energy. This way we can avoid confusion when dealing withmore general situations, where these dimensionless quantities and those with thedimension of physical energy may coexist. For instance, for systems in contact

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with multiple heat baths modeled as a Markov process, we can still define the di-mensionless mathematical ‘internal energy’ (the cross entropy) in terms of Ps andPt, but we may still be able to define the physical internal energy (if interactionenergy is properly accounted for). Yet these two do not necessarily have a simplerelation as that for non-equilibrium isothermal processes in Eq. (3.24).

Besides, using entropy language to describe these dimensionless quantitiesalso has validity on its own right. U and A, as with S , are indeed of entropy(or information) nature, since all three are constructed purely from the ensembleprobability distributions Pt and Ps, with the transient entropy S = ⟨− lnPt⟩t, thecross entropy U = ⟨− lnPs⟩t and the relative entropy A = ⟨ln(Pt/Ps)⟩t. We canregard U , S and A as three distinct yet related aspects of non-equilibrium entropy,differentiated from a single concept of equilibrium entropy, similar to the differ-entiation of a single concept of heat in equilibrium thermodynamics into threeaspects of heat in non-equilibrium isothermal processes (i.e., the total heat, theexcess heat and the housekeeping heat). In equilibrium systems where only theequilibrium ensemble distribution Pe is of concern, these three aspects of entropyare degenerate. More specifically, in equilibrium states Pt = Ps = Pe, the relativeentropy A vanishes while the transient entropy S and the cross entropy U becomeidentical with the equilibrium entropy. In non-equilibrium systems, however, boththe stationary distribution Ps and the transient distribution Pt capture certain non-equilibrium aspects of the system and thus both are required in the description ofnon-equilibrium systems. This is why the non-equilibrium entropy becomes dif-ferentiated. If the microstate vector q takes on discrete values, then the transiententropy, cross entropy and relative entropy have direct physical meanings in termsof the information theory [143]. The transient entropy S measures the averageamount of information needed to identify a microstate of the system when thesystem is described by an ensemble distribution Pt. The cross entropy U mea-sures the average amount of information needed to identify a microstate of thesystem from the transient ensemble distribution Pt, using a coding scheme basedon the stationary ensemble distribution Ps. The relative entropy A measures theaverage amount of extra information needed to identify a microstate of the systemfrom the transient ensemble distribution Pt, using a coding scheme based on thestationary ensemble distribution Ps. If the microstate vector q takes on continuousvalues as in the case of Fokker-Planck dynamics, there are difficulties interpretingthese entropies directly, as has been encountered in information theory in the so-called differential entropy or continuous entropy [143]. Yet these interpretationaldifficulties can be circumvented as far as entropy difference is concerned, whichmeans only the relative values of entropy are considered. In particular, the tran-

98

sient entropy S and the cross entropy U are both defined up to a common additiveconstant, while the relative entropy A = U − S does not have this freedom. Innon-equilibrium processes all three aspects of entropy, the transient entropy S ,cross entropy U and relative entropy A, are required to fully characterize non-equilibrium entropy. (The stationary entropy Ss = ⟨− lnPs⟩s is a special case ofS and U when Pt = Ps.)

The set of non-equilibrium thermodynamic equations in Eq. (3.62) govern-ing various entropic quantities thus represent the pure information dynamics (or‘infodynamics’) of the non-equilibrium stochastic dynamical system. The non-equilibrium infodynamics is part of the non-equilibrium thermodynamics (specif-ically, the second law of thermodynamics), but it does not deal with quantitieswith the dimension of physical energy directly (thus the first law of thermody-namics). The formulation of the first law representing energy conservation re-quires additional thermodynamic input apart from the microscopic internal dy-namics [121, 130, 135]. For systems in an isothermal environment, when the en-vironment temperature and the stationary free energy are given, the first law ofthermodynamics can be recovered from the cross entropy balance equation. Itis still an open question whether or not, in more general cases, given certain ex-tra thermodynamic content of the system, the non-equilibrium internal energy andfree energy can still be defined, with which the first law in terms of internal energyand the second law in terms of free energy can be formulated and recovered fromthe infodynamic equations. Without further information of the system’s thermo-dynamic characterization, the best we can do is extract the infodynamics of thenon-equilibrium thermodynamics from the stochastic dynamics.

3.2 Non-Equilibrium Thermodynamics for Spatial-ly Homogeneous Stochastic Systems with Multi-ple State Transition Mechanisms

In this section we consider systems described by Langevin and Fokker-Planckequations with multiple state transition mechanisms (e.g., multiple heat or parti-cle reservoirs) in a multidimensional state space. We also expand the potential-flux landscape framework to accommodate such an extended stochastic dynamics.We then generalize the non-equilibrium thermodynamics obtained in the previoussection (Sec. 3.1) for one mechanism, specifically, the infodynamic equations andexpressions in (3.62)-(3.71), to systems with multiple mechanisms. Then we clar-

99

ify under what conditions the results for multiple mechanisms reduce to those forfor one mechanism. We conclude this section with an illustration of the generalformalism using the Ornstein-Uhlenbeck process and a more specific example.

3.2.1 Stochastic Dynamics for Multiple State Transition Mech-anisms

We consider the following Langevin equation describing the stochastic dy-namics of systems with multiple state transition mechanisms labeled by the indexm, as an extension of the Langevin equation for systems with one state transitionmechanism given by Eq. (3.1):

dq =∑m

dq (m) =∑m

[F (m)(q, t)dt+

∑s

G(m)s (q, t)dWs(t)

]. (3.72)

We assume the state space is an n-dimensional Euclidean space Rn. We alsoassume for each transition mechanism labeled by the fixed index m, the vectorsF (m)(q, t) and G

(m)s (q, t) (s = 1, 2, ...) all belong to the same vector space R(m)

that is a linear subspace of Rn. We call R(m) the state transition space of mech-anism m. R(m) for different transition mechanisms are not assumed to have thesame dimension and they are not necessarily orthogonal to each other. Mathemat-ically, if e (m)

i (i = 1, 2, ...) is an orthonormal basis of R(m) ⊆ Rn, we have:

F (m)(q, t) =∑i

e(m)i F

(m)i (q, t), (3.73)

G(m)s (q, t) =

∑i

e(m)i G

(m)i s (q, t), (3.74)

where F(m)i (q, t) = e

(m)i · F (m)(q, t) and G

(m)i s (q, t) = e

(m)i · G(m)

s (q, t) are thecomponents of F (m)(q, t) and G

(m)s (q, t) in the basis e (m)

i , respectively. Weintroduce a projection operator (or a projection matrix) for each state transitionspace R(m):

Π(m) =∑i

e(m)i e

(m)i , (3.75)

which projects a vector α ∈ Rn onto R(m) via the projection operation:

Π(m) · α =

(∑i

e(m)i e

(m)i

)· α =

∑i

e(m)i

(e(m)i · α

).

100

Π(m) can also operate on a vector from the right side. When α ∈ R(m), we haveΠ(m) · α = α. The projection operator Π(m) also has the idempotent propertyΠ(m) ·Π(m) = Π(m). Yet we do not require Π(m) ·Π(m′) = 0 for m = m′, sinceR(m) and R(m′) do not have to be orthogonal.

The Fokker-Planck equation corresponding to Eq. (3.72) (in Ito’s sense) reads:

∂tPt = −∇ ·(FPt −∇ · (DPt)

), (3.76)

where

F =∑m

F (m), D =∑m

D(m) =∑m

[1

2

∑s

G(m)s G(m)

s

]. (3.77)

The argument (q, t) has been suppressed to simplify notations. The diffusion ma-trix D(m) of mechanism m and the total diffusion matrix D are all nonnegativedefinite symmetric by construction. D(m) is a matrix within the space R(m) ⊆ Rn.When the dimension of R(m) is smaller than Rn, D(m) is not invertible in Rn. Yetwe require D(m) is invertible within R(m). Also we require D is invertible withinRn. Thus D(m) and D are positive definite symmetric, respectively, within R(m)

and Rn. The Fokker-Planck equation (Eq. (3.76)) still has the form of a continuityequation ∂tPt = −∇ · Jt. The (collective) transient probability flux is given by:

Jt = F ′Pt −D · ∇Pt, (3.78)

where F ′ = F −∇ ·D is the effective driving force. We can also define the indi-vidual transient probability flux for each mechanism J

(m)t , which is related to the

collective transient probability flux via Jt =∑

m J(m)t . According to Eqs. (3.77)

and (3.78), we have:

J(m)t = F ′(m)Pt −D(m) · ∇Pt, (3.79)

where F ′(m) = F (m) −∇ ·D(m) is the effective driving force of mechanism m.The stationary Fokker-Planck equation reads:

∇ ·(FPs −∇ · (DPs)

)= 0. (3.80)

It has the form ∇ · Js = 0, where the (collective) stationary probability flux is:

Js = F ′Ps −D · ∇Ps. (3.81)

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Js can be decomposed into components of each mechanism Js =∑

m J(m)s ,

where the stationary probability flux of mechanism m is given by:

J (m)s = F ′(m)Ps −D(m) · ∇Ps. (3.82)

As with systems having one transition mechanism, we assume that the transientdistribution Pt is always positive and normalized to 1 and that at each instant oftime the stationary distribution Ps is unique, positive and normalized to 1 in theaccessible state space.

3.2.2 Potential-Flux Landscape Framework for Multiple StateTransition Mechanisms

The stationary dynamical decomposition equation for each mechanism, fromEq. (3.82), reads:

F ′(m) = −D(m) · ∇U + V (m)s , (3.83)

where the stationary potential landscape is U = − lnPs and the stationary fluxvelocity for mechanism m is V (m)

s = J(m)s /Ps. The transient dynamical decom-

position equation for each mechanism from Eq. (3.79) reads:

F ′(m) = −D(m) · ∇S + V(m)

t , (3.84)

where the transient potential landscape is S = − lnPt and the transient flux ve-locity for mechanism m is V (m)

t = J(m)t /Pt. From Eq. (3.83) and (3.84) we also

have the relative dynamical constraint equation for each mechanism:

V (m)r = −D(m) · ∇A, (3.85)

where the relative potential landscape is A = U − S = ln(Pt/Ps) and the relativeflux velocity for mechanism m is V (m)

r = V(m)t − V

(m)s . The flux decomposition

equation for each mechanism is:

V(m)t = V (m)

s + V (m)r . (3.86)

The collective form of Eqs. (3.83)-(3.86) as those for systems with one mechanismgiven in Eqs. (3.6)-(3.9) can be recovered by summing over the mechanism indexm.

We discuss some points specific to systems with multiple mechanisms. First,considering that detailed balance means each microscopic process is balanced by

102

its own reverse process, for systems with multiple mechanisms each process with-in a mechanism should be balanced by its own reverse process in that mechanism.This implies when detailed balance condition holds, the stationary flux for eachmechanism should be zero, i.e. J

(m)s = 0 (or V (m)

s = 0) for each m. This is astronger condition than the total stationary flux being zero, Js = 0 (or Vs = 0). Inprinciple, there could be such situations in which J

(m)s = 0 (for at least two dif-

ferent m’s) while Js =∑

m J(m)s = 0. In that case, detailed balance is broken in

the steady state although the total stationary flux is zero. Hence, we expect thereis non-zero adiabatic entropy production and thus also total entropy production inthat scenario, since there is irreversibility due to detailed balance breaking in thesteady state. This clearly indicates that the expressions of the adiabatic and totalentropy production rates for one mechanism in Eqs. (3.68) and (3.70), in general,do not apply to systems with multiple mechanisms. What we can expect is that theadiabatic and total entropy productions, at least, have contributions from the irre-versibility within each mechanism (though cross effects of different mechanismsmay also contribute). Thus each non-zero J

(m)s (or V (m)

s ) should have a strictlypositive contribution to the adiabatic and total entropy productions.

Second, although we still have ∇ · Js = 0, this is not generally true for eachmechanism, that is, ∇ · J (m)

s = 0 in general. Further, since D(m) in Eq. (3.85)is not necessarily invertible in Rn, A = 0 (i.e., Pt = Ps) is not equivalent toV

(m)r = 0 (i.e., V (m)

t = V(m)s ) for each m. Yet since we have assumed D is

invertible in Rn, A = 0 (i.e., Pt = Ps) is still equivalent to Vr = 0 (i.e., Vt =

Vs or∑

m V(m)t =

∑m V

(m)s ). Moreover, since D(m) is invertible in R(m) but

not necessarily in Rn, modifications are required when inverting D(m) in (3.83)-(3.85). By first projecting these equations onto R(m) using the projection matrixΠ(m) and then inverting the matrix D(m), we obtain the following three dynamicaldecomposition equations for multiple mechanisms (compare with Eq. (3.26) forone mechanism):

Π(m) · ∇U = [D(m)]−1 · V (m)s − [D(m)]−1 · F ′(m)

Π(m) · ∇S = [D(m)]−1 · V (m)t − [D(m)]−1 · F ′(m)

Π(m) · ∇A = −[D(m)]−1 · V (m)r

(3.87)

where [D(m)]−1 means the inverse of the matrix D(m) in the subspace R(m) ⊆ Rn.In addition, we also have the flux decomposition equation for each mechanismV

(m)t = V

(m)s + V

(m)r .

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The dynamical quantities in these dynamical decomposition equations con-structed directly from the Fokker-Planck equation also have thermodynamic mean-ings, which will be revealed shortly in Sec. 3.2.3. Here we mention that the sta-tionary, transient and relative potential landscapes U , S and A are, respectively,the microscopic stationary, transient and relative entropies. They are also, re-spectively, the potentials of the thermodynamic forces generating excess entropyflow, transient entropy change and nonadiabatic entropy production. The tran-sient, stationary and relative flux velocities V (m)

t , V (m)s and V

(m)r are, respectively,

the thermodynamic forces generating total, adiabatic and nonadiabatic entropyproductions from mechanism m. The effective driving force F ′(m) is the ther-modynamic force generating entropy flow from mechanism m. The dynamicaldecomposition equations act as a bridge connecting the stochastic dynamics withthe non-equilibrium thermodynamics of systems with multiple transition mecha-nisms.

3.2.3 Non-Equilibrium Thermodynamics for Multiple State Tran-sition Mechanisms

The microscopic dynamics of the system with multiple mechanisms is gov-erned by the Langevin equation in Eq. (3.72). The ensemble distribution is gov-erned by the corresponding Fokker-Planck equation in Eq. (3.76). The definitionsof the three basic non-equilibrium entropies for systems with multiple mecha-nisms are the same as those for one mechanism: the cross entropy U = ⟨− lnPs⟩t,the transient entropy S = ⟨− lnPt⟩t and the relative entropy A = U − S =⟨ln(Pt/Ps)⟩t. Taking the time derivative of these three entropies and using Eq.(3.87), we can derive the same set of non-equilibrium thermodynamic (infody-namic) equations in Eq. (3.62), namely the cross, transient and relative entropybalance equations and the entropy production and entropy flow decompositionequations, which we copy here for completeness:

U = Sed − Sex

S = Spd − Sfl

A = Sed − Sna

Spd = Sad + Sna

Sfl = Sad + Sex

(3.88)

104

with the sign properties Sad ≥ 0, Sna ≥ 0, Spd ≥ 0. The difference for systemswith multiple transition mechanisms lies in the specific expressions of the variousthermodynamic quantities in Eq. (3.88), which are given below, as an extension ofEqs. (3.63)-(3.71) for one mechanism:

U = ⟨∂tU⟩+⟨Vt · ∇U

⟩, (3.89)

S =⟨Vt · ∇S

⟩, (3.90)

A = ⟨∂tU⟩+⟨Vt · ∇A

⟩, (3.91)


⟨∂λiU⟩λi, (3.92)

Sex = −⟨Vt · ∇U⟩ =∑m

−⟨V (m)t · ∇U⟩, (3.93)

Spd =∑m

⟨V (m)t · [D(m)]−1 · V (m)

t ⟩, (3.94)

Sfl =∑m

⟨V (m)t · [D(m)]−1 · F ′(m)⟩, (3.95)

Sad =∑m

⟨V (m)t · [D(m)]−1 · V (m)

s ⟩ =∑m

⟨V (m)s · [D(m)]−1 · V (m)

s ⟩,(3.96)

Sna = ⟨Vt ·D−1 · Vr⟩ =∑m

⟨V (m)t · [D(m)]−1 · V (m)

r ⟩

= −⟨Vt · ∇A⟩ =∑m

−⟨V (m)t · ∇A⟩

= ⟨Vr ·D−1 · Vr⟩ =∑m

⟨V (m)r · [D(m)]−1 · V (m)

r ⟩

= −⟨Vr · ∇A⟩ =∑m

−⟨V (m)r · ∇A⟩

= ⟨∇A ·D · ∇A⟩ =∑m

⟨∇A ·D(m) · ∇A⟩. (3.97)

We use the transient entropy balance equation S = Spd−Sfl as an example toillustrate how this equation and the expressions of the thermodynamic quantities

105

involved are obtained. We calculate the rate of change of the transient entropy:

S = ⟨Vt · ∇S⟩ =∑m

⟨V (m)t · ∇S⟩ =

∑m

⟨V (m)t ·Π(m) · ∇S⟩

=∑m

⟨V (m)t · [D(m)]−1 · V (m)

t ⟩ −∑m

⟨V (m)t · [D(m)]−1 · F ′(m)⟩,(3.98)

where we have used V(m)t = V

(m)t · Π(m) (since V

(m)t is a vector in R(m), its

projection into R(m) is itself) and the dynamical decomposition equation for Π(m)·∇S in Eq. (3.87). By identifying the first term (nonnegative) in the last equationas the entropy production rate and the second term as the entropy flow rate, weobtain the entropy balance equation S = Spd − Sfl, together with the expressionsin Eqs. (3.94) and (3.95). Other equations and expressions can be constructedsimilarly. An extension of the results in Eqs. (3.88)-(3.97) as well as the wayto construct them for spatially inhomogeneous systems will be given in Sec. 3.3using the functional language. Therefore, we shall not further elaborate along thisline in this section in order to reduce redundancy.

The expressions of the total entropy production rate Spd in Eq. (3.94) and thatof the adiabatic entropy production rate Sad in Eq. (3.96), as a sum over the mech-anism index m, agree with our analysis that they should have contributions fromeach individual transition mechanism. According to Eq. (3.96), for each non-zeroV

(m)s , there is a strictly positive contribution ⟨V (m)

s · [D(m)]−1 · V (m)s ⟩ to the adi-

abatic entropy production rate, which is also part of the total entropy productionrate. One may wonder about the cross effects of different mechanisms. It seem-s that Eqs. (3.94) and (3.96) as well as the rest show no sign of cross effect, asthere is no term involving two different mechanism indexes. However, this ismerely an illusion. In fact, the fluxes V

(m)t , V (m)

s and V(m)r of different mecha-

nisms in those equations are not independent of each other; they are coordinatedby the global potential landscapes U = − lnPs, S = − lnPt and A = (lnPt/Ps),which are determined by the collective inputs of F (m) and D(m) from individ-ual mechanisms via the Fokker-Planck equation. Put it the other way around.F (m) and D(m) from each individual mechanism collectively determine F and Din the Fokker-Planck equation and thus collectively determine its transient solu-tion Pt and stationary solution Ps. Pt and Ps then determine the potential land-scapes S, U and A, which in turn instructs the individual fluxes V

(m)t , V (m)

s andV

(m)r of each mechanism through the dynamical decomposition equations in E-

qs. (3.83)-(3.85). Therefore, the individual flux of each mechanism emerges asthe collective effect of all the mechanisms, with the potential landscapes play-

106

ing the role of a ‘self-consistent field’ in some sense. Thus the individual fluxof each mechanism has already incorporated the cross effects of different mecha-nisms. Further, thermodynamic equilibrium indicated by zero entropy production,Spd = Sad = Sna = 0, is characterized by all the fluxes of all the individual mech-anisms vanishing, V (m)

t = V(m)s = V

(m)r = 0 for all m, according to Eqs. (3.94),

(3.96) and (3.97). Non-zero fluxes break reversibility of equilibrium states andcreate entropy production in non-equilibrium processes. This also justifies theirroles as the thermodynamic forces generating entropy production.

Then we investigate the individual and collective forms of the expressionsof thermodynamics quantities in Eqs. (3.92)-(3.97). From those expressions wecan see that the external driving entropic power Sed, the excess entropy flow rateSex and the nonadiabatic entropy production rate Sna have expressions withinwhich the mechanism index m does not appear. Therefore these three quantitiesare not dependent on or not sensitive to the recognition of individual transitionmechanisms. In other words, they can be defined collectively (i.e., expressedby collective rather than individual quantities). On the other hand, the entropyproduction rate Spd, the adiabatic entropy production rate Sad and the entropyflow rate Sfl can only be expressed as a sum over the index m, which means theyare sensitive to the correct recognition of individual transition mechanisms.

Equations (3.93)-(3.95) show that Sex, Spd and Sfl have only one unique ex-pression that is decomposed into each individual mechanism. Hence, they can bedefined uniquely for each mechanism:

S(m)ex = −⟨V (m)

t · ∇U⟩, (3.99)

S(m)pd = ⟨V (m)

t · [D(m)]−1 · V (m)t ⟩, (3.100)

S(m)fl = ⟨V (m)

t · [D(m)]−1 · F ′(m)⟩. (3.101)

However, there are subtleties in defining the adiabatic and nonadiabatic entropyproduction rates Sad and Sna for each individual mechanism. For Sad there aretwo inequivalent expressions for mechanism m, ⟨V (m)

t · [D(m)]−1 · V (m)s ⟩ and

⟨V (m)s · [D(m)]−1 · V (m)

s ⟩, which generally are not equal to each other. If weadditionally require that the adiabatic entropy production rate for each mechanismis nonnegative, then we can fix its definition as:

S(m)ad = ⟨V (m)

s · [D(m)]−1 · V (m)s ⟩. (3.102)

The difference between those two inequivalent expressions of Sad for mechanism

107

m can be defined as the mixing entropy production rate for mechanism m:

S(m)mix = ⟨V (m)

s · [D(m)]−1 · V (m)r ⟩ = −⟨V (m)

s · ∇A⟩. (3.103)

The word ‘mixing’ in its name is motivated by the observation that ⟨V (m)s ·[D(m)]−1·

V(m)r ⟩ is a ‘mixing’ of V (m)

s and V(m)r . The stationary flux V

(m)s is an indicator

of detailed balance breaking in the steady state from mechanism m. When it isnon-zero, it has a positive contribution to the adiabatic entropy production rate inthe form of ⟨V (m)

s · [D(m)]−1 · V (m)s ⟩. The relative flux V

(m)r is an indicator of non-

stationarity from mechanism m. When it is non-zero, it has a positive contributionto the nonadiabatic entropy production rate in the form of ⟨V (m)

r · [D(m)]−1 ·V (m)r ⟩.

Therefore S(m)mix characterizes the ‘mixing’ or ‘cross effect’ of these two basic as-

pects of entropy production in the same mechanism m. Although called an en-tropy production rate, S(m)

mix does not have a definite sign. In fact, when summedover m,

∑m S(m)

mix = −∑

m⟨V(m)s · ∇A⟩ = −⟨Vs · ∇A⟩ = 0. (This goes back to

the fact that ∇ · Js = 0, but ∇ · J (m)s = 0 in general.) This means the contribu-

tions of S(m)mix from different mechanisms cancel each other and thus do not have

an effect on the collective level. This property ensures that the two inequivalentexpressions of S(m)

ad , when summed over m, give the same collective expressionof Sad. A necessary (but not sufficient) condition for S(m)

mix = 0 is that V (m)r = 0

and V(m)s = 0. In other words, in order to have a non-zero S(m)

mix, the mechanismm must create both the non-stationary condition and detailed balance breakingcondition in the steady state, which has a positive contribution to both the adiabat-ic and nonadiabatic entropy production rates; if either one of them is zero, S(m)

mix

would be zero.The situation for the nonadiabatic entropy production rate Sna of each mech-

anism is similar. Among the five mechanism-wise expressions of Sna, two ofthem ⟨V (m)

t · [D(m)]−1 · V (m)r ⟩ = −⟨V (m)

t · ∇A⟩, which do not have a definitesign property, are not equivalent to the other three: ⟨V (m)

r · [D(m)]−1 · V (m)r ⟩ =

−⟨V (m)r · ∇A⟩ = ⟨∇A ·D(m) · ∇A⟩ ≥ 0. Their difference is again S(m)

mix, whichdoes not have an effect on the collective level. If we also require that Sna for eachmechanism is nonnegative, then we fix its definition as:

S(m)na = ⟨V (m)

r · [D(m)]−1 · V (m)r ⟩ = −⟨V (m)

r ·∇A⟩ = ⟨∇A ·D(m) ·∇A⟩. (3.104)

Because of the presence of S(m)mix, the decomposition equations of entropy pro-

duction and entropy flow, Spd = Sad + Sna and Sfl = Sad + Sex, generally do not

108

hold for each individual mechanism. The amended equations for each individualmechanism read as follows:

S(m)pd = S(m)

ad + S(m)na + 2S(m)

mix, (3.105)

S(m)fl = S(m)

ad + S(m)ex + S(m)

mix. (3.106)

Due to the property∑

m S(m)mix = 0, the term S(m)

mix does not appear in the collectiveequation.

3.2.4 Necessary and Sufficient Condition for the Collective Def-inition Property

We investigate under what conditions the expressions of thermodynamic quan-tities derived for systems with one state transition mechanism also apply to sys-tems with multiple state transition mechanisms. To be more specific, we study theconditions under which the thermodynamic quantities calculated using combinedcollective quantities from all the mechanisms give the same results as those calcu-lated using individual quantities from each mechanism and then combined togeth-er. This can be termed the collective definition property. For example, the nona-diabatic entropy production rate has this collective definition property accordingto Eq. (3.97): Sna = ⟨Vr ·D−1 · Vr⟩ =

∑m⟨V

(m)r · [D(m)]−1 · V (m)

r ⟩. The resultof Sna calculated using the combined collective quantities Vr and D from all themechanisms is the same as that calculated using the individual quantities V (m)

r andD(m) from each mechanism and then combined together. This property, however,is generally not true for the adiabatic or total entropy production rates. In otherwords, we do not always have Sad =

∑m⟨V

(m)s ·[D(m)]−1 ·V (m)

s ⟩ = ⟨Vs ·D−1 ·Vs⟩or Spd =

∑m⟨V

(m)t · [D(m)]−1 · V (m)

t ⟩ = ⟨Vt · D−1 · Vt⟩. It is therefore worth-while to investigate under what conditions they do hold true. An obvious startingpoint is to see what makes Sna have such a property. By investigating Eq. (3.97)one will find that this collective definition property originates from the relativedynamical constraint equation: V (m)

r = −D(m) · ∇A. This observation leads tothe discovery of a necessary and sufficient condition for the collective definitionproperty, which is proven in Appendix B. We state the result as a theorem in thefollowing.

Theorem: For given vectors β(m) ∈ R(m) ⊆ Rk, β =∑

m β(m) ∈ Rk andinvertible matrixes D(m) ∈ M (m)×(m), D =

∑mD(m) ∈ M k×k, the necessary

109

and sufficient condition for∑

m α(m) · [D(m)]−1 · β(m) = α ·D−1 · β to hold forall α(m) ∈ R(m) with α =

∑m α(m) ∈ Rk is that there exists γ ∈ Rk such that

β(m) = D(m) · γ (m = 1, 2, ...). When γ does exist, it is given by D−1 · β.The above theorem is stated in its most general form. We need to adapt it

to our particular situation. We first specify Rk to be the entire state space Rn.Also we notice that the vectors and matrixes involved in the theorem can also bedependent on the state vector q and time t, in which case ‘hold’ or ‘exist’ meanshold or exist for all q and t. Furthermore, the theorem is applicable when averagedover the ensemble distribution. With these specifications applied, the theoremshows that the collective definition property of Sna is a result of the dynamicalconstraint equation V

(m)r = −D(m) · ∇A, where −∇A plays the role of γ in the

theorem, which exists by definition. The reason why Sad and Spd in general donot have the collective definition property is due to the structure of the stationaryand transient dynamical decomposition equations, V (m)

s = D(m) · ∇U + F ′(m)

and V(m)t = D(m) · ∇S + F ′(m), both of which have an additional term F ′(m)

that cannot be absorbed into the first term without changing the structure of theequation.

We further investigate why γ ∈ Rk satisfying β(m) = D(m) · γ (m = 1, 2, ...)does not always exist and under what conditions it does exist. The conditionβ(m) = D(m) · γ is equivalent to Π(m) · γ = [D(m)]−1 · β(m), which specifies theprojection of γ onto each space R(m). Because the spaces R(m) (m = 1, 2, ...)could have non-trivial overlaps (the collection of the base vectors of these s-paces are linearly dependent), specification of the projection of a vector ontothese spaces can be contradictory to each other, resulting in the vector γ non-existent. If the collection of the base vectors of R(m) (m = 1, 2, ...) are linearlyindependent, the vector γ always exists and is given by D−1 · β. In that case∑

m α(m) · [D(m)]−1 · β(m) = α ·D−1 · β according to the theorem.Placing this analysis into the context of transition mechanisms, it means the

thermodynamic expressions for one transition mechanism also apply to multipletransition mechanisms under the conditions specified in the following. If thereis a subset of transition subspaces R(m), with the collection of their base vec-tors linearly independent, then the thermodynamic expressions for one transitionmechanism also apply to the collection of that particular subset of transition mech-anisms. Recognizing or not the individual transition mechanisms within that par-ticular subset does not influence the result. In particular, if the collection of thebase vectors of all the transition subspaces R(m) (m = 1, 2, ...) form a complete(not necessarily orthogonal) basis of the entire state space Rn, then the expres-

110

sions of thermodynamic quantities are insensitive to the (non-)recognition of theindividual transition mechanisms of the entire system. To put it another way,for a system with multiple transition mechanisms, whenever there are transitionmechanisms with the collection of the base vectors of the corresponding transitionspaces linearly independent, they can be grouped together as one effective transi-tion mechanism, in terms of calculating the thermodynamic quantities using theformula for one transition mechanism given in Eqs. (3.63)-(3.71).

The above understanding of the collective definition property is mainly fromthe abstract mathematical point of view. There is a more intuitive approach tounderstand this property by considering the paths on the state space. The variousthermodynamic quantities introduced so far, such as those in Eq. (3.88), can alsobe defined on the trajectory level on the microstate space [128–130]. The averageover the paths on the microstate space then gives the macroscopic thermodynamicquantities. A trajectory on the microstate space is a depiction of the evolutionof the system’s microstate (a process the system goes through) that is composedof successive state transitions. When there are multiple physically different statetransition mechanisms, a phenomenon emerges that does not exist for systemswith only one state transition mechanism, which is configuration degeneracy ofphysically different paths. That means physically different paths that are real-ized via different transition mechanisms (or different combinations of successivetransition mechanisms) can share the same path configuration on the microstatespace, resulting in degeneracy of the configuration of physically different paths.The correct expressions of those various quantities, defined via paths in the statespace, are dependent on taking into account the path degeneracy phenomenon.If physically different paths realized via different (combinations of) mechanismsthat share the same path configuration are not identified correctly, they would betreated as if they are a single path without degeneracy, as in systems with only onestate transition mechanism. This has an influence on the derived expressions ofthe various quantities defined via paths. However, there are also situations wherethe paths on the state space are not degenerate, even if there are multiple statetransition mechanisms. In that case, the expressions for one transition mechanismalso apply to those of multiple transition mechanisms. The condition we have de-rived for the entire state space, that the collection of the base vectors within R(m)

form a complete basis of Rn, can be understood as a statement of no path degen-eracy. This is because each elementary transition vector dq along a specific pathconfiguration can be uniquely decomposed, in that basis, into components in eachstate transition subspace R(m). Therefore the contribution from each transitionmechanism to each path element dq along the path configuration is specific and

111

has no ambiguity. A path on the state space therefore can be uniquely identifiedby its configuration. In other words, there are no physically different paths sharingthe same path configuration leading to the path degeneracy phenomenon. This isthe reason why the expressions for systems with one transition mechanism alsoapply to systems with multiple transition mechanisms in such situations.

3.2.5 Ornstein-Uhlenbeck Processes of Spatially HomogeneousSystems

We apply the formalism developed so far to Ornstein-Uhlenbeck processes(OU process for short), processes with a linear force and additive (state-independent)noise, which can be studied analytically [15, 16]. We first consider the general ndimensional OU process with one effective state transition mechanism. Then weconsider a more specific example of a two dimensional OU process with three s-tate transition mechanisms, which cannot be reduced to one effective mechanism.

General OU processes with one state transition mechanism

The Langevin equation of the OU process has the following Gaussian white-noise form:

˙q = −γ · q + ξ(t), (3.107)

where the fluctuating force ξ(t) is Gaussian white noise with the following statis-tical property:

⟨ξ(t)⟩ = 0, ⟨ξ(t)ξ(t′)⟩ = 2Dδ(t− t′). (3.108)

The matrices γ and D do not depend on the state vector q. Yet, for generality,we allow them to be time-dependent (e.g., via dependence on external controlparameters). We also assume that γ is invertible and D is positive definite. TheFokker-Planck equation corresponding to the above Langevin equation is:

∂tPt = ∇ · (γ · q Pt +∇ · (DPt)) . (3.109)

Due to the particular form of the drift vector (linear in q) and the diffusion matrix(independent of q), the transient distribution Pt(q) is a Gaussian distribution allthe time as long as the initial condition is a Gaussian distribution [15, 16]. Thisis still true even when γ and D are time-dependent. We focus on such Gaussiansolutions:

Pt(q) =1√

det(2πσ)exp

−1

2(q − µ) · σ−1 · (q − µ)

, (3.110)

112

where the mean vector µ and the covariance matrix σ are time-dependent anddetermined by the following equations [16]:

˙µ = −γ · µ (3.111)σ = −γ · σ − σ · γᵀ + 2D, (3.112)

where γᵀ is the transpose of the matrix γ. With these two equations one can verifythat the Gaussian distribution in Eq. (3.110) is indeed the solution of Eq. (3.109).When γ and D are time-dependent, we can introduce the instantaneous stationarysolution by setting the right side of Eq. (3.109) as zero. It turns out that this in-stantaneous stationary solution is still a Gaussian distribution, whose mean vectorand covariance matrix are the instantaneous stationary solutions of Eqs. (3.111)and (3.112), respectively, by setting the right side of these two equations as zero.Since γ is invertible by assumption, the instantaneous stationary mean vector van-ishes according to Eq. (3.111). Therefore the instantaneous stationary distributionis given by:

Ps(q) =1√

det(2πσ)exp

−1

2q · σ−1 · q

, (3.113)

where σ is the instantaneous stationary covariance matrix, determined by, accord-ing to Eq. (3.112):

γ · σ + σ · γᵀ = 2D. (3.114)

Ps(q) in Eq. (3.113) can be verified to be the instantaneous stationary distributionby plugging it into the right side of Eq. (3.109) and proving it to be zero usingEq. (3.114).

From the transient and stationary distributions given by Eqs. (3.110) and (3.113),we are able to derive the expressions of the stationary, transient and relative poten-tial landscapes of OU processes, using the definitions U = − lnPs, S = − lnPt

and A = U − S:

U =1

2q · σ−1 · q + 1

2tr (ln(2πσ)) , (3.115)

S =1

2(q − µ) · σ−1 · (q − µ) +

1

2tr (ln(2πσ)) , (3.116)

A =1

2q · σ−1 · q − 1

2(q − µ) · σ−1 · (q − µ) +

1

2tr (ln σ − lnσ)(3.117)

where tr(·) represents the trace of the matrix in the bracket and we have usedln(detB) = tr(lnB) which holds for positive-definite matrices σ and σ. Their

113

gradients are given, respectively, by:

∇U = σ−1 · q, (3.118)∇S = σ−1 · (q − µ), (3.119)∇A = σ−1 · q − σ−1 · (q − µ). (3.120)

Then using the dynamical decomposition equations in Eqs. (3.6)-(3.8), we can fur-ther derive the expressions of the stationary, transient and relative flux velocitiesfor the OU process:

Vs = −γ · q +D · σ−1 · q, (3.121)Vt = −γ · q +D · σ−1 · (q − µ), (3.122)

Vr = D · σ−1 · (q − µ)−D · σ−1 · q. (3.123)

Therefore for OU processes we have obtained the explicit expressions of eachterm in the stationary dynamical decomposition equation, F ′ = −D · ∇U + Vs,with the effective driving force (also the deterministic driving force in this case)F ′ = −γ · q, the gradient-like potential term −D · ∇U = −D · σ−1 · q andthe curl-like flux term Vs = −γ · q + D · σ−1 · q. For the transient dynamicaldecomposition equation F ′ = −D · ∇S + Vt, we have explicitly the effectivedriving force F ′ = −γ · q, the potential term −D · ∇S = −D · σ−1 · (q − µ)

and the flux term Vt = −γ · q+D ·σ−1 · (q− µ). And for the relative dynamicalconstraint equation Vr = −D · ∇A, the expression of both sides of the equationis given explicitly by D · σ−1 · (q − µ)−D · σ−1 · q. These explicit expressionsfacilitate the study of global stability and dynamics of OU process in the potential-flux landscape framework [9, 10, 13, 14].

So far we have not specified the state transition mechanisms which are nec-essary physical inputs for the calculation of certain thermodynamic quantities. Inthe following we consider systems with only one effective state transition mecha-nism. This includes situations where the system actually has multiple physicallydifferent transition mechanisms, but the collection of the base vectors of the cor-responding state transition subspaces are linearly independent, thus enabling themto be treated as one effective transition mechanism. The cross entropy U , transiententropy S and relative entropy A can be calculated explicitly from their definition-s, as the averages of U , S, A in Eqs. (3.115)-(3.117) over the transient ensemblePt in Eq. (3.110), respectively. Their time derivatives as well as the rest of thethermodynamic quantities can be calculated using (3.63)-(3.71), supplemented by

114

Eqs. (3.118)-(3.123) specific to OU processes. We list the results in the followingand leave the details of calculation in Appendix C:

U = tr

(1

2σ−1(σ + µµ) +

1

2ln(2πσ)

)(3.124)

S = tr

(1

2I +

1

2ln(2πσ)

)(3.125)

A = tr

(1

2

[σ−1(σ + µµ)− I

]+

1

2(ln σ − lnσ)

)(3.126)

U = tr

(1

2(σ + µµ− σ)

d

dtσ−1 − σ−1Dσ−1(σ + µµ) + γ

)(3.127)

S = tr(σ−1D − γ

)(3.128)

A = tr

(1

2(σ + µµ− σ)

d

dtσ−1 − σ−1Dσ−1(σ + µµ)

−σ−1D + 2γ)

(3.129)

Sed = tr

(1

2(σ + µµ− σ)

d

dtσ−1

)(3.130)

Sex = tr(σ−1Dσ−1(σ + µµ)− γ

)(3.131)

Sfl = tr(γᵀD−1γ(σ + µµ)− γ

)(3.132)

Spd = tr(γᵀD−1γ(σ + µµ) + σ−1D − 2γ

)(3.133)

Sad = tr((γᵀD−1γ − σ−1Dσ−1)(σ + µµ)

)(3.134)

Sna = tr(σ−1Dσ−1(σ + µµ) + σ−1D − 2γ

)(3.135)

where I represents the identity matrix and µµ is the matrix with entries [µµ]ij =µiµj . It is easy to verify that these expressions satisfy the set of non-equilibriumthermodynamic equations in Eq. (3.88). Therefore, for OU processes with oneeffective transition mechanism, the set of non-equilibrium thermodynamic equa-tions is realized explicitly, with the expressions of thermodynamic quantities givenexplicitly by (3.124)-(3.135).

When there are multiple transition mechanisms that cannot be treated as oneeffective mechanism, some of the results above need to be amended. Yet only thetotal entropy production rate Spd, the adiabatic entropy production rate Sad andthe total entropy flow rate Sfl will be different; the expressions of other thermo-dynamic quantities in Eqs. (3.124)-(3.135) still apply since they can be definedcollectively without referring to individual mechanisms.

115

A two dimensional OU process with three state transition mechanisms

We consider a more specific example of an OU process, with a two dimension-al state space and three different state transition mechanisms. Each mechanisminduces a one dimensional OU process in the state space. Thus the collection ofthe base vectors of these three state transition subspaces are necessarily linearlydependent. They cannot be reduced to one effective mechanism. This examplecan model, for example, two Brownian particles in contact with three heat reser-voirs. The state vector is represented by q = (q+, q−)

ᵀ. For convenience we callthe subspace corresponding to the component q+ the upper space and that corre-sponding to the component q− the lower space. We assume that both mechanism1 and mechanism 2 induce an OU process only in the upper space, while mecha-nism 3 induces an OU process only in the lower space. To be more specific, formechanism 1 the drift vector is F (1) = (−γ1q+, 0)

ᵀ and the diffusion coefficientD1 is in the upper space. For mechanism 2, the drift vector is F (2) = (−γ2q+, 0)

ᵀ

and the diffusion coefficient D2 is in the upper space. For mechanism 3, the driftvector is F (3) = (0,−γ3q−)

ᵀ and the diffusion coefficient D3 is in the lower s-pace. Therefore, the total linear drift matrix and the total diffusion matrix are bothdiagonal, given by:

γ = diag(γ+, γ−), D = diag(D+, D−), (3.136)

where γ+ = γ1 + γ2, γ− = γ3, D+ = D1 +D2, and D− = D3. The mean vectorµ = (µ+, µ−)

ᵀ of the transient distribution is determined by Eq. (3.111), which inthis case becomes:

µ+ = −γ+µ+, µ− = −γ−µ−. (3.137)

The solutions are given by:

µ+(t) = e−

∫ tt0

γ+(t1)dt1µ+(t0),

µ−(t) = e−

∫ tt0

γ−(t1)dt1µ−(t0). (3.138)

The covariance matrix σ of the transient distribution is determined by Eq. (3.112),which in the component form reads:

σ+ = −2γ+σ+ + 2D+,

σ− = −2γ−σ− + 2D−,

σo = −(γ+ + γ−)σo, (3.139)

116

where σ+ and σ− are diagonal elements of the covariance matrix σ while σo isthe off-diagonal element, which means σ+ = var(q+), σ− = var(q−) and σo =cov(q+, q−) of the transient probability distribution. These are all first-order linearODEs. The general solutions are given by:

σ+(t) = e−2

∫ tt0

γ+(t1)dt1

[∫ t

t0

2D+(t2)e2∫ t2t0

γ+(t1)dt1dt2 + σ+(t0)

],

σ−(t) = e−2

∫ tt0

γ−(t1)dt1

[∫ t

t0

2D−(t2)e2∫ t2t0

γ−(t1)dt1dt2 + σ−(t0)

],

σo(t) = e−

∫ tt0[γ+(t1)+γ−(t1)]dt1σo(t0). (3.140)

They can be simplified when γ± and D± are time-independent. The instanta-neous stationary covariance matrix σ is the instantaneous stationary solution ofEq. (3.139), solved algebraically by

σ+ = D+/γ+, σ− = D−/γ−, σo = 0, (3.141)

where σ+ = var(q+), σ− = var(q−) and σo = cov(q+, q−) of the instantaneousstationary probability distribution.

The thermodynamic quantities in Eqs. (3.124)-(3.135) can then be evaluatedfor this 2D case, except Spd, Sad and Sfl which need to be treated differently byconsidering each individual mechanism. Since only algebraic manipulations of2D matrices are involved, we simply list the results below.

U =1

2

[σ+ + µ2

+

σ+

+σ− + µ2

−

σ−

]+

1

2ln(σ+σ−) + ln(2π) (3.142)

S =1

2ln(σ+σ− − σ2

0) + ln(2π) + 1 (3.143)

A =1

2

[σ+ + µ2

+

σ+

+σ− + µ2

−

σ−

]+

1

2ln

(σ+σ−

σ+σ− − σ20

)− 1 (3.144)

117

U =1

2

[(σ+ − σ+ − µ2

+)˙σ+

σ2+

+ (σ− − σ− − µ2−)

˙σ−

σ2−

]

−[D+

σ2+

(σ+ + µ2+) +

D−

σ2−(σ− + µ2

−)− (γ+ + γ−)

](3.145)

S =σ−D+ + σ+D−

σ+σ− − σ2o

− (γ+ + γ−) (3.146)

A =1

2

[(σ+ − σ+ − µ2

+)˙σ+

σ2+

+ (σ− − σ− − µ2−)

˙σ−

σ2−

]−[D+

σ2+

(σ+ + µ2+)

+D−

σ2−(σ− + µ2

−) +σ−D+ + σ+D−

σ+σ− − σ2o

− 2(γ+ + γ−)

](3.147)

Sed =1

2

[(σ+ − σ+ − µ2

+)˙σ+

σ2+

+ (σ− − σ− − µ2−)

˙σ−

σ2−

](3.148)

Sex =D+

σ2+

(σ+ + µ2+) +

D−

σ2−(σ− + µ2

−)− (γ+ + γ−) (3.149)

Sna =D+

σ2+

(σ+ + µ2+) +

D−

σ2−(σ− + µ2

−) +σ−D+ + σ+D−

σ+σ− − σ2o

−2(γ+ + γ−) (3.150)

For comparison, we also list the results of Sfl, Spd and Sad calculated using theformula for one effective mechanism in Eqs. (3.124)-(3.135).

S∗fl =

γ2+

D+

(σ+ + µ2+) +

γ2−

D−(σ− + µ2

−)− (γ+ + γ−) (3.151)

S∗pd =

γ2+

D+

(σ+ + µ2+) +

γ2−

D−(σ− + µ2

−) +σ−D+ + σ+D−

σ+σ− − σ2o

−2(γ+ + γ−) (3.152)

S∗ad =

(γ2+

D+

− D+

σ2+

)(σ+ + µ2

+) +

(γ2−

D−− D−

σ2−

)(σ− + µ2

−)

= 0 (3.153)

The star ∗ indicates that these results are not true for the case we consider here. Inderiving the result S∗

ad = 0 in Eq. (3.153) we have used Eq. (3.141). This meansthat if the three individual mechanisms are not identified correctly, there would

118

seem to be no detailed balance breaking in the steady state indicated by non-zero adiabatic entropy production. Results obtained with proper identification ofdifferent mechanisms are different, which will be given below.

First we need the results in Eqs. (3.118) and (3.119), which in the componentform reads:

∂q+U = γ+q+/D+

∂q−U = γ−q−/D−

∂q+S = [σ−(q+ − µ+)− σo(q− − µ−)]/[σ+σ− − σ2o ]

∂q−S = [σ+(q− − µ−)− σo(q+ − µ+)]/[σ+σ− − σ2o ] (3.154)

For mechanism 1 we work in the upper space. Since the diffusion coefficients areindependent of q, the (effective) drift coefficient (recall F ′ = F −∇·D) is givenby

F ′(1) = F (1) = −γ1q+. (3.155)

The stationary flux velocity of mechanism 1 is calculated using Eq. (3.83), whichreads:

V (1)s = −γ1q+ +D1∂q+U =

[−γ1 +

D1

D1 +D2

(γ1 + γ2)

]q+. (3.156)

Using Eq. (3.84) the transient flux velocity of mechanism 1 given by:

V(1)t = −γ1q+ +D1∂q+S

= −γ1q+ +D1[σ−(q+ − µ+)− σo(q− − µ−)]/[σ+σ− − σ2o ].(3.157)

Similarly, for mechanism 2, we have

F ′(2) = −γ2q+ (3.158)

V (2)s =

[−γ2 +

D2

D1 +D2

(γ1 + γ2)

]q+. (3.159)

V(2)t = −γ2q+ +D2[σ−(q+ − µ+)− σo(q− − µ−)]/[σ+σ− − σ2

o ](3.160)

For mechanism 3, we work in the lower space and have

F ′(3) = −γ3q− (3.161)V (3)s = 0. (3.162)

V(3)t = −γ3q− +D3[σ+(q− − µ−)− σo(q+ − µ+)]/[σ+σ− − σ2

o ](3.163)

119

The stationary flux V(3)s = 0 means there is no detailed balance breaking in the

steady state from mechanism 3. Detailed balance breaking in the steady state ispossible from mechanism 1 and 2 when D1/γ1 = D2/γ2. If D1/γ1 = D2/γ2,then V

(1)s = 0 and V

(2)s = 0. Therefore the detailed balance condition for this

particular case is D1/γ1 = D2/γ2. If these mechanisms represent heat reservoirs,then D1/γ1 = D2/γ2 is simply T1 = T2, i.e., the condition of thermal equilibrium.

Then we can calculate Sfl, Spd and Sad with different mechanisms identifiedproperly. The total entropy flow rate is calculated using Eq. (3.95) for systemswith multiple mechanisms. With the effective drift coefficients of the three mech-anisms given in Eqs. (3.155), (3.158) and (3.161), we have:

Sfl =3∑

i=1

⟨V

(i)t D−1

i F ′(i)⟩

=

(γ21

D1

+γ22

D2

)(σ+ + µ2

+) +γ23

D3

(σ− + µ2−)

−(γ1 + γ2 + γ3). (3.164)

The total entropy production rate is calculated using Eq. (3.94) for systems withmultiple mechanisms. With the transient flux velocities given by (3.157), (3.160)and (3.163), we have

Spd =3∑

i=1

⟨V

(i)t D−1

i V(i)t

⟩=

(γ21

D1

+γ22

D2

)(σ+ + µ2

+) +γ23

D3

(σ− + µ2−)

+σ−(D1 +D2) + σ+D3

σ+σ− − σ2o

− 2(γ1 + γ2 + γ3). (3.165)

The adiabatic entropy production rate is calculated using Eq. (3.96) for systemswith multiple mechanisms, with the stationary flux velocities given by (3.156),(3.159) and (3.162):

Sad =3∑

i=1

⟨V (i)s D−1

i V (i)s

⟩=

(γ21

D1

+γ22

D2

− (γ1 + γ2)2

D1 +D2

)(σ+ + µ2

+). (3.166)

The true adiabatic entropy production rate Sad is not identically zero, in contrast

with S∗ad = 0 in Eq. (3.153). Instead, Sad ≥ 0 due to the inequality

γ21

D1

+γ22

D2

≥

120

(γ1 + γ2)2

D1 +D2

. The necessary and sufficient condition for Sad = 0 is D1/γ1 =

D2/γ2, which is also the detailed balance condition indicated by V(i)s = 0 (i =

1, 2, 3).We notice that the offsets of S∗

fl, S∗pd, and S∗

ad calculated using the formula forone effective mechanism from their true values for multiple mechanisms Sfl, Spd,and Sad, in this particular case, are such that: Sfl − S∗

fl = Sad, Spd − S∗pd = Sad,

Sad − S∗ad = Sad. Therefore, the balance equation of the transient entropy holds

for both set of quantities: S = Spd − Sfl = S∗pd − S∗

fl, yet individually S∗pd and

S∗fl are both off from the true values Spd and Sfl. In general, it is not necessarily

such that Sad − S∗ad = Sad (i.e., S∗

ad = 0). But the offset of the adiabatic entropyproduction rate, due to the incorrect identification of different mechanisms, Sad −S∗ad = ∆Sad, will also be the offset of the total entropy production rate and the

total entropy flow rate: Spd − S∗pd = ∆Sad, Sfl − S∗

fl = ∆Sad. This can beseen from the two decomposition equations: Spd = Sad + Sna and Sfl = Sad +Sex, where Sna and Sex are not influenced by the (non-)recognition of individualmechanisms since they can be defined using collective quantities. Therefore, theoffset of Sad is also the offset of Spd and Sfl, which in turn ensures that S in thebalance equation S = Spd − Sfl is not influenced by the (non-)recognition ofindividual mechanisms.

3.3 Non-Equilibrium Thermodynamics for Spatial-ly Inhomogeneous Stochastic Dynamical System-s

In this section (Sec. 3.3), we extend the major results in Sec. 3.1 and Sec. 3.2for spatially homogeneous systems to spatially inhomogeneous systems. Spatial-ly inhomogeneous systems are systems with infinite degrees of freedom (infinitedimensional systems). A certain level of mathematical accuracy is necessary indealing with such infinite dimensional systems; yet being caught up in too manytechnical details at the initial stage of establishing the formalism would also becounterproductive. Thus we seek to find a balance between physical heuristic mo-tivations and mathematical rigorous treatments. We first look at the description ofspatially inhomogeneous systems. Then we introduce the functional Langevin andFokker-Planck equations. Accordingly, the potential-flux landscape framework is

121

generalized to spatially inhomogeneous systems. With its assistance we formulatethe non-equilibrium thermodynamics for spatially inhomogeneous systems. Gen-eral OU processes are studied using the established formalism. Finally, the spatialstochastic neuronal model, which can also describe a reaction diffusion process,serves as the testing ground and an illustration of the practical application of thegeneral theory.

3.3.1 Description of Spatially Inhomogeneous SystemsWe assume the (micro)state of a spatially inhomogeneous system at each mo-

ment is described by a ℓ-component function of the physical space (a vector field):ϕ(x) = ϕ1(x), ..., ϕa(x), ..., ϕℓ(x). For example, in the context of chemical re-action systems, ϕ(x) may represent the local concentrations of chemical speciesinvolved at any given moment. If the vector ϕ has only one component, the state ofthe system is then described by a scalar field ϕ(x). In the biological context, ϕ(x)may represent, for instance, the local electric potential on the neuron membraneat any given moment.

To specify a state of the spatially inhomogeneous system (a global state inthe physical space; also called a field configuration), the value of the field ϕ(x)need to be given at every location in a domain of the physical space. In practice,the system is always observed at some finite spatial scale, the degrees of freedombelow which are averaged out or coarse-grained, resulting in an effective descrip-tion [137, 158]. In accord with this consideration, we can discretize the relevantregion of the physical space into spatial cells with volume ∆V characterizing theresolution scale and introduce a discrete space representation [15, 50–52]. We la-bel the spatial cells by a discrete space index λ. The average values of ϕ(x) ineach spatial cell, denoted collectively by ϕa

λ, represent the effective degrees offreedom (i.e., state variables) of the system:

ϕaλ =

1

∆V

∫Vλ

dx ϕa(x). (3.167)

Each state of the system is now specified by giving the values of ϕaλ for both

the index a and the discrete space index λ. Therefore, the pair of discrete index-es a and λ in ϕa

λ together labels the effective degrees of freedom of the spatiallyinhomogeneous system. This description is analogous to spatially homogeneoussystems with finite degrees of freedom, labeled by a discrete index i of its statevector q = (q1, ..., qi, ..., qn). This index correspondence (a, λ) ↔ i allows for

122

a formal extension of spatially homogeneous systems with finite degrees of free-dom, to spatially inhomogeneous systems with infinite degrees of freedom in thediscrete space representation [50, 51, 158]. An effective continuous space repre-sentation can be obtained by taking the continuum limit ∆V → 0. What thislimit practically means is that we are studying the system on a scale much largerthan the resolution scale ∆V , so that ∆V is, relatively speaking, very small [15].Mathematically,

lim∆V→0

ϕaλ = lim

∆V→0

1

∆V

∫Vλ

dx ′ ϕa(x ′) = ϕa(x). (3.168)

Equations (3.167) and (3.168) are the prescriptions to switch between the discreteand continuous space representations of the state of the system. We will use thisapproach later to introduce and study OU processes of spatially inhomogeneoussystems.

To allow for appropriate mathematical treatments, we introduce further math-ematical structures for the state and state space of spatially inhomogeneous sys-tems. We assume the physical space (space of x) is modeled as a k-dimensionalEuclidean space Rk, while the vector ϕ lies in an ℓ-dimensional Euclidean spaceRℓ. The region of the physical space relevant for the description of the system ismodeled as a suitable domain V in Rk. Therefore, a state of the spatially inho-mogeneous system is represented by a vector field ϕ(x) : V → Rℓ . The statespace of the system Ω, as the space of the field ϕ(x), is also called the field con-figuration space, with each ‘point’ or ‘element’ in this space representing a fieldconfiguration ϕ(x) (a global state in the physical space). The state space (the fieldconfiguration space) as a function space is an infinite-dimensional space, whichwe assume is a (real) Hilbert space (a linear space with an inner product that iscomplete) [42,43]. The real-valued inner product between two states in the Hilbertspace is defined as:

(φ|ϕ) ≡∫

dx φ(x) · ϕ(x) =∫

dx∑a

φa(x)ϕa(x), (3.169)

where the integral is taken over the domain V and the dot between φ(x) and ϕ(x)is the standard Euclidean dot product in Rℓ. We have used a variation of theDirac notation borrowed from quantum mechanics [159], where the states φ(x)

and ϕ(x) in the Hilbert space within the inner product are represented abstractly inthe bra-ket notation as (φ| and |ϕ), respectively, with round brackets replacing an-gle brackets conventional in quantum mechanics (angle brackets are reserved for

123

ensemble average in this work). The interpretation here, however, is quite differ-ent from quantum mechanics. Here an element in the real Hilbert space representsa definite classical state of the spatially inhomogeneous system (a classical field),while in quantum mechanics an element in the complex Hilbert space represents aquantum state associated with probabilistic interpretations (a quantum wave func-tion). The inner product in Eq. (3.169) represents a geometrical relation betweentwo states, allowing for the introduction of angle and distance on the Hilbert statespace, as a generalization of the Euclidean state space. We note that the flux-forcebilinear form (integrated over space) in classical irreversible thermodynamics [94]has a similar form as Eq. (3.169).

With the inner product defined, an orthonormal basis en(x) of the Hilbert s-tate space can be characterized by the following orthonormality and completenessconditions:∫

dx en(x) · em(x) = δmn,∑n

en(x)en(x′) = I δ(x− x ′), (3.170)

where I is the identity matrix in Rℓ. In this basis a state in the Hilbert space canbe expanded as:

ϕ(x) =∑n

ϕnen(x), (3.171)

whereϕn =

∫dx en(x) · ϕ(x). (3.172)

The coefficient ϕn can be seen as the component of an infinite dimensional vectorϕn. The field ϕ(x) and the infinite dimensional vector ϕn are merely twodifferent representations of the same state |ϕ) in the Hilbert space. ϕ(x) is thestate |ϕ) in the space configuration representation, while ϕn is the state |ϕ) inthe representation of the orthonormal basis en(x). A linear operator B on theHilbert space, mapping a state |ϕ) into another state |φ) , can be represented by amatrix-valued integral kernel, B(x, x ′), in the space configuration representation,with the operation of linear mapping given by the integral transform:

φ(x) =

∫dx ′B(x, x ′) · ϕ(x ′). (3.173)

In an orthonormal basis en(x), B(x, x ′) can be expanded as:

B(x, x ′) =∑mn

Bmnem(x)en(x′), (3.174)

124

whereBmn =

∫dx

∫dx ′em(x) ·B(x, x ′) · en(x ′). (3.175)

The coefficient Bmn can be regarded as the entry of an infinite dimensional squarematrix [Bmn]. The integral kernel B(x, x ′) and the infinite dimensional squarematrix [Bmn] are two different representations of the same linear operator B on theHilbert space. In the abstract representation using the Dirac notation Eq. (3.173)is written as |φ) = B|ϕ), while in the representation of en(x) it reads φm =∑

n Bmnϕn, which can be obtained using Eqs. (3.171)-(3.174).All these different representations are mathematically equivalent and related

by certain transformation rules. Yet one representation may be more convenient orsuitable than another in a particular situation. Generally, the space configurationrepresentation is more intuitive as it works with the physical space; the abstrac-t representation has the advantage of being very compact; the representation ofen(x) is convenient for practical calculations. These statements are not to betaken as absolute, though. In the following, we work in the space configurationrepresentation in the course of establishing the non-equilibrium thermodynamicformalism for spatially inhomogeneous systems. In Appendix D we give the ab-stract representation of the major equations that will be developed in the main textas well as the transformation rules to switch between different representations.When it comes to practical examples we will transform into the representation ofen(x). Occasionally, we also use the abstract representation in the main text.

3.3.2 Stochastic Dynamics of Spatially Inhomogeneous System-s

The spatial-temporal stochastic dynamics of many spatially inhomogeneoussystems can be described by functional Langevin and Fokker-Planck equation-s [15,31,37,42–51,54–60]. We consider stochastic spatially inhomogeneous sys-tems governed by the following functional Langevin equation (a stochastic differ-ential equation on the infinite dimensional Hilbert state space Ω) [15, 31, 42–44,50, 51]:

dϕ(x, t) = F (x, t)[ϕ]dt+∑s

Gs(x, t)[ϕ] dWs(t), (3.176)

where F (x, t)[ϕ] and Gs(x, t)[ϕ] are, respectively, the deterministic and stochasticdriving force fields. They are maps from Ω into Ω, parameterized by t, allowingthem to be time-dependent. For any t, F (x, t) and Gs(x, t) are elements in Ω. The

125

square bracket [ϕ] denotes functional dependence, indicating they are functionsof the field ϕ(x, t) ∈ Ω. In most applications, F (x, t)[ϕ] and Gs(x, t)[ϕ] can beexpressed as (nonlinear) partial differential operators or integro-differential opera-tors acting on the field ϕ(x, t) [31,42–44]. In the former case, the values of F (x, t)

and Gs(x, t) at location x depend only on the local value of the field ϕ(x, t) at xor its immediate neighborhood; in the latter case, which is more general, they de-pend on the field ϕ(x, t) as a whole. Ws(t) (s = 1, 2, ...), with s allowed to go toinfinity, are independent one-dimensional standard Wiener processes. The formof the fluctuation term in Eq. (3.176) is a little different from, yet is equivalen-t to, that usually presented in the mathematical literature in terms of cylindricalBrownian motions [43]. The index s labels statistically independent fluctuationsources. Yet these fluctuation sources do not necessarily represent different statetransition mechanisms. Thus we introduce the state transition mechanism indexm and further specify Eq. (3.176) as follows:

dϕ(x, t) =∑m

[dϕ(x, t)

](m)

=∑m

[F (m)(x, t)[ϕ]dt+

∑s

G(m)s (x, t)[ϕ] dW (m)

s (t)

].(3.177)

For each mechanism m, the state transition[dϕ(x, t)

](m)

has the form of a func-

tional Langevin equation in Eq. (3.176). F (m)(x, t)[ϕ] and G(m)s (x, t)[ϕ] are, re-

spectively, the deterministic and stochastic driving force fields of mechanism m.W

(m)s (t) (m = 1, 2, ...; s = 1, 2, ...) are statistically independent one-dimensional

standard Wiener processes. We assume for each mechanism m the vectors F (m)

and G(m)s (s = 1, 2, ...) are in the same linear subspace R(m) ⊆ Rℓ, which means

they have the following component forms:

F (m)(x, t)[ϕ] =∑i

e(m)i F

(m)i (x, t)[ϕ], (3.178)

G(m)s (x, t)[ϕ] =

∑i

e(m)i G

(m)i s (x, t)[ϕ], (3.179)

where e (m)i (i = 1, 2, ...) are the base vectors of R(m) ⊆ Rℓ. The projection opera-

tor Π(m) defined in Eq. (3.75) and its properties still apply here. The restriction ofRℓ to R(m) defines a Hilbert subspace Ω(m) of the entire Hilbert state space Ω. The

126

projection operator Π(m) also induces the projection from Ω to Ω(m). F (m)(x, t)[ϕ]

and G(m)s (x, t)[ϕ] are therefore maps from the Hilbert state space Ω into its sub-

space Ω(m) ⊆ Ω. Equations (3.177)-(3.179) are extensions of Eqs. (3.72)-(3.74)to spatially inhomogeneous systems. We mention that boundary conditions mayhave important effects on the non-equilibrium dynamics of spatially inhomoge-neous systems [160].

The functional Fokker-Planck equation (usually called the forward Kolmogrovequation in the mathematical literature) on the Hilbert state space Ω, correspond-ing to the functional Langevin equation (Eq. (3.177); interpreted as an Ito stochas-tic differential equation), reads [15, 31, 43, 45–47, 50, 51]:

∂tPt[ϕ] = −∫

dx δϕ(x) ·(F (x, t)[ϕ]Pt[ϕ]

)+

∫dx δϕ(x) ·

∫dx ′δϕ(x ′) · (D(x, x ′, t)[ϕ]Pt[ϕ]) , (3.180)

where δϕ(x) is a short notation for the multi-component functional derivative, with

the component (δϕ(x))a ≡ δ/δϕa(x). The drift vector field and the diffusion matrix

field in Eq. (3.180) are given by:

F (x, t)[ϕ] =∑m

F (m)(x, t)[ϕ], (3.181)

D(x, x ′, t)[ϕ] =∑m

D(m)(x, x ′, t)[ϕ]

=∑m

[1

2

∑s

G(m)s (x, t)[ϕ]G(m)

s (x ′, t)[ϕ]

]. (3.182)

D(x, x ′, t) and D(m)(x, x ′, t) are the counterparts of nonnegative symmetric ma-trices in the infinite dimensional space (i.e., nonnegative self-adjoint operators onthe Hilbert space). By construction they have the following symmetry property:

Dab(x, x ′, t)[ϕ] = Dba(x ′, x, t)[ϕ], [D(m)]ab(x, x ′, t)[ϕ] = [D(m)]ba(x ′, x, t)[ϕ].(3.183)

They also have the following nonnegative property according to Eq. (D.5). Forany φ(x) ∈ Ω, ∫∫

dxdx ′ φ(x) ·D(x, x ′, t)[ϕ] · φ(x ′) ≥ 0, (3.184)

127

∫∫dxdx ′ φ(x) ·D(m)(x, x ′, t)[ϕ] · φ(x ′) ≥ 0. (3.185)

We impose a stronger condition on D and D(m), requiring them to be positivedefinite (we use this term in the strict sense in contrast to nonnegative definite),which means that for any φ(x) ∈ Ω (for any φ(x) ∈ Ω(m)) the equality in E-q. (3.184) (the equality in Eq. (3.185)) holds only when φ(x) ≡ 0. We also assumethat D(x, x ′, t)[ϕ] is invertible in the sense that there exists M (x, x ′, t)[ϕ] on Ωwhich has the symmetry property as in Eq. (3.183) and satisfies the following:∫

dx ′′∑c

Mac(x, x ′′, t)[ϕ]Dcb(x ′′, x ′, t)[ϕ] = δabδ(x− x ′), (3.186)

where the indexes a, b, c are vector indexes of the space Rℓ. Define M (x, x ′, t)[ϕ] ≡[D(x, x ′, t)]−1 [ϕ], where the notation [D(x, x ′, t)]−1 indicates the inverse is notonly with respect to the matrix indexes of D but also the space indexes x andx ′. Note that [D(x, x ′, t)]−1 does not mean 1/D(x, x ′, t). Similarly, we assumeD(m)(x, x ′, t)[ϕ] is invertible on Ω(m) in the sense that there exists M (m)(x, x ′, t)[ϕ]on Ω(m), with the symmetry property in Eq. (3.183) and the condition:∫

dx ′′∑c

[M (m)]ac(x, x ′′, t)[ϕ] [D(m)]cb(x ′′, x ′, t)[ϕ] = δabδ(x− x ′), (3.187)

where the indexes a, b, c are restricted to the vector indexes of the space R(m) ⊆Rℓ. We also define M (m)(x, x ′, t)[ϕ] ≡

[D(m)(x, x ′, t)

]−1[ϕ]. In practice,

[D(x, x ′, t)]−1 [ϕ] and[D(m)(x, x ′, t)

]−1[ϕ] can be constructed by decomposing

them into their eigenvalues and eigenfunctions and then inverting the non-zeroeigenvalues while keeping the eigenfunctions unchanged. A rigorous mathemati-cal treatment of the (pseudo-)inverse of linear operators on the Hilbert space canbe found in Ref. [161].

3.3.3 Potential-Flux Field Landscape Framework for SpatiallyInhomogeneous Systems

The functional Fokker-Planck equation (Eq. (3.180)) can be interpreted as acontinuity equation on the field configuration space (Hilbert state space) [50]:

∂tPt[ϕ] = −∫

dx δϕ(x) · Jt(x)[ϕ], (3.188)

128

where the transient probability flux field is given by:

Jt(x)[ϕ] = F ′(x, t)[ϕ]Pt[ϕ]−∫

dx ′D(x, x ′, t)[ϕ] · δϕ(x ′)Pt[ϕ], (3.189)

with the effective drift vector field defined as:

F ′(x, t)[ϕ] = F (x, t)[ϕ]−∫

dx ′δϕ(x ′) ·D(x, x ′, t)[ϕ]. (3.190)

Equation (3.189) can be reformulated into the transient field dynamical decompo-sition equation:

F ′(x, t)[ϕ] = −∫

dx ′D(x, x ′, t)[ϕ] · δϕ(x ′)S[ϕ] + Vt(x)[ϕ], (3.191)

where the transient potential field landscape is S[ϕ] = − lnPt[ϕ] and the transientflux velocity field is Vt(x)[ϕ] = Jt(x)[ϕ]/Pt[ϕ]. The (instantaneous) stationaryprobability functional Ps[ϕ] satisfies the stationary functional Fokker-Planck e-quation:

∫dx δϕ(x) · Js(x)[ϕ] = 0. We assume Ps[ϕ] at each moment is unique (up

to a multiplication constant explained later) and positive. Accordingly, we havethe stationary field dynamical decomposition equation:

F ′(x, t)[ϕ] = −∫

dx ′D(x, x ′, t)[ϕ] · δϕ(x ′)U [ϕ] + Vs(x)[ϕ], (3.192)

where the stationary potential field landscape is U [ϕ] = − lnPs[ϕ] and the sta-tionary flux velocity field is Vs(x)[ϕ] = Js(x)[ϕ]/Ps[ϕ]. From Eqs. (3.191) and(3.192) we can also derive the relative field dynamical constraint equation:

Vr(x)[ϕ] = −∫

dx ′D(x, x ′, t)[ϕ] · δϕ(x ′)A[ϕ], (3.193)

where the relative potential field landscape is defined as A[ϕ] = U [ϕ] − S[ϕ] =

ln(Pt[ϕ]/Ps[ϕ]) and the relative flux velocity field is Vr(x)[ϕ] = Vt(x)[ϕ]−Vs(x)[ϕ].The flux field decomposition equation then reads:

Vt(x)[ϕ] = Vs(x)[ϕ] + Vr(x)[ϕ]. (3.194)

The probability flux field can also be defined for each individual mechanism:

J(m)t (x)[ϕ] = F ′(m)(x, t)[ϕ]Pt[ϕ]

−∫

dx ′D(m)(x, x ′, t)[ϕ] · δϕ(x ′)Pt[ϕ], (3.195)

129

where the effective drift vector field of mechanism m is given by:

F ′(m)(x, t)[ϕ] = F (m)(x, t)[ϕ]−∫

dx ′δϕ(x ′) ·D(m)(x, x ′, t)[ϕ].(3.196)

Accordingly, Eqs. (3.191)-(3.193) also have their counterparts for each mechanis-m:

F ′(m)(x, t)[ϕ] = −∫

dx ′D(m)(x, x ′, t)[ϕ] · δϕ(x ′)U [ϕ]

+V (m)s (x)[ϕ], (3.197)

F ′(m)(x, t)[ϕ] = −∫

dx ′D(m)(x, x ′, t)[ϕ] · δϕ(x ′)S[ϕ]

+V(m)t (x)[ϕ], (3.198)

V (m)r (x)[ϕ] = −

∫dx ′D(m)(x, x ′, t)[ϕ] · δϕ(x ′)A[ϕ]. (3.199)

The sum over the index m gives the combined collective quantity:∑

m J(m)t = Jt,∑

m F ′(m) = F ′,∑

m V(m)t = Vt,

∑m V

(m)s = Vs,

∑m V

(m)r = Vr. The diffusion

matrix field in (3.197)-(3.199) can be inverted using Eq. (3.187) to produce analternative form of these equations:

Π(m) · δϕ(x)U [ϕ] =

∫dx ′ [D(m)(x, x ′, t)

]−1[ϕ] · V (m)

s (x ′)[ϕ]

−∫

dx ′ [D(m)(x, x ′, t)]−1

[ϕ] · F ′(m)(x ′, t)[ϕ],

Π(m) · δϕ(x)S[ϕ] =

∫dx ′ [D(m)(x, x ′, t)

]−1[ϕ] · V (m)

t (x ′)[ϕ]

−∫

dx ′ [D(m)(x, x ′, t)]−1

[ϕ] · F ′(m)(x ′, t)[ϕ],

Π(m) · δϕ(x)A[ϕ] = −∫

dx ′ [D(m)(x, x ′, t)]−1

[ϕ] · V (m)r (x ′)[ϕ],(3.200)

where Π(m) is the projection operator onto space R(m) and[D(m)(x, x ′, t)

]−1[ϕ]

is the inverse of D(m)(x, x ′, t)[ϕ] defined in Eq. (3.187). Besides, we also havethe flux field decomposition equation for each mechanism:

V(m)t (x)[ϕ] = V (m)

s (x)[ϕ] + V (m)r (x)[ϕ]. (3.201)

130

We discuss some particularities of spatially inhomogeneous systems. Thefunctional Fokker-Planck equation, as a continuity equation in Eq. (3.188), de-scribes information (probability) transport process in the field configuration s-pace, rather than matter or energy transport in the physical space. For spatiallyinhomogeneous systems, this distinction has become drastic, as the field config-uration space is an infinite dimensional space, while the physical space is finitedimensional (usually assumed to be three dimensional). The various quantitiesintroduced, the potential field landscapes, U [ϕ], S[ϕ], A[ϕ], and the flux velocityfields, V (m)

s (x)[ϕ], V (m)t (x)[ϕ], V (m)

r (x)[ϕ], are all functionals of the field ϕ(x).They are all defined on the field configuration space, with each ‘element’, ‘point’or ‘vector’ in this space representing a global state of the system in the physicalspace (i.e., a field configuration). The (thermo)dynamical interpretations of thesequantities as ‘potentials’ and ‘forces’ thus should also be understood in the contextof information transport in the infinite dimensional field configuration space. Withthis distinction from spatially homogeneous systems taken into account, these var-ious quantities can be interpreted similarly as those introduced in Sec. 3.2. Thestationary, transient and relative potential field landscapes, U [ϕ], S[ϕ], A[ϕ], rep-resent, respectively, the microscopic stationary, transient and relative entropies ofthe system; they are also the potentials of the thermodynamic force fields gener-ating, respectively, the excess entropy flow, transient entropy change and nona-diabatic entropy production. The transient, stationary and relative flux velocityfields, V (m)

t (x)[ϕ], V (m)s (x)[ϕ], V (m)

r (x)[ϕ], are the thermodynamic force fields ofmechanism m generating the total, adiabatic and nonadiabatic entropy produc-tions, respectively.

3.3.4 Non-Equilibrium Thermodynamics of Spatially Inhomo-geneous Systems

In the thermodynamic context, the functional Langevin equation (Eq. (3.177))is interpreted as the dynamical equation of the microstate of the spatially inhomo-geneous system. The functional Fokker-Planck equation (Eq. (3.180)) is interpret-ed as the dynamical equation governing an ensemble of systems under the samemacroscopic external conditions. These macroscopic external conditions (i.e.,macrostates) can vary both in time and space, thus represented by a set of space-time functions (i.e., time-dependent fields) λ(x, t) ≡ λ1(x, t), ..., λi(x, t), ....For instance, an inhomogeneous environmental temperature distribution also chang-ing with time can be represented by T (x, t), which regulates the microscopic and

131

ensemble dynamics. The time dependence in F (x, t)[ϕ] and D(x, t)[ϕ] and thatwhich can be traced back to them can be replaced by the functional dependence onλ(x, t) (e.g., F (x)[ϕ, λ], D(x, x ′)[ϕ, λ]). To simplify notations, however,we still use an explicit time dependence t instead of λ.

The cross, transient and relative entropies of the spatially inhomogeneous sys-tem are defined, respectively, as the averages of the microstate functional U , Sand A over the transient ensemble:

U = ⟨U [ϕ]⟩ = −∫

Pt[ϕ] lnPs[ϕ]D[ϕ], (3.202)

S = ⟨S[ϕ]⟩ = −∫

Pt[ϕ] lnPt[ϕ]D[ϕ], (3.203)

A = ⟨A[ϕ]⟩ =∫

Pt[ϕ] ln (Pt[ϕ]/Ps[ϕ])D[ϕ], (3.204)

where the integrals are functional integrals taken over the field configuration s-pace (Hilbert state space) [45–47, 158]. We stress that the definitions of thesenon-equilibrium entropies are not expressed as integrals over the physical space,in contrast with classical irreversible thermodynamics based on the local equi-librium assumption which requires entropy to be locally defined in the physicalspace [94]. Here, in Eqs. (3.202)-(3.204), ‘local’ is expressed in the field configu-ration space, within which each local point represents a global state in the physicalspace. Therefore, we are not using a local description in the physical space; theglobal description in the physical space is disguised in the local form in the fieldconfiguration space. It is important to understand this point.

Then we calculate time derivatives of Eqs. (3.202)-(3.204) to derive their re-spective entropy balance equations and the expressions of thermodynamic quan-tities. For cross entropy we have:

U =d

dt

∫Pt[ϕ]U [ϕ]D[ϕ]

=

∫Pt[ϕ]∂tU [ϕ]D[ϕ] +

∫(∂tPt[ϕ])U [ϕ]D[ϕ]

= ⟨∂tU⟩+⟨∫

dxVt(x) · δϕ(x)U⟩, (3.205)

132

where we have used∫(∂tPt[ϕ])U [ϕ]D[ϕ] = −

∫ [∫dx δϕ(x) · Jt(x)[ϕ]

]U [ϕ]D[ϕ]

=

∫ [∫dxJt(x)[ϕ] · δϕ(x)U [ϕ]

]D[ϕ] =

⟨∫dxVt(x) · δϕ(x)U

⟩(3.206)

which is a result of the continuity equation, integration by parts and the definitionJt = PtVt. We have also suppressed the functional dependence [ϕ] in the ensembleaverage to simplify notations, but it is better to keep in mind it is still there. Thefirst term in the last equation of Eq. (3.205) is due to the time dependence in thestationary distribution functional Ps[ϕ], which comes from λ(x, t) representingspace-time dependent external conditions. Thus the first term is identified as theexternal driving entropic power for spatially inhomogeneous systems:


∫dx⟨δλi(x)U

⟩[∂tλi(x)], (3.207)

where we have used the chain rule for functional derivatives [50,158] and δλi(x)Uis the short notation for δU/δλi(x). The second term (with a negative sign) inEq. (3.205) is identified as the rate of excess entropy flow from the system to theenvironment for spatially inhomogeneous systems:

Sex = −⟨∫

dxVt(x) · δϕ(x)U⟩

= −∑m

⟨∫dxV

(m)t (x) · δϕ(x)U

⟩, (3.208)

where the summand of the last equation with the negative sign before it can bedefined as S(m)

ex , the excess entropy flow rate of mechanism m. Thus we have thefollowing cross entropy balance equation for spatially inhomogeneous systems:

U = Sed − Sex. (3.209)

The time derivative of the transient entropy is given by:

S = −∫

(∂tPt[ϕ]) lnPt[ϕ]D[ϕ] =

⟨∫dxVt(x) · δϕ(x)S

⟩, (3.210)

where we have used results similar to Eq. (3.206). Using Eq. (3.200) for S, we

133

further have:

S =∑m

⟨∫dxV

(m)t (x) · δϕ(x)S

⟩=∑m

⟨∫dxV

(m)t (x) ·Π(m) · δϕ(x)S

⟩=

∑m

⟨∫∫dxdx ′V

(m)t (x) ·

[D(m)(x, x ′, t)

]−1 · V (m)t (x ′)

⟩−

∑m

⟨∫∫dxdx ′V

(m)t (x) ·

[D(m)(x, x ′, t)

]−1 · F ′(m)(x ′, t)

⟩. (3.211)

The first term of the last equation is always nonnegative. It is identified as theentropy production rate for the spatially inhomogeneous system:

Spd =∑m

⟨∫∫dxdx ′V

(m)t (x) ·

[D(m)(x, x ′, t)

]−1 · V (m)t (x ′)

⟩, (3.212)

where the summand is defined as S(m)pd , the entropy production rate of mechanis-

m m. Since[D(m)(x, x ′, t)

]−1[ϕ] is positive definite and Pt[ϕ] is assumed to be

positive, the necessary and sufficient condition for Spd ≡ 0 is V(m)t (x)[ϕ] ≡ 0

(m = 1, 2, ...), that is, the transient flux velocity field of each mechanism is iden-tically 0. The second term is identified as the entropy flow rate (from the systemto the environment) for spatially inhomogeneous systems:

Sfl =∑m

⟨∫∫dxdx ′V

(m)t (x) ·

[D(m)(x, x ′, t)

]−1 · F ′(m)(x ′, t)

⟩, (3.213)

with the summand defined as S(m)fl , the entropy flow rate of mechanism m. Thus

we have the transient entropy balance equation for spatially inhomogeneous sys-tems:

S = Spd − Sfl. (3.214)

The time derivative of the relative entropy is given by:

A = U − S = ⟨∂tU⟩+⟨∫

dxVt(x) · δϕ(x)A⟩, (3.215)

where we have used Eqs. (3.205) and (3.210) and the definition A = U − S. Thefirst term in the last equation of Eq. (3.215) is the external driving entropic powerSed. The second term with a negative sign is identified as the nonadiabatic entropy

134

production rate, which has multiple alternative expressions, several of which areexplicitly nonnegative:

Sna =

⟨∫∫dxdx ′Vt(x) · [D(x, x ′, t)]

−1 · Vr(x′)

⟩=

∑m

⟨∫∫dxdx ′V

(m)t (x) ·

[D(m)(x, x ′, t)

]−1 · V (m)r (x ′)

⟩= −

⟨∫dxVt(x) · δϕ(x)A

⟩= −

∑m

⟨∫dxV

(m)t (x) · δϕ(x)A

⟩=

⟨∫∫dxdx ′Vr(x) · [D(x, x ′, t)]

−1 · Vr(x′)

⟩=

∑m

⟨∫∫dxdx ′V (m)

r (x) ·[D(m)(x, x ′, t)

]−1 · V (m)r (x ′)

⟩= −

⟨∫dxVr(x) · δϕ(x)A

⟩= −

∑m

⟨∫dxV (m)

r (x) · δϕ(x)A⟩

=

⟨∫∫dxdx ′

[δϕ(x)A

]·D(x, x ′, t) ·

[δϕ(x ′)A

]⟩=

∑m

⟨∫∫dxdx ′

[δϕ(x)A

]·D(m)(x, x ′, t) ·

[δϕ(x ′)A

]⟩. (3.216)

These expressions can be obtained using the property⟨∫

dxVs(x) · δϕ(x)A⟩= 0

(due to the stationary functional Fokker-Planck equation) and Eq. (3.200) for A.The summands in the 6th, 8th and 10th expressions in Eq. (3.216), which areequal to each other and non-negative, can be defined as S(m)

na , the nonadiabaticentropy production rate of mechanism m. The necessary and sufficient conditionfor Sna ≡ 0 is Vr(x)[ϕ] ≡ 0, which is also equivalent to V

(m)r (x)[ϕ] ≡ 0 for all

m. Then Eq. (3.215) becomes the relative entropy balance equation for spatiallyinhomogeneous systems:

A = Sed − Sna. (3.217)

From Sfl in Eq. (3.213), Sex in Eq. (3.208) and the entropy flow decompositionequation:

Sfl = Sad + Sex, (3.218)

the adiabatic entropy production rate Sad for spatially inhomogeneous systems is

135

given by:

Sad =∑m

⟨∫∫dxdx ′V

(m)t (x) ·

[D(m)(x, x ′, t)

]−1 · V (m)s (x ′)

⟩=

∑m


s (x) ·[D(m)(x, x ′, t)

]−1 · V (m)s (x ′)

⟩(3.219)

where in deriving the second line we have used Eq. (3.200) for A and the property⟨∫dxVs(x) · δϕ(x)A

⟩= 0. The second expression of Sad in Eq. (3.219) is man-

ifestly nonnegative. Its summand, which is also nonnegative, can be defined asS(m)ad , the adiabatic entropy production rate of mechanism m. The necessary and

sufficient condition for Sad ≡ 0 is V (m)s (x)[ϕ] ≡ 0 for all m.

Then we can also verify that the following entropy production decompositionequation holds for spatially inhomogeneous systems:

Spd = Sad + Sna, (3.220)

where the adiabatic entropy production rate Sad and nonadiabatic entropy produc-tion rate Sna are individually nonnegative; together they ensure the total entropyproduction rate Spd is also nonnegative. The adiabatic entropy production Sad

is generated by non-zero stationary flux velocity field V(m)s (x)[ϕ] (m = 1, 2, ...)

which characterizes detailed balance breaking in the steady state of spatially inho-mogeneous systems by mechanism m, while the nonadiabatic entropy productionSna is generated by non-zero relative flux velocity field V

(m)r (x)[ϕ] (m = 1, 2, ...)

which characterizes the irreversible relaxational process from the transient stateto the steady state of spatially inhomogeneous systems contributed by mechanismm. These two forms of non-equilibrium irreversibility together constitute the totalirreversibility generated by the spatially inhomogeneous system indicated by non-zero total entropy production Spd, which is characterized by non-zero transientflux velocity field V

(m)t (x)[ϕ] (m = 1, 2, ...).

In accord with the mixing entropy production rate for mechanism m intro-duced in Eq. (3.103), we can define its counterpart for spatially inhomogeneoussystems:

S(m)mix =


r (x) ·[D(m)(x, x ′, t)

]−1 · V (m)s (x ′)

⟩=

⟨∫dxV (m)

s (x) · δϕ(x)A⟩, (3.221)

136

which has the property∑

m S(m)mix = 0. Then we also have the decomposition

equations of the entropy production and entropy flow for each mechanism m forspatially inhomogeneous systems, which has the same form of Eqs. (3.105) and(3.106) for spatially homogeneous systems:

S(m)fl = S(m)

ad + S(m)ex + S(m)

mix. (3.222)

S(m)pd = S(m)

ad + S(m)na + 2S(m)

mix, (3.223)

When summed over m, S(m)mix does not appear in Eqs. (3.218) and (3.220).

Therefore, for spatially inhomogeneous systems described by functional Langevinand Fokker-Planck equations with multiple state transition mechanisms, we havealso constructed the non-equilibrium thermodynamic equations and the expres-sions of various thermodynamic functions in these equations, given in (3.205)-(3.223). These equations are generalizations of those for spatially homogeneoussystems given in Eqs. (3.88)-(3.106). For completeness and simplicity, in the fol-lowing we also list the set of non-equilibrium thermodynamic equations and theexpressions of the thermodynamic quantities for spatially inhomogeneous system-s with one component in one dimensional physical space and with only one statetransition mechanism:

U = Sed − Sex

S = Spd − Sfl

A = Sed − Sna

Spd = Sad + Sna

Sfl = Sad + Sex

(3.224)

with the definitions U = ⟨− lnPs⟩, S = ⟨− lnPt⟩, A = ⟨ln(Pt/Ps)⟩ and theexpressions:

U = ⟨∂tU⟩+⟨∫

dxVt(x)δϕ(x)U

⟩(3.225)

S =

⟨∫dxVt(x)δϕ(x)S

⟩(3.226)

A = ⟨∂tU⟩+⟨∫

dxVt(x)δϕ(x)A

⟩(3.227)

137


∫dx⟨δλi(x)U

⟩[∂tλi(x)] (3.228)

Sex = −⟨∫

dxVt(x)δϕ(x)U

⟩(3.229)

Spd =

⟨∫∫dxdx′Vt(x)[D(x, x′, t)]−1Vt(x

′)

⟩(3.230)

Sfl =

⟨∫∫dxdx′Vt(x)[D(x, x′, t)]−1F ′(x′, t)

⟩(3.231)

Sad =

⟨∫∫dxdx′Vt(x)[D(x, x′, t)]−1Vs(x

′)

⟩=

⟨∫∫dxdx′Vs(x)[D(x, x′, t)]−1Vs(x

′)

⟩(3.232)

Sna =

⟨∫∫dxdx′Vt(x)[D(x, x′, t)]−1Vr(x

′)

⟩= −

⟨∫dxVt(x)δϕ(x)A

⟩=

⟨∫∫dxdx′Vr(x)[D(x, x′, t)]−1Vr(x

′)

⟩= −

⟨∫dxVr(x)δϕ(x)A

⟩=

⟨∫∫dxdx′ [δϕ(x)A]D(x, x′, t)

[δϕ(x′)A

]⟩(3.233)

On the one hand, Eqs. (3.224)-(3.233) for spatially inhomogeneous systems withone component in one dimension with one mechanism are generalizations of E-qs. (3.62)-(3.71) for spatially homogeneous systems with one mechanism. On theother hand, they are special cases of Eqs. (3.205)-(3.223) for spatially inhomo-geneous systems with multiple components in multiple dimensions with multiplemechanisms. The equations for cases in between (e.g., spatially inhomogeneoussystems with multiple components in multiple dimensions with one mechanism)can be easily deduced from Eqs. (3.205)-(3.223) or extended from Eqs. (3.224)-(3.233).

We discuss some particular points in these results. The ensemble average inEqs. (3.225)-(3.233), when spelled out, is a functional integral in the field config-uration space. If we assume the order of the functional integral and the spatialintegral can be interchanged, they can be expressed as spatial integrals. For ex-ample, Eq. (3.230) can be written as Spd =

∫∫dxdx′Spd(x, x

′, t), with

Spd(x, x′, t) =

∫D[ϕ]Pt[ϕ]Vt(x)[ϕ][D(x, x′, t)]−1[ϕ]Vt(x

′)[ϕ], (3.234)

138

which may be termed the entropy production (spatial) correlation function. Thetotal entropy production rate Spd is thus expressed as a double spatial integral ofthe entropy production correlation function. In the special case when fluctuationsare spatially uncorrelated, i.e., D(x, x′, t) ∝ δ(x− x′), entropy production is alsospatially uncorrelated according to Eq. (3.234), which has the form Spd(x, x

′, t) =Spd(x, t)δ(x − x′). Thus Spd =

∫dxSpd(x, t), where Spd(x, t) can be identified

as the local entropy production rate in the physical space. In general cases, onemay play the following trick, Spd =

∫dx[∫

dx′Spd(x, x′, t)]≡∫dxSpd(x, t),

with Spd(x, t) being the effective local entropy production rate. Yet this is doneat the cost of coarse-graining the detailed information of spatial correlations ofentropy production. Moreover, according to Eq. (3.234), the entropy productioncorrelation function Spd(x, x

′, t), thus also the effective local entropy productionrate Spd(x, t), is expressed as a functional integral in the field configuration space.This means they are global quantities in the field configuration space. (Thus ‘lo-cal’ in the physical space does not imply ‘local’ in the field configuration spaceeither.) The treatment in classical irreversible thermodynamics based on the lo-cal equilibrium assumption [94] is different, as the total entropy production rateis generically expressed as a single spatial integral of the local entropy produc-tion rate, without considering the spatial correlation of entropy production and itsunderlying statistical origin. These distinctions from classical irreversible ther-modynamics imply that our treatment and results are not limited by the local e-quilibrium assumption. Further, we note that from Eqs. (3.225)-(3.233), the totalentropy production rate Spd, the adiabatic entropy production rate Sad and the to-tal entropy flow rate Sfl have the form of a double spatial integral of correlationfunctions, while the rest can be expressed as a single spatial integral of local den-sities in the physical space, determined by a functional integral (thus global) inthe field configuration space.

The non-equilibrium thermodynamic formalism for spatially inhomogeneoussystems developed so far from Sec. 3.3.2 to Sec. 3.3.4 are presented in the spaceconfiguration representation. In Appendix D we present the major results in thesesections in the abstract representation and explain how to switch between differentrepresentations; we also mention how to recover results of spatially homogeneoussystems from those of spatially inhomogeneous systems.

139

3.3.5 Ornstein-Uhlenbeck Processes of Spatially InhomogeneousSystems

We apply the formalism developed so far for general spatially inhomogeneoussystems to more specific systems that are analytically tractable (to a certain exten-t). For spatially homogeneous systems, we have studied the Ornstein-Uhlenbeckprocess (OU process). Here we investigate its counterpart for spatially inhomoge-neous systems. To fully utilize the available results of the OU process for spatiallyhomogeneous systems, we proceed as follows. First we discretize the physical s-pace into spatial cells and work in the discrete space representation, whereby themethods and results for the OU process of spatially homogeneous systems can beported in. Then we take the continuum limit and return to the continuous spacerepresentation. We also present the expressions of the thermodynamic quantitiesfor one state transition mechanism.

Discrete Space Representation

We study the stochastic dynamics and potential-flux field landscape of OUprocesses for spatially inhomogeneous systems in the discrete space representa-tion. The state of the system is described by the set of variables ϕa

λ defined inEq. (3.167), which we assume follows an OU process. In the Langevin dynamics,this means the deterministic force is linear in the state variables ϕa

λ, while thestochastic force is independent of the state variables ϕa

λ. Thus the Langevinequation has the following form (see Eq. (3.107) for spatially homogeneous sys-tems):

d

dtϕaλ = −γab

λµϕbµ + ξaλ(t), (3.235)

where we have used Einstein summation convention with repeated indexes in aterm summed over (e.g., both b and µ are summed over in the above equation).The Gaussian white noise ξaλ(t) has the following statistical property:

⟨ξaλ(t)⟩ = 0, ⟨ξaλ(t)ξbµ(t′)⟩ = 2Dabλµδ(t− t′). (3.236)

γabλµ and Dab

λµ do not depend on ϕaλ, but they can be time-dependent (e.g., through

depending on external control parameters). We also require [γabλµ] to be invertible

and [Dabλµ] to be symmetric and positive definite, when seen as a matrix with a

double row index a λ and a double column index b µ. The corresponding Fokker-

140

Planck equation, as the counterpart of Eq. (3.109), is then:

∂

∂tPt(ϕ) = γab

λµ

∂

∂ϕaλ

(ϕbµPt(ϕ)

)+Dab

λµ

∂2

∂ϕaλ∂ϕ

bµ

Pt(ϕ), (3.237)

where we have used the short notation ϕ for the collection of state variables.We consider the Gaussian solutions of the above Fokker-Planck equation:

Pt(ϕ) =1√

det([2πσabλµ])

exp

−1

2(ϕa

λ −Kaλ)[σ−1]abλµ

(ϕbµ −Kb

µ)

, (3.238)

where [2πσabλµ] is the matrix whose entries are given by 2πσab

λµ. The means Kaλ and

the covariances σabλµ are determined by the following equations (see Eqs. (3.111)

and (3.112)):

Kaλ = −γab

λµKbµ,

σabλµ = −γac

λνσcbνµ − γbc

µνσcaνλ + 2Dab

λµ, (3.239)

The instantaneous stationary distribution of Eq. (3.237) is a Gaussian distributionwith mean zero:

Ps(ϕ) =1√

det([2πσabλµ])

exp

−1

2ϕaλ

[σ−1]abλµ

ϕbµ

, (3.240)

where the instantaneous stationary covariance matrix σ is the instantaneous sta-tionary solution of Eq. (3.239), satisfying:

γacλν σ

cbνµ + γbc

µν σcaνλ = 2Dab

λµ. (3.241)

With the transient and stationary distributions given by Eqs. (3.238) and (3.240),the expressions of the stationary, transient and relative potential landscapes canbe obtained using the definitions U = − lnPs, S = − lnPt and A = U − S (seeEqs. (3.115)-(3.117)):

U =1

2ϕaλ

[σ−1]abλµ

ϕbµ +

1

2tr(ln[2πσab

λµ])

(3.242)

S =1

2(ϕa


(ϕbµ −Kb

µ) +1

2tr(ln[2πσab

λµ])

(3.243)

A =1

2ϕaλ

[σ−1]abλµ

ϕbµ −

1

2(ϕa


(ϕbµ −Kb

µ)

+1

2tr(ln[σab

λµ]− ln[σabλµ])

(3.244)

141

Their gradients are given by (see Eqs. (3.118)-(3.120)):

∂U

∂ϕaλ

=[σ−1]abλµ

ϕbµ (3.245)

∂S

∂ϕaλ

=[σ−1]abλµ

(ϕbµ −Kb

µ) (3.246)

∂A

∂ϕaλ

=[σ−1]abλµ

ϕbµ −

[σ−1]abλµ

(ϕbµ −Kb

µ) (3.247)

We can further derive the stationary, transient and relative flux velocities for thesystem (see Eqs. (3.121)-(3.123)) using the dynamical decomposition equations:

(Vs)aλ = F ′a

λ +Dabλµ

∂U

∂ϕbµ

= −γabλµϕ

bµ +Dab

λµ

[σ−1]bcµν

ϕcν (3.248)

(Vt)aλ = F ′a

λ +Dabλµ

∂S

∂ϕbµ

= −γabλµϕ

bµ +Dab

λµ

[σ−1]bcµν

(ϕcν −Kc

ν) (3.249)

(Vr)aλ = −Dab

λµ

∂A

∂ϕbµ

= Dabλµ

[σ−1]bcµν

(ϕcν −Kc

ν)−Dabλµ

[σ−1]bcµν

ϕcν(3.250)

Continuous Space Representation

We transform the above results in the discrete space representation into thecontinuous space representation by taking the continuum limit. There is a direc-t correspondence between the notations in the discrete space representation andthose in the continuous space representation. A brief dictionary to translate nota-tions in one representation to the other is given below [50, 51, 158]:

ϕaλ ⇐⇒ ϕa(x), P (ϕ) ⇐⇒ P [ϕ],

F aλ (ϕ) ⇐⇒ F a(x)[ϕ], Dab

λµ(ϕ) ⇐⇒ Dab(x, x ′)[ϕ],∑λ ⇐⇒

∫ dx

∆V, δλµ ⇐⇒ ∆V δ(x− x ′),

∂

∂ϕaλ

⇐⇒ ∆Vδ

δϕa(x),

∏aλ

∫dϕa

λ ⇐⇒∫D[ϕ].

(3.251)

We first consider the continuum limit of the deterministic force in Eq. (3.235),F aλ (ϕ) = −

∑µ

∑b γ

abλµϕ

bµ, where we have spelled out the sum. In the continu-

um limit, F aλ (ϕ) → F a(x)[ϕ], ϕb

µ → ϕb(x ′) and∑

µ →∫dx ′/∆V . Plugging

them in, we have F a(x)[ϕ] = −∫dx ′∑

b(γabλµ/∆V )ϕb(x ′). To get rid of the

142

volume element ∆V , we must have γabλµ → γab(x, x ′)∆V in the continuum lim-

it. Thus the deterministic force of the OU process for spatially inhomogeneoussystems has the form:

F a(x)[ϕ] = −∫

dx ′∑b

γab(x, x ′)ϕb(x ′). (3.252)

This is an integral transform of (or an integral operator acting on) the vectorfield ϕ(x) in the Hilbert state space, returning another vector field F (x) in theHilbert state space, with the ab entry of the matrix-valued integral kernel given by−γab(x, x ′). If the kernel function is allowed to be generalized functions (e.g., theDirac delta function δ(x− x ′) and its derivatives), which we assume so, then theintegral operator in Eq. (3.252) can also represent differential operators [15, 162].In the most general sense, Eq. (3.252) is simply a statement that the deterministicforce field F (x)[ϕ] is a linear map from the Hilbert state space into the Hilbert s-tate space. Also, in the continuum limit the Gaussian white-noise stochastic forceξaλ(t) in Eq. (3.235) becomes ξa(x, t).

Thus in the continuum limit Eq. (3.235) becomes a functional Langevin equa-tion, describing the OU process of spatially inhomogeneous systems :

∂

∂tϕa(x, t) = −

∫dx ′

∑b

γab(x, x ′)ϕb(x ′, t) + ξa(x, t), (3.253)

where ξa(x, t) has the following statistical property as the continuum limit of E-q. (3.236):

⟨ξa(x, t)⟩ = 0, ⟨ξa(x, t)ξb(x ′, t′)⟩ = 2Dab(x, x ′)δ(t− t′). (3.254)

Equation (3.253) is the white-noise form of the functional Langevin equation; itcan also be reformulated into the more rigorous form in Eq. (3.176) [43]. Wenote that this functional Langevin equation has an important subclass, name-ly linear stochastic partial differential equations with additive noise. γab(x, x ′)

and Dab(x, x ′) do not depend on the state of the system ϕ(x), but they are al-lowed to be time-dependent. We require them to be invertible in the sense ofEq. (3.186) when seen as matrices with the row index a, x and the column indexb, x ′. Dab(x, x ′) is symmetric in the sense that Dab(x, x ′) = Dba(x ′, x) and posi-tive definite as defined in Eq. (3.184). The condition for the existence and unique-ness of the solution of Eq. (3.253) and even more general functional Langevinequations on Hilbert spaces (Eq. (3.176)) has already been studied [43, 163].

143

The corresponding functional Fokker-Planck equation is the continuum limitof Eq. (3.237):

∂

∂tPt[ϕ] =

∫∫dxdx ′

∑ab

γab(x, x ′)δ

δϕa(x)

(ϕb(x ′)Pt[ϕ]

)+

∫∫dxdx ′

∑ab

Dab(x, x ′)δ2

δϕa(x)δϕb(x ′)Pt[ϕ]. (3.255)

Accordingly, the Gaussian solution in Eq. (3.238) becomes a Gaussian distributionfunctional:

Pt[ϕ] =1√

det([2πσab(x, x ′)])exp

−1

2

∫∫dxdx ′

∑ab

[ϕa(x)−Ka(x)]

×[σ−1]ab

(x, x ′)[ϕb(x ′)−Kb(x ′)

]. (3.256)

[σ−1]ab(x, x ′) as the inverse of σab(x, x ′) is defined in the sense of Eq. (3.186).In practice, it can be found by inverting the eigenvalues of σab(x, x ′). The deter-minant in the denominator det([2πσab(x, x ′)]) is a functional determinant [158],defined on an infinite dimensional Hilbert space. It can be formally defined asthe product of all the eigenvalues of [2πσab(x, x ′)]. Yet the problem is that thefunctional determinant may be divergent or approach zero, which also makes theprobability distribution functional in Eq. (3.256) not well-defined. This is also asituation often encountered in quantum field theory; the solution often proposed isto demand that only the ratio of two functional determinants are meaningful [158].Here we also adopt this solution. This also means only the ratio of two Gaussianprobability distribution functionals are meaningful; alternatively, these probabili-ty distribution functionals are only defined up to a multiplicative constant. A morerigorous yet also more technically demanding treatment is to resort to the Gaussianmeasure theory on an infinite dimensional Hilbert space [42]. The mean Ka(x)and covariance σab(x, x ′) of the transient Gaussian distribution in Eq. (3.256) aredetermined by the following equations as the continuum limit of Eq. (3.239):

Ka(x) = −∫

dx ′∑b

γab(x, x ′)Kb(x ′), (3.257)

σab(x, x ′) = −∫

dx ′′∑c

[γac(x, x ′′)σcb(x ′′, x ′) + γbc(x ′, x ′′)σca(x ′′, x)

]+2Dab(x, x ′). (3.258)

144

We assume that with the initial conditions of Ka(x) and σab(x, x ′) given, the solu-tions of these two equations exist and are unique when γab(x, x ′) and Dab(x, x ′)satisfy certain conditions [43, 163]. The instantaneous stationary distribution inthe continuous space representation reads:

Ps[ϕ] =

exp

−1

2

∫∫dxdx ′∑

ab ϕa(x) [σ−1]

ab(x, x ′)ϕb(x ′)

√det([2πσab(x, x ′)])

, (3.259)

where the instantaneous stationary covariance σab(x, x ′) satisfies the followingequation:∫

dx ′′∑c

[γac(x, x ′′)σcb(x ′′, x ′) + γbc(x ′, x ′′)σca(x ′′, x)

]= 2Dab(x, x ′).

(3.260)These two equations are the continuous limit of Eqs. (3.240) and (3.241). Weassume the solution σcb(x, x ′) of Eq. (3.260) is unique when γab(x, x ′) andDab(x, x ′) satisfy appropriate conditions [43, 163].

The potential field landscapes U = − lnPs, S = − lnPt and A = ln (Pt/Ps)are then given by

U [ϕ] =1

2

∫∫dxdx ′

∑ab

ϕa(x)[σ−1]ab

(x, x ′)ϕb(x ′)

+1

2tr(ln[2πσab(x, x ′)]

)(3.261)

S[ϕ] =1

2

∫∫dxdx ′

∑ab

[ϕa(x)−Ka(x)][σ−1]ab

(x, x ′)[ϕb(x ′)−Kb(x ′)

]+

1

2tr(ln[2πσab(x, x ′)]

)(3.262)

A[ϕ] =1

2

∫∫dxdx ′

∑ab

ϕa(x)[σ−1]ab

(x, x ′)ϕb(x ′)

− 1

2

∫∫dxdx ′

∑ab

[ϕa(x)−Ka(x)][σ−1]ab

(x, x ′)[ϕb(x ′)−Kb(x ′)

]+

1

2tr(ln[σab(x, x ′)]− ln[σab(x, x ′)]

). (3.263)

U and S both contain a term coming from a functional determinant, given thatln det(B) = tr(lnB). Hence they may be divergent. Yet since U and S are only

145

defined up to a common additive constant, the divergence may be removed. Thelast term in A comes from a ratio between two functional determinants, which maycancel the divergence. Their functional derivatives, however, are all independentof the functional determinants:

δU

δϕa(x)=

∫dx ′

∑b

[σ−1]ab

(x, x ′)ϕb(x ′) (3.264)

δS

δϕa(x)=

∫dx ′

∑b

[σ−1]ab

(x, x ′)[ϕb(x ′)−Kb(x ′)

](3.265)

δA

δϕa(x)=

∫dx ′

∑b

[σ−1]ab

(x, x ′)ϕb(x ′)

−∫

dx ′∑b

[σ−1]ab

(x, x ′)[ϕb(x ′)−Kb(x ′)

](3.266)

The expressions of the stationary, transient and relative flux velocity fields can beobtained from the dynamical decomposition equations in Eqs. (3.191)-(3.193):

V as (x)[ϕ] = −

∫dx ′

∑b

γab(x, x ′)ϕb(x ′)

+

∫∫dx ′dx ′′

∑bc

Dab(x, x ′)[σ−1]bc

(x ′, x ′′)ϕc(x ′′) (3.267)

V at (x)[ϕ] = −

∫dx ′

∑b

γab(x, x ′)ϕb(x ′) +

∫∫dx ′dx ′′

∑bc

Dab(x, x ′)

×[σ−1]bc

(x ′, x ′′)(ϕc(x ′′)−Kc(x ′′)) (3.268)

V ar (x)[ϕ] =

∫∫dx ′dx ′′

∑bc


(x ′, x ′′)(ϕc(x ′′)−Kc(x ′′))

−∫∫

dx ′dx ′′∑bc


(x ′, x ′′)ϕc(x ′′). (3.269)

Therefore, for OU processes of spatially inhomogeneous systems, we have de-rived the explicit expressions of the various terms in the dynamical decompo-sition equations (Eqs. (3.191)-(3.193)), with the (effective) force field given byEq. (3.252), the potential field landscapes given by Eq. (3.261)-(3.263), the func-tional gradient of the potential field landscapes given by Eqs. (3.264)-(3.266), andthe flux velocity fields given by Eqs. (3.267)-(3.269). These results can either be

146

obtained by taking the continuum limit of the results in the discrete space repre-sentation or by directly working in the continuous space representation using thefunctional language from the start. These explicit expressions can be used to studythe global stability and dynamics of spatially inhomogeneous OU processes in thepotential-flux landscape framework using the methods presented in chapter 2.

These equations and expressions in the continuous space representation canalso be written compactly in the abstract representation, which are given in Ap-pendix D. Here we mention a few. Equation (3.262) in the abstract representa-tion reads S(ϕ) = (ϕ − K|σ−1|ϕ − K)/2 + tr(ln(2πσ))/2, Eq. (3.265) reads|δϕS(ϕ)) = σ−1|ϕ−K) and Eq. (3.268) reads |Vt(ϕ)) = −γ|ϕ) + Dσ−1|ϕ−K).

Non-Equilibrium Thermodynamics for One State Transition Mechanism

We consider OU processes of spatially inhomogeneous system with one ef-fective state transition mechanism. The thermodynamic quantities in the non-equilibrium thermodynamic equations can be calculated using those for one mech-anism (extension of Eqs. (3.225)-(3.233) to multiple dimensional physical space),by plugging in the specific expressions of the potential field landscapes and theflux velocity fields for OU processes derived in (3.261)-(3.269). The results canalso be obtained by formally generalizing those already worked out for spatiallyhomogeneous systems in Eqs. (3.124)-(3.135). More specifically, if we replacethe matrices I , γ, D, σ, σ and µµ with their counterparts in spatially inhomoge-neous systems (linear operators on the Hilbert space), I , γ, D, ˆσ, σ and |K)(K|,then Eqs. (3.124)-(3.135) formally carry over to spatially inhomogeneous systems,as long as the operations involved of finite-dimensional matrices are also legiti-mate for these linear operators on the infinite dimensional Hilbert space. This,however, is not a trivial issue.

We first write down the formal results and then touch upon some technicalissues involved. The thermodynamic expressions in (3.124)-(3.135), when for-mally generalized to spatially inhomogeneous systems, are given by the followingexpressions in the abstract representation:

U = tr

(1

2ˆσ−1(σ + |K)(K|) + 1

2ln(2π ˆσ)

)(3.270)

S = tr

(1

2I +

1

2ln(2πσ)

)(3.271)

A = tr

(1

2

(ˆσ−1(σ + |K)(K|)− I

)+

1

2(ln ˆσ − ln σ)

)(3.272)

147

U = tr

(1

2(σ + |K)(K| − ˆσ)

d

dtˆσ−1

−ˆσ−1D ˆσ

−1(σ + |K)(K|) + γ

)(3.273)


)(3.274)

A = tr

(1

2(σ + |K)(K| − ˆσ)

d

dtˆσ−1

− ˆσ−1D ˆσ

−1(σ + |K)(K|)

−σ−1D + 2γ)

(3.275)

Sed = tr

(1

2

(σ + |K)(K| − ˆσ

) d

dtˆσ−1)

(3.276)

Sex = tr(ˆσ−1D ˆσ

−1(σ + |K)(K|)− γ

)(3.277)

Sfl = tr(γᵀD−1γ (σ + |K)(K|)− γ

)(3.278)

Spd = tr(γᵀD−1γ (σ + |K)(K|) + σ−1D − 2γ

)(3.279)

Sad = tr((

γᵀD−1γ − ˆσ−1D ˆσ

−1)(σ + |K)(K|)

)(3.280)

Sna = tr(ˆσ−1D ˆσ

−1(σ + |K)(K|) + σ−1D − 2γ

)(3.281)

It is easy to verify that Eqs. (3.273)-(3.281) formally satisfy the set of non-equilibriumthermodynamic equations (Eq. (3.224)). The expressions of the thermodynamicquantities in Eqs. (3.273)-(3.281) are also given in the space configuration rep-resentation in Appendix E, written in the explicit functional language. But thespace configuration representation is not practical for actual calculations. Thesethermodynamic quantities are most conveniently calculated by working in an or-thonormal basis of the Hilbert space, such as the eigenfunctions of the operator γ(if its eigenfunctions form an orthonormal basis), within which the linear opera-tors are represented by infinite-dimensional matrices and the calculations can bemuch simplified if they are diagonal.

Then we take a look at a technical issue in these expressions. On an infi-nite dimensional Hilbert space, not all linear operators have a well-defined trace(e.g., the trace of the identity operator I diverges); those that do have a trace arecalled trace-class operators [162]. Therefore, it is possible that the thermody-namic quantities in Eqs. (3.270)-(3.281) may diverge. At this moment they areonly formal expressions. For these expressions to be physically meaningful, theyhave to produce finite results themselves or we have to use some techniques such

148

as renormalization to remove the infinities. Shortly in Sec. 3.3.6 we shall studya specific example, within which the expressions in Eqs. (3.270)-(3.281) can beworked out explicitly to be checked whether they are divergent or not. We specu-late from that example that, in general, when the operators γ and D (they governthe dynamics of the system and determine K, σ and ˆσ) satisfy certain condition-s [43, 163], the thermodynamic quantities in Eqs. (3.272)-(3.281) produce finiteresults themselves, while the cross entropy U in Eq. (3.270) and the transient en-tropy S in Eq. (3.271) may still be divergent, but the divergence can be removedby a simple ‘renormalization’ considering they are only defined up to a commonadditive constant. We illustrate these ideas in the following example.

3.3.6 Spatial Stochastic Neuronal ModelWe study a spatial stochastic neuronal model described by the linear stochastic

cable equation [43, 54, 55], which governs the stochastic evolution of the mem-brane potential of a spatially extended neuron. In the context of chemical reac-tions, the same equation can describe the stochastic degradation-diffusion process.Mathematically, the linear stochastic cable equation is a particular case of the s-patially inhomogeneous OU process just studied in Sec. 3.3.5. Thus it can serveas a testing ground for the non-equilibrium thermodynamic framework we haveformulated. This equation has been solved previously from the perspective of astochastic partial differential equation (functional Langevin equation) [43]. Herewe solve it from the perspective of the functional Fokker-Planck equation. Moreimportantly, we use the solution to calculate the potential-flux field landscapesand the expressions of the various non-equilibrium thermodynamic quantities fora specific initial condition. Discussions as well as a picture illustrating the resultsare also given.

Stochastic Dynamics

Let ϕ(x, t) represent the electric potential on the membrane of a neuron mod-eled as a one dimensional infinitely thin cylinder extended from x = 0 to x = π.The linear stochastic cable equation (a functional Langevin equation) governingthe evolution of ϕ(x, t) reads [43]:

∂

∂tϕ(x, t) =

∂2

∂x2ϕ(x, t)− ϕ(x, t) + ξ(x, t), 0 ≤ x ≤ π, t > 0, (3.282)

149

where the space-time Gaussian white noise has the statistical property:

⟨ξ(x, t)⟩ = 0, ⟨ξ(x, t)ξ(x′, t′)⟩ = δ(x− x′)δ(t− t′). (3.283)

On the right side of Eq. (3.282), the first term represents diffusion of the electricpotential over the neuron fiber; the second term represents ions leaking acrossthe membrane; the third term represents space-time random disturbance to themembrane potential [43]. We remark that Eq. (3.282) can also be interpreted inthe context of chemical reactions, where ϕ(x, t) represents the local concentrationof a chemical species and the three terms on the right side represent, respectively,diffusion of the chemical species across space, degradation of the chemical speciesand space-time random fluctuations of the local concentration. In the following wework in the context of the neuron model, but the results also apply to the stochasticchemical degradation-diffusion process. The boundary condition is assumed to beNeumann:

∂

∂xϕ(0, t) =

∂

∂xϕ(π, t) = 0, ∀t > 0, (3.284)

which means the neuron fiber is insulated at both ends all the time. The initialcondition is:

ϕ(x, 0) = f(x), 0 ≤ x ≤ π. (3.285)

Equation (3.282) can be written in an operator form:

∂

∂tϕ(x, t) = −γϕ(x, t) + ξ(x, t), 0 ≤ x ≤ π, t > 0, (3.286)

where γ is a differential operator given by:

γ = 1− ∂2

∂x2. (3.287)

To establish a connection with our formalism, we notice that this differential op-erator γ also has the following representation as an integral operator:

γ =

∫dx′[δ(x− x′)− ∂2

∂x2δ(x− x′)

]∗, (3.288)

with the integral kernel

γ(x, x′) = δ(x− x′)− ∂2

∂x2δ(x− x′). (3.289)

150

The action of the integral operator on ϕ(x′, t) gives the same result as the differen-tial operator 1− ∂2/∂x2 acting on ϕ(x, t). The eigenvalues and eigenfunctions ofthe differential operator γ has been worked out [43], which is also easy to verify.The eigenfunctions are given by:

e0(x) =1√π, en(x) =

√2

πcosnx (n ≥ 1), (3.290)

with the corresponding eigenvalues

γn = n2 + 1 (n ≥ 0). (3.291)

These eigenfunctions are orthonormal and complete, satisfying Eq. (3.170) adapt-ed to this particular case of one dimensional space:∫ π

0

dx en(x)em(x) = δmn,+∞∑n=0

en(x)en(x′) = δ(x− x ′). (3.292)

Therefore en(x) form an orthonormal basis of the Hilbert space Ω = L2([0, π])(square integrable functions on the interval [0, π]). Using Eq. (3.292), the integralkernel γ(x, x′) can be decomposed using the eigenvalues and eigenfunctions:

γ(x, x′) =+∞∑n=0

γnen(x)en(x′). (3.293)

Then we consider the functional Fokker-Planck equation corresponding to thefunctional Langevin equation (Eq. (3.282)). The statistical property of the fluctua-tion in Eq. (3.283) means the diffusion coefficient in the functional Fokker-Planckequation is given by (see Eq. (3.254))

D(x, x′) =1

2δ(x, x′). (3.294)

Therefore, according to Eq. (3.255), the functional Fokker-Planck equation reads:

∂

∂tPt[ϕ] =

∫dx

δ

δϕ(x)

[(ϕ(x)− ∂2

∂x2ϕ(x)

)Pt[ϕ]

]+

1

2

∫dx

(δ

δϕ(x)

)2

Pt[ϕ].

(3.295)The transient Gaussian solution has the following form:

Pt[ϕ] =1√

det([2πσ(x, x ′)])exp

−1

2

∫∫dxdx ′ [ϕ(x)−K(x)]

× σ−1(x, x ′) [ϕ(x ′)−K(x ′)], (3.296)

151

where the mean and covariance are now determined by (see Eqs. (3.257) and(3.258)):

K(x) = −∫

dx ′γ(x, x ′)K(x ′), (3.297)

σ(x, x ′) = −∫

dx ′′ [γ(x, x ′′)σ(x ′′, x ′) + γ(x ′, x ′′)σ(x ′′, x)]

+2D(x, x ′). (3.298)

Yet we are not going to solve these two equations directly in their current forms.Rather, we work in the orthonormal basis en(x). In this basis, γ(x, x ′) is givenby Eq. (3.293). According to Eq. (3.292) the space-time diffusion coefficient inEq. (3.294) has the following form:

D(x, x ′) =+∞∑n=0

1

2en(x)en(x

′), (3.299)

which means in the basis en(x) the diffusion matrix is diagonal, with the matrixelements given by Dmn = δmn/2 (see Eq. (3.174)). We also expand K(x) andσ(x, x′) in this basis (see Eqs. (3.171)-(3.175)), which are given respectively by:

K(x) =+∞∑n=0

Knen(x), σ(x, x′) =+∞∑

m,n=0

σmnem(x)en(x′), (3.300)

where σmn = σnm due to the symmetry property σ(x, x′) = σ(x′, x). Thus in thebasis en(x), Eqs. (3.297) and (3.298) take on the following simple form:

Kn = −γnKn, (3.301)σmn = −(γm + γn)σmn + δmn. (3.302)

The initial condition Eq. (3.285) is a definite state, which means the mean K(x)|t=0 =ϕ(x, 0) = f(x) and the covariance σ(x, x′)|t=0 = 0. In the basis en(x), thismeans

Kn(t = 0) = fn, σmn(t = 0) = 0, (3.303)

where fn is the expansion coefficient in f(x) =∑+∞

n=0 fnen(x), given by (seeEq. (3.172))

fn =

∫ π

0

en(x)f(x)dx. (3.304)

152

The solutions of Eqs. (3.301) and (3.302) with the initial condition Eq. (3.303) aregiven by:

Kn(t) = e−γntfn, σmn(t) =1− e−2γnt

2γnδmn. (3.305)

Plugging them back into Eq. (3.300), we thus have solved the mean and covarianceof the transient Gaussian distribution functional:

E[ϕ(x)]t = K(x, t) =+∞∑n=0

e−γntfnen(x), (3.306)

Cov[ϕ(x), ϕ(x′)]t = σ(x, x′, t) =+∞∑n=0

1− e−2γnt

2γnen(x)en(x

′). (3.307)

The stationary solution of Eq. (3.305) can be obtained by taking the limit t →+∞, giving:

Kn = 0, σmn =δmn

2γn. (3.308)

Therefore the mean and covariance of the stationary Gaussian distribution func-tional Ps[ϕ] are, respectively, given by:

E[ϕ(x)]s = K(x) = 0, (3.309)

Cov[ϕ(x), ϕ(x′)]s = σ(x, x′) =+∞∑n=0

1

2γnen(x)en(x

′). (3.310)

These results agree with those obtained in Ref. [43] by working with the functionalLangevin equation. Here we have obtained the same results using the functionalFokker-Planck equation, without resorting to the technique of stochastic calculuson the Hilbert space.

Potential-Flux Field landscape

The expressions of the potential field landscapes, their functional gradients andthe flux velocity fields of OU processes in the space configuration representationare given by Eqs. (3.261)-(3.269). They can be calculated conveniently in thebasis of en(x). The state of the system ϕ(x) is expanded in this basis as ϕ(x) =∑∞

n=0 ϕnen(x), with the coefficient given by ϕn =∫ π

0en(x)ϕ(x)dx. We leave

153

the details of calculation in Appendix F and list the results in the following. Thepotential field landscapes are given by:

U [ϕ] =∞∑n=0

[(n2 + 1)ϕ2

n +1

2ln

π

n2 + 1

](3.311)

S[ϕ] =∞∑n=0

[(n2 + 1)

1− e−2(n2+1)t

(ϕn − fne

−(n2+1)t)2

+1

2ln(1− e−2(n2+1)t

)+1

2ln

π

n2 + 1

](3.312)

A[ϕ] =∞∑n=0

[(n2 + 1)ϕ2

n −(n2 + 1)

1− e−2(n2+1)t

(ϕn − fne

−(n2+1)t)2

−1

2ln(1− e−2(n2+1)t

)](3.313)

Their functional gradients are:

δU

δϕ(x)=

∞∑n=0

2(n2 + 1)ϕn en(x) (3.314)

δS

δϕ(x)=

∞∑n=0

2(n2 + 1)

1− e−2(n2+1)t(ϕn − fne

−(n2+1)t) en(x) (3.315)

δA

δϕ(x)=

∞∑n=0

[2(n2 + 1)ϕn −

2(n2 + 1)

1− e−2(n2+1)t(ϕn − fne

−(n2+1)t)

]×en(x) (3.316)

The flux velocity fields are calculated to be:

Vs(x)[ϕ] = 0 (3.317)Vt(x)[ϕ] = Vr(x)[ϕ]

=∞∑n=0

[−(n2 + 1)ϕn +

(n2 + 1)

1− e−(n2+1)t(ϕn − fne

−(n2+1)t)

]en(x)(3.318)

In the expressions of U [ϕ] and S[ϕ] in Eqs. (3.311) and (3.312), there is a commondivergent constant

∑∞n=0(ln[π/(n

2 + 1)])/2 which can be removed, consideringthat U [ϕ] and S[ϕ] are defined only up to a common additive constant. With

154

these expressions given explicitly, we can verify the field dynamical decomposi-tion equations given in Eqs. (3.191)-(3.193). In our current case, the state of thesystem ϕ(x) has only one component and the physical space is one dimension-al. The (effective) driving force field F ′(x)[ϕ] = F (x)[ϕ]. The diffusion matrixfield D(x, x′) = δ(x − x′)/2. Therefore, the stationary, transient and relativefield decomposition equations in this particular case take on the following simpli-fied form: F (x)[ϕ] = −δϕ(x)U/2 + Vs(x)[ϕ]; F (x)[ϕ] = −δϕ(x)S/2 + Vt(x)[ϕ];Vr(x)[ϕ] = −δϕ(x)A/2. They can be verified easily by using the expressionF (x)[ϕ] =

∑∞n=0 −(n2 + 1)ϕnen(x) and the expressions given in Eqs. (3.314)-

(3.318).According to Eq. (3.317), the stationary flux velocity field Vs(x)[ϕ] is identi-

cally zero. This means there is no non-equilibrium irreversibility from detailedbalance breaking in the steady state. Therefore the adiabatic entropy productionrate Sad should be identically zero. The only contribution to entropy production isthe nonadiabatic entropy production Sna, indicated by non-zero relative flux ve-locity field Vr(x)[ϕ], which comes from the non-equilibrium irreversible processof relaxing from the transient state to the equilibrium steady state, if the system isinitially not already in that state. Eventually, the system will reach the equilibri-um steady state and the total entropy production should be zero. This can also beseen from Eq. (3.318), Vt(x)[ϕ] = Vr(x)[ϕ] = 0 when t → +∞, which indicatesSpd = Sna = 0 (t → +∞). These observations are confirmed by calculating thenon-equilibrium thermodynamic quantities directly in the following.

Non-Equilibrium Thermodynamics

Since this particular system is a special case of the spatially inhomogeneousOU process, it also obeys the set of non-equilibrium thermodynamic equations inEq. (3.224). We calculate the expressions of the thermodynamic quantities us-ing Eqs. (3.270)-(3.281), by working in the basis en(x). The linear operatorsinvolved, I , γ, D, ˆσ, σ and |K)(K|, are represented by infinite dimensional ma-trices in this basis. According to Eqs. (3.292), (3.293), (3.299), (3.306), (3.307)and (3.310), the matrix elements of these linear operators in the basis en(x) aregiven, respectively, by:

Imn = δmn, γmn = γnδmn,

Dmn = δmn/2, σmn = δmn/(2γn),

σmn = δmn(1− e−2γnt)/(2γn), KmKn = e−(γm+γn)tfmfn,

(3.319)

155

where γn = n2 + 1 and fn is given by Eq. (3.304) determined by the initial con-dition. For specificity and simplicity, we consider the particular initial conditionϕ(x, 0) = f(x) ≡ 0 (thus fn = 0). That means the electric potential on the mem-brane is initially 0 everywhere; the evolution of the membrane potential is purelyinitiated by stochastic fluctuations. With this initial condition fn = 0, the operator|K)(K| becomes the zero operator as its matrix elements KmKn are all zero.

First we have calculate the cross, transient and relative entropies. The crossentropy is calculated using Eq. (3.270):

U = tr

(1

2ˆσ−1(σ + |K)(K|) + 1

2ln(2π ˆσ)

)=

∞∑n=0

[1

2

σnn +K2n

σnn

+1

2ln(2πσnn)

]=

∞∑n=0

1

2

[−e−2(n2+1)t + 1 + ln

π

n2 + 1

]. (3.320)

Using Eq. (3.271), we have the transient entropy given by:

S = tr

(1

2I +

1

2ln(2πσ)

)=

∞∑n=0

[1

2δnn +

1

2ln(2πσnn)

]=

∞∑n=0

1

2

[ln(1− e−2(n2+1)t) + 1 + ln

π

n2 + 1

]. (3.321)

The relative entropy is then:

A = U − S =∞∑n=0

1

2

[−e−2(n2+1)t − ln(1− e−2(n2+1)t)

]. (3.322)

We examine whether these quantities are divergent or not. Define the summand inthe infinite series of the expression of U , S and A in Eqs. (3.320)-(3.322) as un(t),sn(t) and an(t), respectively. Thus U =

∑∞n=0 un(t), S =

∑∞n=0 sn(t) and A =∑∞

n=0 an(t). It is not difficult to check that for any t ∈ (0,+∞), un(t) → −∞(n → +∞), sn(t) → −∞ (n → +∞) while an(t) → 0 (n → +∞). Therefore,the cross entropy U =

∑∞n=0 un(t) and the transient entropy S =

∑∞n=0 sn(t) are

both divergent for any t ∈ (0,+∞). The fact that an(t) → 0 (n → +∞) alone

156

does not guarantee that A =∑∞

n=0 an(t) is convergent. Yet further informationdoes. The inequality ln(1−x)+x ≤ 0 (x ≤ 1) shows that an(t) ≥ 0. And one canalso prove n2an(t) = an(t)/(1/n

2) → 0 (n → +∞) for any t ∈ (0,+∞). Theconvergence of the series

∑∞n=1 1/n

2 ensures that A =∑∞

n=0 an(t) is convergentand thus finite for any t ∈ (0,+∞). Therefore, in this case, the cross entropy Uand the transient entropy S are divergent, while the relative entropy A is finite.

We notice from Eqs. (3.320)-(3.322) that there is a common term cn = (1 +ln[π/(n2 + 1)])/2 in un(t) and sn(t) preventing them from approaching zero inthe limit n → +∞, which is canceled in the expression of A. We exploit thisobservation and the fact that U and S are only defined up to a common additiveconstant, to implement the ‘renormalization’ of the cross entropy and the transiententropy. We define the following constant in terms of an infinite series:

SC =∞∑n=0

cn =∞∑n=0

1

2

(1 + ln

π

n2 + 1

). (3.323)

Since cn → −∞ (n → +∞), SC is a divergent constant. We subtract this commondivergent constant from U and S and define the ‘renormalized’ U and S . Therenormalized cross entropy is:

U ren = U − SC =∞∑n=0

−1

2e−2(n2+1)t, (3.324)

and the renormalized transient entropy reads:

Sren = S − SC =∞∑n=0

1

2ln(1− e−2(n2+1)t

). (3.325)

The relative entropy remains the same: A = U − S = U ren − Sren. (One canstill add a finite constant to both U ren and Sren, which is simply a change inthe reference point of entropy.) Define the summand in the series expression ofU ren and Sren as uren

n (t) and srenn (t), respectively. One can prove that for anyt ∈ (0,+∞), we have uren

n (t) ≤ 0 and n2urenn (t) → 0 (n → +∞) as well

as srenn (t) ≤ 0 and n2srenn (t) → 0 (n → +∞) . Using the same argument forthe convergence of A, we conclude that the renormalized cross entropy U ren andtransient entropy Sren are both finite and thus well-defined for any t ∈ (0,+∞).In fact, U ren has an analytical expression, U ren = −e−2t[ϑ3(0, e

−2t)+1]/4, whereϑ3(a, q) = 1 + 2

∑+∞n=1 q

n2cos(2na) is the third elliptic theta function. Thus we

have successfully removed the divergence of U and S through ‘renormalization’.

157

The rate of change of U , S and A can either be calculated directly from theirexpressions (or the renormalized expressions) in Eqs. (3.320)-(3.325) or usingthe formula in Eqs. (3.273)-(3.275). We take the former approach to check theirconsistency with the rest of the thermodynamic quantities. The rate of change ofthe cross entropy calculated using Eq. (3.320) or Eq. (3.324) is given by:

U =∞∑n=0

(n2 + 1)e−2(n2+1)t. (3.326)

Denote the summand of the infinite series as un(t). One can prove that un(t) ≥ 0and n2un(t) → 0 (n → +∞) for any t ∈ (0,+∞). Therefore, U is finite for anyt ∈ (0,+∞). According to Eq. (3.321) or Eq. (3.325), the rate of change of thetransient entropy is:

S =∞∑n=0

(n2 + 1)e−2(n2+1)t

1− e−2(n2+1)t. (3.327)

Using the same argument for U , we can prove S is finite for any t ∈ (0,+∞).The rate of change of the relative entropy is given by:

A = U − S = −∞∑n=0

(n2 + 1)e−4(n2+1)t

1− e−2(n2+1)t, (3.328)

which is also finite for any t ∈ (0,+∞) as U and S are both finite.Then we calculate the rest of the thermodynamic quantities. Since we did

not consider external driving in this particular case, the external driving entropicpower is zero:

Sed = 0, (3.329)

which can also be seen by noticing that in Eq. (3.276) dˆσ/dt = 0 since ˆσ given byEq. (3.310) does not depend on time. Then we calculate the excess entropy flowrate using Eq. (3.277):


−1(σ + |K)(K|)− γ

)=

∞∑n=0

[Dnn

σ2nn

(σnn +K2n)− γnn

]= −

∞∑n=0

(n2 + 1)e−2(n2+1)t. (3.330)

158

We notice that this expression of Sex is simply the negative of U . This is consistentwith the cross entropy balance equation U = Sed − Sex, within which Sed = 0in this particular case. The total entropy flow rate calculated using Eq. (3.278) isgiven by


)=

∞∑n=0

[γ2nn

Dnn

(σnn +K2n)− γnn

]= −

∞∑n=0

(n2 + 1)e−2(n2+1)t. (3.331)

However, this is exactly Sex. In other words, in this particular case Sfl = Sex. Theexpression in Eq. (3.331) shows that the entropy flow rate Sfl is negative for anyt ∈ (0,+∞) and goes to 0 as t → ∞. This means that during the entire processof relaxing back to the equilibrium state, there is an entropy flow from the envi-ronment into the system (e.g., by absorbing heat from the environment), whichdisappears eventually as the system reaches equilibrium. The fact that Sfl = Sex

also implies that the adiabatic entropy production rate Sad is identically zero, ac-cording to the entropy flow decomposition equation Sfl = Sad + Sex. Indeed, thiscan be proven directly, using the expression in Eq. (3.280):

Sad = tr((


−1)(σ + |K)(K|)

)=

∞∑n=0

(γ2nn

Dnn

− Dnn

σ2nn

)(σnn +K2

n

)=

∞∑n=0

(2γ2

n − 2γ2n)(σnn +K2

n

)= 0. (3.332)

This agrees with our analysis based on the observation that Vs(x)[ϕ] = 0. Thismeans there is no detailed balance breaking in the steady state in this particularcase. The only source of irreversibility comes from the relaxation process due todeviation from the steady state, which is characterized by nonadiabatic entropy

159

production rate. According to Eq. (3.281), it is given by:


−1(σ + |K)(K|) + σ−1D − 2γ

)=

∞∑n=0

[Dnn

σ2nn

(σnn +K2n) +

Dnn

σnn

− 2γnn

]=

∞∑n=0

(n2 + 1)e−4(n2+1)t

1− e−2(n2+1)t. (3.333)

According to this expression Sna is positive for any t ∈ (0,+∞) and goes to 0as t → +∞ in the process of relaxing back to equilibrium. This also agrees withour analysis that Vr(x)[ϕ] → 0 (t → +∞). Notice that the expression of Sna isalso the negative of A in Eq. (3.328). This is consistent with the relative entropybalance equation A = Sed − Sna and the fact that Sed = 0 in this case. SinceSad = 0, we expect that the total entropy production rate Spd = Sna, according tothe entropy production decomposition equation Spd = Sad + Sna. This is indeedtrue:


)=

∞∑n=0

[γ2nn

Dnn

(σnn +K2n) +

Dnn

σnn

− 2γnn

]=

∞∑n=0

(n2 + 1)e−4(n2+1)t

1− e−2(n2+1)t. (3.334)

As with Sna, the total entropy production Spd is also positive for any t ∈ (0,+∞)and goes to zero as t → +∞. That means eventually the system reaches equilib-rium where there is no irreversibility contributing to non-zero entropy production.This also agrees with the fact that Vt(x)[ϕ] → 0 (t → +∞). We can also verifythe transient entropy balance equation S = Spd − Sfl from the expressions inEqs. (3.327), (3.331) and (3.334).

Discussion

We have proven in this particular case that the thermodynamic expressions inEqs. (3.326)-(3.334) and the relative entropy A in Eq. (3.322), calculated from E-qs. (3.272)-(3.281), are all finite for any t ∈ (0,+∞). Although the cross entropy

160

U and the the transient entropy S calculated directly from Eqs. (3.270) and (3.271)are divergent, they can be simply ‘renormalized’ by subtracting a common diver-gent constant that has no physical significance. This shows the thermodynamicexpressions in Eqs. (3.270)-(3.281) for spatially inhomogeneous OU processes arenot merely formal results. When applied to concrete systems, they can producesensible, quantitative, physical results.

This particular example is somewhat special in that Sed = Sad = 0, that is,there is neither external driving nor detailed balance breaking in the steady state.In this case the set of non-equilibrium thermodynamic equations (Eq. (3.224)) re-duces to:

S = Spd − Sfl (3.335)Spd = Sna = −A (3.336)Sfl = Sex = −U (3.337)

The first equation is the usual (transient) entropy balance equation, stating thatthe system entropy is increased by entropy production within the system and de-creased by entropy flow into the environment. The other two equations are spe-cific to the situation Sed = Sad = 0. Equation (3.336) indicates that the non-equilibrium nature of this particular case is merely due to the non-equilibriumrelaxation process from the transient state to the equilibrium steady state (no a-diabatic entropy production generated by detailed balance breaking in the steadystate) induced by the initial state preparation (no external-driving induced non-equilibrium relaxation). Equation (3.337) states that the total entropy flow is justthe excess entropy flow (no entropy flow associated with detailed balance break-ing in the steady state) into the environment, which is also equal to the decreaseof cross entropy (no change of cross entropy caused by external driving).

We have also shown that throughout the entire relaxation process there is anon-zero entropy production within the system (Spd > 0) and an entropy flowfrom the environment into the system (Sfl < 0). Thus the entropy of the systemincreases during the entire relaxation process (S > 0). Eventually when the sys-tem has reached equilibrium, both the entropy production and entropy flow stopand the system entropy remains constant. In fact, the entropy change, entropyproduction and entropy flow in this process between any two finite time pointst = t1 and t = t2 (t2 > t1) can be calculated quantitatively. The change ofentropy S(t2) − S(t1) can be calculated using S(t) in Eq. (3.321) or its renor-malized form in Eq. (3.325). According to Eq. (3.336), the entropy productionSpd(t1 → t2) = A(t1) − A(t2), with A(t) given by Eq. (3.322). According to

161

Eq. (3.337), the entropy flow Sfl(t1 → t2) = U(t1) − U(t2), with U(t) given byEq. (3.320) or its renormalized form in Eq. (3.324). This means in this process theamount of entropy produced in the system, Spd(t1 → t2), is equal to the amountof relative entropy decreased, −[A(t2) − A(t1)]; the amount of entropy trans-ferred into the system, −Sfl(t1 → t2), is equal to the amount of cross entropyincreased, U(t2) − U(t1); the system entropy change is a result of both factors:S(t2)−S(t1) = Spd(t1 → t2)−Sfl(t1 → t2) = −[A(t2)−A(t1)]+[U(t2)−U(t1)].This is clearly seen in Fig. 3.1. The relative entropy A decreases with time, in-

0.5 1.0 1.5 t

-0.6

-0.4

-0.2

0.2

0.4

0.6

Non-Equilibrium Thermodynamic Functions

AHtL

UHtL

SHtL

Figure 3.1: Temporal profile of the system’s (renormalized) transient entropy S ,cross entropy U and relative entropy A in the process of relaxing to the equilibri-um state in the spatial stochastic neuronal model.

dicating entropy production in the system. The cross entropy U increases withtime, indicating entropy flow into the system. The system entropy S thereforeincreases with time, until the equilibrium state is reached in the limit t → +∞,where both entropy production and entropy flow stop as A and U become constan-t. We note that if the temperature of the environment is constant, this process isa non-equilibrium isothermal process. The cross entropy and the relative entropyare then related, respectively, to the non-equilibrium internal energy U and freeenergy A as in Eqs. (3.24) and (3.25). As mentioned before, in a non-equilibriumisothermal process without external driving, the change of cross entropy and rel-ative entropy are, respectively, proportional to the change of internal energy and

162

free energy: ∆U = T−1∆U and ∆A = T−1∆A. The increase of the systementropy can thus also be understood in terms of the free energy dissipation gener-ating entropy production in the system [115] and the increase of internal energydue to heat flow into the system that also increases the system entropy. Anotherperspective to look at the increase of system entropy is as follows. The systemwas initially prepared in a definite (micro)state ϕ(x, 0) = 0 (this, however, is anidealization; in reality, the prepared state still has some, though maybe small, un-certainty; this is also why the exact point t = 0 seems unphysical in Fig. 3.1 andother analytical expressions). The final equilibrium state is a Gaussian probabilitydistribution of (micro)states (see Eqs. (3.309) and (3.310)). Therefore the systemstate becomes more uncertain and random as it evolves to equilibrium, increasingthe system entropy.

The successful application of the non-equilibrium thermodynamic formalismfor spatially inhomogeneous systems to the spatial neuron model gives us confi-dence in applying the formalism to other similar or even more interesting spatiallyinhomogeneous systems, which may also include effects of spatially correlatedfluctuations, detailed balance breaking in the steady state, time-dependent exter-nal driving and nonlinearity that were not present in the particular example stud-ied. The results in Sec. 3.3.5 for general spatially inhomogeneous OU processesare applicable to systems described by those processes. In particular, stochasticchemical reaction diffusion processes [15,50–52], with first-order (linear) chemi-cal reactions and general diffusions [51], under the influence of external additivespace-time fluctuations [15,50], can be described by spatially inhomogeneous OUprocesses; thus these systems can be studied in a similar fashion illustrated in thespatial neuron model. For general non-linear stochastic systems (even for spatial-ly homogeneous systems), analytical solutions of the (functional) Fokker-Planckequation are generally not available; thus approximation techniques and numer-ical simulations are usually required in studying these systems [15, 16, 52, 53].The general formalism of the potential-flux landscape and non-equilibrium ther-modynamic equations, however, still apply to these general non-linear stochasticsystems, even though analytical expressions cannot be obtained. The formalis-m established in this work has potential applications in the study of the spatial-temporal non-equilibrium (thermo)dynamics of various spatially inhomogeneousstochastic systems displaying self-organization and pattern formation behaviors,such as convection flow in fluids, Turing pattern, cell differentiation, as well aspopulation dynamics in ecological systems [1, 35–41]. It may also be applied tostudy some particular (types of) spatially inhomogeneous stochastic systems, de-scribed by, for instance, stochastic Ginzburg-Landau equation in superconductivi-

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ty [56], Kardar-Parisi-Zhang equation studying surface growth [58] and stochasticspatial Hodgkin-Huxley model in neurobiology [54].

3.4 SummaryIn this chapter, utilizing the potential-flux landscape framework as a bridge

to connect stochastic dynamics with non-equilibrium thermodynamics, we haveestablished a general non-equilibrium thermodynamic formalism consistently ap-plicable to both spatially homogeneous and inhomogeneous systems, governedby the Langevin and Fokker-Planck stochastic dynamics. We first constructedthe non-equilibrium thermodynamics within the (extended) potential-flux land-scape framework for spatially homogeneous systems in an isothermal environ-ment and extended the results to systems with one general state transition mech-anism. We further expanded the potential-flux landscape framework and non-equilibrium thermodynamics to accommodate spatially homogeneous systems withmultiple state transition mechanisms and then spatially inhomogeneous system-s. General Ornstein-Uhlenbeck processes were worked out systematically in thecontext of the non-equilibrium thermodynamic formalism established. The spa-tial stochastic neuronal model was studied in detail as both an application and avalidation of the general formalism.

A conceptual distinction is made between the information transport process inthe state space and the matter or energy transport process in the physical space,with their possible connections discussed as well. The potential(s) and flux(es) inthe potential-flux landscape framework are in nature ‘informational’, as they char-acterize the information transport process in the state space, associated with thematter or energy transport process in the physical space. The construction of thenon-equilibrium thermodynamics within the potential-flux landscape frameworkfrom the stochastic dynamics is facilitated by identifying the double role that thepotential landscape and the flux velocity play. On the one hand, they are dynami-cal quantities constructed directly from the probability distribution and probabilityflux of the stochastic dynamics. On the other hand, they also have thermody-namic meanings connected directly to the non-equilibrium thermodynamics. Thepotential landscape is the microscopic entropy, which also acts as the potentialof certain thermodynamic force. The flux velocity is the thermodynamic forcegenerating entropy production. The flux velocity plays a central role in character-izing the non-equilibrium nature of the system’s (thermo)dynamics. It representsa force breaking detailed balance on the dynamic level, entailing the dynamical

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decomposition equations; it also represents a force creating entropy productionon the thermodynamic level, manifested in the non-equilibrium thermodynamicequations.

It is recognized that there are two fundamental aspects of non-equilibriumprocesses. One aspect is detailed balance breaking in the steady state throughconstant system-environment interaction and exchange, characterized by the sta-tionary flux velocity which is also the thermodynamic force generating the adia-batic entropy production. The other aspect is relaxation from the transient state tothe steady state due to the non-stationary condition, characterized by the relativeflux velocity which is also the thermodynamic force generating the nonadiabaticentropy production. (The second aspect can be further identified as containing twofacets, namely initial preparation of the transient state and time-dependent exter-nal driving of the steady state.) The combined non-equilibrium effects of the twofundamental aspects are characterized by the transient flux velocity, which is alsothe thermodynamic force generating the total entropy production. The decompo-sition of the non-equilibrium process into these two fundamental aspects is repre-sented by the flux decomposition equation on the dynamic level and mapped intothe entropy production decomposition equation on the thermodynamic level. As aresult of this fundamental decomposition, the potential landscape is differentiatedinto three distinct yet related aspects, that is, the stationary potential landscape asthe microscopic stationary entropy, the relative potential landscape as the micro-scopic relative entropy, and the transient potential landscape as the microscopictransient entropy. This conceptual differentiation is further manifested as a struc-tural differentiation in the (thermo)dynamic equations. On the dynamic level wehave the stationary, transient and relative dynamical decomposition equations inaddition to the flux decomposition equation, forming a set of dynamical decom-position equations. On the thermodynamic level we have the cross, transient andrelative entropy balance equations as well as the entropy flow and entropy produc-tion decomposition equations, forming a set of non-equilibrium thermodynamicequations. The set of non-equilibrium thermodynamic equations is a reflectionand manifestation of the set of dynamical decomposition equations, with the fluxvelocities playing a crucial role on both levels.

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Chapter 4

Conclusion

In this dissertation we have established a potential and flux field landscapetheory, quantifying the global stability and dynamics as well as facilitating the es-tablishment of the non-equilibrium thermodynamics of spatially inhomogeneousnon-equilibrium systems governed by Langevin and Fokker-Planck dynamics.1

In chapter 2 we studied the global stability and dynamics of non-equilibriumsystems (spatially inhomogeneous systems in particular) using the potential-fluxlandscape theory. We developed a general method to construct Lyapunov func-tionals that quantify the global stability and robustness of deterministic and s-tochastic spatially inhomogeneous non-equilibrium systems. We found the in-trinsic potential field landscape is the Lyanpunov functional of the deterministicspatially inhomogeneous system quantifying its global stability. The topographyof the intrinsic potential field landscape can be characterized by the basins of at-tractions and barrier heights that are directly related to the global stability of thesystem. The relative entropy functional is found to be a Lyapunov functionalquantifying the global stability of the stochastic spatially inhomogeneous non-equilibrium system. We discovered that the global dynamics of spatially inhomo-geneous non-equilibrium systems is determined by both the functional gradient ofthe potential field landscape and the curl probability flux field. Vanishing prob-ability flux field indicates the spatially inhomogeneous system is in detailed bal-ance everywhere in the physical space. In such cases, the potential field landscapeis usually known a priori, given by the interaction potential or energy function-

1Most of the material in this chapter was originally co-authored with Jin Wang. Reprintedwith permission from W. Wu and J. Wang, The Journal of Chemical Physics, 139, 121920 (2013).Copyright 2013, AIP Publishing LLC. Reprinted with permission from W. Wu and J. Wang, TheJournal of Chemical Physics, 141, 105104 (2014). Copyright 2014, AIP Publishing LLC.

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al of the system. Therefore, in spatially inhomogeneous equilibrium systems theglobal dynamics is determined by the potential field landscape alone. For non-equilibrium spatially inhomogeneous dynamical systems, in general, an energyfunctional governing the equilibrium gradient dynamics cannot be found. Thusboth the potential field landscape and the curl probability flux field have to be con-sidered. The curl probability flux field breaks detailed balance and characterizeshow far the system is away from equilibrium. The potential field landscape andthe probability flux field form a dual pair that gives a complete characterizationof the global stability and dynamics of spatially inhomogeneous non-equilibriumsystems. We applied our general framework to reaction diffusion systems and theBrusselator reaction diffusion model in particular to illustrate the general theory.

In chapter 3 we established a general non-equilibrium thermodynamic formal-ism consistently applicable to both spatially homogeneous and, more importantly,spatially inhomogeneous systems, governed by Langevin and Fokker-Planck dy-namics with multiple state transition mechanisms, using the potential-flux land-scape framework as a bridge to connect stochastic dynamics with non-equilibriumthermodynamics. A set of non-equilibrium thermodynamic equations, quantify-ing the relations of the non-equilibrium entropy, entropy flow, entropy production,and other thermodynamic quantities, is constructed from a set of dynamical de-composition equations associated with the potential-flux landscape framework.The flux velocity plays a pivotal role on both the dynamic and thermodynamiclevels. On the dynamic level, it represents a dynamic force breaking detailed bal-ance, entailing the dynamical decomposition equations. On the thermodynamiclevel, it represents a thermodynamic force generating entropy production, man-ifested in the non-equilibrium thermodynamic equations. The non-equilibriumthermodynamic equations are reflections and manifestations of the dynamical de-composition equations, with the link between them given quantitatively by the fluxvelocity. The Ornstein-Uhlenbeck process and, in particular, the spatial stochasticneuronal model are studied to test and illustrate the general theory. The non-equilibrium thermodynamic formalism established in this work is not limited tothe linear regime or bound by the local equilibrium assumption. It can be appliedto study the spatial-temporal non-equilibrium dynamics and thermodynamics of avariety of physical, chemical and biological systems in nature.

This work is part of a series of works we are preparing on the non-equilibriumtheory of spatially inhomogeneous stochastic dynamical systems. In future studieswe will explore the generalized fluctuation dissipation theorem, the generalizedfluctuation theorem, the gauge field representation, and the kinetic paths of non-equilibrium spatially inhomogeneous systems.

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178

Appendix A

Proof of Several Equations

This appendix is supplementary to Sec. 3.1 of the main text, proving equationsmentioned there without proof.

Proof of ∂λiAs = ⟨∂λiU⟩s and ∂T As = ⟨∂T U⟩s − Ss

These two equations are in Sec. 3.1.4 of the main text. We list the followingtwo equations from the main text that will be used in the proof:

As(λ) = −T lnZ(λ), (A.1)

Ps(q, λ) = exp

[1

T

(As(λ)− U(q, λ)

)]. (A.2)

Also, notice the definitions Z(λ) =∫exp

[−U(q, λ)/T

]dq and Ss = ⟨− lnPs⟩s.

We first prove the first equation ∂λiAs = ⟨∂λi

U⟩s, where λi is not the temperatureT .

∂λiAs = −TZ−1(λ)∂λi

Z(λ)

= −TZ−1(λ)∫

∂λiexp

[− 1

TU(q, λ)

]dq

= Z−1(λ)∫ [

∂λiU(q, λ)

]exp

[− 1

TU(q, λ)

]dq

=

∫ [∂λi

U(q, λ)]Ps(q, λ)dq

= ⟨∂λiU⟩s. (A.3)

179

Then we prove the second equation ∂T As = ⟨∂T U⟩s − Ss.

∂T As = − lnZ(λ)− TZ−1(λ)∂TZ(λ)

=As

T− TZ−1(λ)

∫∂T exp

[− 1

TU(q, λ)

]dq

=As

T− TZ−1(λ)

∫ (U(q, λ)

T 2− 1

T

∂U(q, λ)∂T

)

× exp

[− 1

TU(q, λ)

]dq

=⟨As⟩sT

− ⟨U⟩sT

+

⟨∂U

∂T

⟩s

=

⟨As − U

T

⟩s

+

⟨∂U

∂T

⟩s

= ⟨lnPs⟩s +⟨∂T U

⟩s=⟨∂T U

⟩s− Ss. (A.4)

Proof of ⟨Vt·D−1·Vs⟩ = ⟨Vs·D−1·Vs⟩ and ⟨Vs·∇A⟩ = 0

Using the definition Vr = Vt − Vs and the dynamical decomposition equationD−1 · Vr = −∇A, we have ⟨Vt · D−1 · Vs⟩ − ⟨Vs · D−1 · Vs⟩ = ⟨Vr · D−1 · Vs⟩ =−⟨Vs·∇A⟩. Therefore ⟨Vt·D−1·Vs⟩ = ⟨Vs·D−1·Vs⟩ is equivalent to ⟨Vs·∇A⟩ = 0.The relation A = TA+As shows ∇A = T∇A, since As as a macrostate functionis independent of the microstate vector q. Thus ⟨Vs·∇A⟩ = T ⟨Vs·∇A⟩. Therefore,in order to prove ⟨Vt ·D−1 · Vs⟩ = ⟨Vs ·D−1 · Vs⟩ or its equivalence ⟨Vs ·∇A⟩ = 0,we only need to prove ⟨Vs · ∇A⟩ = 0, which is shown as follows:

⟨Vs · ∇A⟩ =

∫PtVs · ∇ ln

(Pt

Ps

)dq

=

∫PtVs ·

Ps

Pt

∇(Pt

Ps

)dq

=

∫Js · ∇

(Pt

Ps

)dq

=

∫ (∇ · Js

) Pt

Ps

dq = 0, (A.5)

where we have used the definition A = ln(Pt/Ps), integration by parts with van-ishing boundary terms for appropriate boundary conditions, and ∇ · Js = 0.

180

Appendix B

Proof of the Necessary andSufficient Condition for theCollective Definition Property

This appendix is supplementary to Sec. 3.2.4 of the main text. We give theproof of the necessary and sufficient condition for the collective definition prop-erty.

Theorem 1: For given vectors β(m) ∈ R(m) ⊆ Rk, β =∑

m β(m) ∈ Rk andinvertible matrixes D(m) ∈ M (m)×(m), D =

∑mD(m) ∈ M k×k, a sufficient

condition for∑

m α(m) ·D(m) −1 · β(m) = α ·D−1 · β to hold for all α(m) ∈ R(m)

with α =∑

m α(m) ∈ Rk is that there exists γ ∈ Rk such that β(m) = D(m) · γ(m = 1, 2, ...).

Proof: On the one hand, β(m) = D(m) · γ = D(m) ·Π(m) · γ. Therefore Π(m) ·γ = D(m) −1 · β(m). On the other hand, β =

∑m β(m) =

∑mD(m) · γ = D · γ.

Therefore γ = D−1 · β. Then∑

m α(m) ·D(m) −1 · β(m) =∑

m α(m) ·Π(m) · γ =∑m α(m) · γ = α · γ = α ·D−1 · β. Theorem 2: For given vectors β(m) ∈ R(m) ⊆ Rk, β =

∑m β(m) ∈ Rk and

invertible matrixes D(m) ∈ M (m)×(m), D =∑

mD(m) ∈ M k×k, if∑

m α(m) ·D(m) −1 · β(m) = α · D−1 · β holds for all α(m) ∈ R(m) (m = 1, 2, ...) withα =

∑m α(m) ∈ Rk, then there exists γ ∈ Rk such that β(m) = D(m) · γ

(m = 1, 2, ...).Proof:

∑m α(m) ·D(m) −1 · β(m) − α ·D−1 · β =

∑m α(m) ·D(m) −1 · β(m) −∑

m α(m) ·Π(m) ·D−1 · β =∑

m α(m) · (D(m) −1 · β(m) −Π(m) ·D−1 · β) = 0for all α(m) ∈ R(m) (m = 1, 2, ...). For any fixed m, take α(m′) = 0 (m′ = m)

181

and α(m) = e(m)i (i = 1, 2, ...), where e (m)

i (i = 1, 2, ...) are the base vectors inR(m). Therefore e

(m)i · (D(m) −1 · β(m) −Π(m) ·D−1 · β) = 0 for all allowed m

and i, which means D(m) −1 · β(m) − Π(m) · D−1 · β = 0 (m = 1, 2, ...). Thusβ(m) = D(m) ·

(D−1 · β

)= D(m) · γ (m = 1, 2, ...), where γ = D−1 · β exists

by construction. Combining theorem 1 and theorem 2 we have the necessary and sufficient

condition for∑

m α(m) ·D(m) −1 · β(m) = α ·D−1 · β to hold for all α(m) ∈ R(m),which is the existence of γ ∈ Rk such that β(m) = D(m) · γ (m = 1, 2, ...). Andwhen γ does exist, it is given by D−1 · β.

182

Appendix C

Ornstein-Uhlenbeck Processes forSpatially Homogeneous Systems

This appendix is supplementary to Sec. 3.2.5 of the main text. We calculatethe explicit expressions of the various thermodynamic quantities of the Ornstein-Uhlenbeck process for spatially homogeneous systems with one effective statetransition mechanism. For convenience, we list the results that have been obtainedin the main text which will be used here.

U =1

2q · σ−1 · q + 1

2tr (ln(2πσ)) , (C.1)

S =1

2(q − µ) · σ−1 · (q − µ) +

1

2tr (ln(2πσ)) , (C.2)

A =1

2q · σ−1 · q − 1

2(q − µ) · σ−1 · (q − µ) +

1

2tr (ln σ − lnσ) ,(C.3)

∇U = σ−1 · q, (C.4)∇S = σ−1 · (q − µ), (C.5)∇A = σ−1 · q − σ−1 · (q − µ). (C.6)

Vs = −γ · q +D · σ−1 · q, (C.7)Vt = −γ · q +D · σ−1 · (q − µ), (C.8)

Vr = D · σ−1 · (q − µ)−D · σ−1 · q. (C.9)

183

The three quantities µ, σ and σ in the above equations are determined by thefollowing equations:

˙µ = −γ · µ (C.10)σ = −γ · σ − σ · γᵀ + 2D (C.11)γ · σ + σ · γᵀ = 2D (C.12)

Two useful equations can be further derived from Eq. (C.12), which will be usedlater. By multiplying Eq. (C.12) by σ−1 from both the left and the right, wederive:

σ−1γ + γᵀσ−1 = 2σ−1Dσ−1. (C.13)

By multiplying Eq. (C.12) by σ−1 from the right and then taking the trace, wederive:

tr(Dσ−1) = tr(γ), (C.14)

where we have used the properties of the trace tr(AB) = tr(BA) and tr(Aᵀ) =tr(A), which gives tr (σ · γᵀ · σ−1) = tr (σ−1 · σ · γᵀ) = tr (γᵀ) = tr (γ).

Then we calculate the expressions of the thermodynamic quantities. The crossentropy U is calculated using Eq. (C.1), given by

U = ⟨U⟩

=

⟨1

2q · σ−1 · q

⟩+

1

2tr (ln(2πσ))

=1

2tr(σ−1 ⟨qq⟩

)+

1

2tr (ln(2πσ))

=1

2tr(σ−1(σ + µµ)

)+

1

2tr (ln(2πσ))

= tr

(1

2σ−1(σ + µµ) +

1

2ln(2πσ)

), (C.15)

where we have used the identity a · B · c = tr(B [ca]), where the dyadic ca isinterpreted as a matrix with the ij entry given by [ca]ij = ciaj . We have alsoused ⟨qq⟩ = σ+ µµ, which comes from the definition of the covariance matrix ofGaussian distributions σ = ⟨(q− µ)(q− µ)⟩ = ⟨qq⟩ − µµ. Then we calculate the

184

transient entropy S, using Eq. (C.2):

S = ⟨S⟩

=

⟨1

2(q − µ) · σ−1 · (q − µ)

⟩+

1

2tr (ln(2πσ))

=1

2tr(σ−1 ⟨(q − µ)(q − µ)⟩

)+

1

2tr (ln(2πσ))

=1

2tr(σ−1σ

)+

1

2tr (ln(2πσ))

= tr

(1

2I +

1

2ln(2πσ)

). (C.16)

The relative entropy A is then given by

A = U − S = ⟨A⟩

= tr

(1

2

[σ−1(σ + µµ)− I

]+

1

2(ln σ − lnσ)

). (C.17)

Next we calculate their time derivatives U , S and A. We first calculate the rateof change of the cross entropy U . Notice that σ can also be time dependent dueto the time dependence of γ and D in the equation determining σ [Eq. (C.12)].Therefore, according to Eq. (C.15), we have

U =d

dttr

(1

2σ−1(σ + µµ) +

1

2ln(2πσ)

)=

d

dttr

(1

2(σ + µµ)σ−1 − 1

2ln(2πσ−1)

)= tr

(1

2(σ + ˙µµ+ µ ˙µ)σ−1 +

1

2(σ + µµ)

d

dtσ−1 − 1

2σd

dtσ−1

)= tr

(1

2(−γσ − σγᵀ + 2D − γµµ− µµγᵀ)σ−1 +

1

2(σ + µµ− σ)

d

dtσ−1

)= tr

(−1

2(σ−1γ + γᵀσ−1)(σ + µµ) +Dσ−1 +

1

2(σ + µµ− σ)

d

dtσ−1

)= tr

(−σ−1Dσ−1(σ + µµ) + γ +

1

2(σ + µµ− σ)

d

dtσ−1

)= tr

(1

2(σ + µµ− σ)

d

dtσ−1 − σ−1Dσ−1(σ + µµ) + γ

), (C.18)

185

where we have used Eqs. (C.10), (C.11), (C.13), (C.14) and the properties of thetrace tr(AB) = tr(BA) and tr(Aᵀ) = tr(A). We have also implicitly utilizedthe so-called Jacobi’s formula of differentiating determinants, in the followingform:

1

det(A)

d

dtdet(A) =

d

dtln det(A) =

d

dttr(lnA)

= tr

(A−1 d

dtA

)= tr

(−A

d

dtA−1

).

If the explicit time-dependence in γ and D and thus σ is realized via a set of time-dependent external control parameters λi(t), then dσ−1/dt =

∑i λi∂λi

σ−1.The rate of change of the transient entropy, according to Eq. (C.16), is calculatedas follows:

S =d

dttr

(1

2I +

1

2ln(2πσ)

)= tr

(1

2σ−1 d

dtσ

)= tr

(1

2σ−1(−γσ − σγᵀ + 2D)

)= tr

(1

2(−γσσ−1 − σ−1σγᵀ) + σ−1D

)= tr

(−1

2(γ + γᵀ) + σ−1D

)= tr(σ−1D − γ), (C.19)

where we have used Eq. (C.11), tr(AB) = tr(BA) and tr(Aᵀ) = tr(A). Therate of change of the relative entropy is then given by

A = U − S

= tr

(1

2(σ + µµ− σ)

d

dtσ−1 − σ−1Dσ−1(σ + µµ)

−σ−1D + 2γ). (C.20)

Then we calculate the rest of the thermodynamic quantities. The external

186

driving entropic power Sed is calculated using Eq. (C.1):

Sed = ⟨∂tU⟩

=

⟨1

2q ·(

d

dtσ−1

)· q⟩+

1

2

d

dttr (ln(2πσ))

= tr

(1

2⟨qq⟩ d

dtσ−1

)− 1

2

d

dttr(ln(2πσ−1)

)= tr

(1

2(σ + µµ)

d

dtσ−1

)− 1

2tr

(σd

dtσ−1

)= tr

(1

2(σ + µµ− σ)

d

dtσ−1

), (C.21)

where we have used the equation ⟨qq⟩ = σ + µµ. The excess entropy flow rate iscalculated as follows from Eqs. (C.4) and (C.8):

Sex = −⟨Vt · ∇U⟩=

⟨(q · γᵀ − (q − µ) · σ−1 ·D

)·(σ−1 · q

)⟩=

⟨q · γᵀσ−1 · q − (q − µ) · σ−1Dσ−1 · q

⟩= ⟨q · 1

2(γᵀσ−1 + σ−1γ) · q − (q − µ) · σ−1Dσ−1 · q⟩

= ⟨q · σ−1Dσ−1 · q − (q − µ) · σ−1Dσ−1 · q⟩= tr

(σ−1Dσ−1⟨qq⟩ − σ−1Dσ−1⟨q(q − µ)⟩

)= tr

(σ−1Dσ−1(σ + µµ)− σ−1Dσ−1σ

)= tr

(σ−1Dσ−1(σ + µµ)− σσ−1Dσ−1

)= tr

(σ−1Dσ−1(σ + µµ)− γ

), (C.22)

where we have used ⟨q(q − µ)⟩ = ⟨qq⟩ − ⟨q⟩µ = σ and Eqs. (C.13) and (C.14).The (total) entropy production rate is calculated using Eq. (C.8):

Spd = ⟨Vt ·D−1 · Vt⟩=

⟨(−q · γᵀ + (q − µ) · σ−1 ·D

)·D−1 ·

(−γ · q +D · σ−1 · (q − µ)

)⟩=

⟨q · γᵀD−1γ · q + (q − µ) · σ−1DD−1Dσ−1 · (q − µ)

−2q · γᵀD−1Dσ−1 · (q − µ)⟩

= tr(γᵀD−1γ⟨qq⟩+ σ−1Dσ−1⟨(q − µ)(q − µ)⟩ − 2γᵀσ−1⟨(q − µ)q⟩

)= tr

(γᵀD−1γ(σ + µµ) + σ−1Dσ−1σ − 2γᵀσ−1σ

)= tr

(γᵀD−1γ(σ + µµ) + σ−1D − 2γ

). (C.23)

187

The (total) entropy flow rate is given by:

Sfl = ⟨Vt ·D−1 · F ′⟩=

⟨(−q · γᵀ + (q − µ) · σ−1 ·D

)·D−1 · (−γ · q)

⟩=

⟨q · γᵀD−1γ · q − (q − µ) · σ−1γ · q

⟩= tr

(γᵀD−1γ⟨qq⟩ − σ−1γ⟨q(q − µ)⟩

)= tr

(γᵀD−1γ(σ + µµ)− σ−1γσ

)= tr

(γᵀD−1γ(σ + µµ)− γ

). (C.24)

The adiabatic entropy production rate is calculated using Eq. (C.7):

Sad = ⟨Vs ·D−1 · Vs⟩=

⟨(−q · γᵀ + q · σ−1 ·D

)·D−1 ·

(−γ · q +D · σ−1 · q

)⟩= tr

((γᵀ − σ−1D)D−1(γ −Dσ−1)⟨qq⟩

)= tr

((γᵀD−1γ − γᵀσ−1 − σ−1γ + σ−1Dσ−1)⟨qq⟩

)= tr

((γᵀD−1γ − 2σ−1Dσ−1 + σ−1Dσ−1)⟨qq⟩

)= tr

((γᵀD−1γ − σ−1Dσ−1)(σ + µµ)

), (C.25)

where we have used Eq. (C.13). The non-adiabatic entropy production rate iscalculated using Eq. (C.9):

Sna = ⟨Vr ·D−1 · Vr⟩=

⟨(−q · σ−1 ·D + (q − µ)σ−1 ·D

)·D−1 ·

(−D · σ−1 · q

+D · σ−1 · (q − µ))⟩

=⟨q · σ−1Dσ−1 · q − 2q · σ−1Dσ−1 · (q − µ)

+(q − µ) · σ−1Dσ−1 · (q − µ)⟩

= tr(σ−1Dσ−1⟨qq⟩+ σ−1Dσ−1⟨(q − µ)(q − µ)⟩

−2σ−1Dσ−1⟨(q − µ)q⟩)

= tr(σ−1Dσ−1(σ + µµ) + σ−1Dσ−1σ − 2σ−1Dσ−1σ

)= tr

(σ−1Dσ−1(σ + µµ) + σ−1D − 2γ

), (C.26)

where we have used Eqs. (C.13) and (C.14).Thus we have derived all the explicit expressions of the thermodynamic quan-

tities in the set of thermodynamic (infodynamic) equations for OU processes of s-patially homogeneous systems with one mechanism, given by Eqs. (C.15)-(C.26).It is easy to verify that these expressions satisfy the set of non-equilibrium ther-modynamic equations.

188

Appendix D

Abstract Representation andRepresentation Transformation

This appendix is supplementary to Sec. 3.3.2 - Sec. 3.3.4 of the main text. Wefirst give the abstract representation using Dirac notation of the major equations inthe non-equilibrium thermodynamic formalism for spatially inhomogeneous sys-tems developed in Sec. 3.3.2 - Sec. 3.3.4 in the main text presented in the spaceconfiguration representation. Then we discuss how to transform the abstract rep-resentation into other representations and how to relate different representations.

D.1 Abstract RepresentationThere is a correspondence between results in the space configuration repre-

sentation and those in the abstract representation in the Dirac bra-ket notation.A space function ϕ(x) representing a state of the spatially inhomogeneous sys-tem in the space configuration representation is represented by a ket |ϕ) or a bra(ϕ| in the abstract representation, depending on its relation to other objects in thesame term. An integral kernel B(x, x ′) in the space configuration representationis represented by a linear operator B in the abstract representation. The function-al derivative δϕ(x) ≡ δ/δϕ(x) is represented by a ket |δϕ) or a bra (δϕ|, whichdoes not only have an algebraic character but also a differential character. Theintegral over space and the sum over the vector index in the space configurationrepresentation can be absorbed into the operation of linear operators acting on aket, or they can be absorbed into the inner product (bra-ket form), so that they notappear explicitly in the abstract representation. Using these prescriptions, we are

189

able to transform results in the space configuration representation into the abstractrepresentation.

Functional Langevin and Fokker-Planck Dynamics of SpatiallyInhomogeneous Systems

The functional Langevin equation, as a stochastic differential equation on theHilbert space, has the following form in the abstract representation:

d|ϕ) = |F (ϕ)) +∑s

|Gs(ϕ))dWs(t), (D.1)

where |F (ϕ)) is the deterministic driving force, |Gs(ϕ)) is the stochastic driv-ing force component from source s, and Ws(t) (s = 1, 2, ...) are independentone-dimensional standard Wiener processes. If there are multiple state transitionmechanisms indexed by m, Eq. (D.1) can be written more specifically as:

d|ϕ) =∑m

[|F (m)(ϕ)) +

∑s

|G(m)s (ϕ))dW (m)

s (t)

]. (D.2)

We assume that |F (m)(ϕ)) and |G(m)s (ϕ)) from the same mechanism m lie in the

same subspace Ω(m) of the entire Hilbert state space Ω. The projection from Ω intoΩ(m) is done through the projection operator Π(m). The corresponding functionalFokker-Planck equation is

∂

∂tPt(ϕ) = −(δϕ|

[|F (ϕ))Pt(ϕ)

]+ tr

(|δϕ)(δϕ|

[D(ϕ)Pt(ϕ)

]). (D.3)

We have a look at each of the two terms on the right side of the equation indi-vidually. In the first term, the algebraic character of (δϕ| acts on |F (ϕ)) forminga bra-ket, producing an algebraically scalar quantity. Yet its differential charac-ter acts not only on |F (ϕ)) but also on Pt(ϕ). That is why |F (ϕ)) and Pt(ϕ)are grouped together. In the second term we can regard |δϕ)(δϕ| as one objec-t constructed from |δϕ) and (δϕ|. As with |δϕ) or (δϕ|, it has both an algebraiccharacter and a differential character. Algebraically, |δϕ)(δϕ| is a linear operatorin the Hilbert space as D(ϕ) is. The product of two linear operators |δϕ)(δϕ| andD(ϕ) is again a linear operator. The notation tr() in this case is taking the traceof the product operator |δϕ)(δϕ|D(ϕ). Yet the differential character of |δϕ)(δϕ|does not only act on D(ϕ) but also on Pt(ϕ). Therefore D(ϕ) and Pt(ϕ) are also

190

grouped together. The drift vector (deterministic driving force) and the diffusionoperator in Eq. (D.3) are given respectively by contributions from each individualmechanism:

|F (ϕ)) =∑m

∣∣F (m)(ϕ)), (D.4)

D(ϕ) =∑m

D(m)(ϕ) =∑m

[1

2

∑s

∣∣G(m)s (ϕ)

) (G(m)

s (ϕ)∣∣] . (D.5)

By construction, D(ϕ) and D(m)(ϕ) are nonnegative real-valued self-adjoint op-erators in the Hilbert space Ω. They have the property that for any |φ) ∈ Ω,(φ|D(ϕ)|φ) ≥ 0 and (φ|D(m)(ϕ)|φ) ≥ 0. We require a stronger condition that forany |φ) ∈ Ω, (φ|D(ϕ)|φ) = 0 only when |φ) = 0. And for any |φ(m)) ∈ Ω(m),(φ(m)|D(m)(ϕ)|φ(m)) = 0 only when |φ(m)) = 0. The (left) inverse of D(ϕ) de-noted by D−1(ϕ) satisfies: D−1(ϕ)D(ϕ) = I . The (left) inverse of D(m)(ϕ) inthe space Ω(m) satisfies: [D(m)(ϕ)]−1D(m)(ϕ) = I(m), where I(m) is the identityoperator in Ω(m).

Potential-Flux Field Landscape of Spatially Inhomogeneous Sys-tems

The functional Fokker-Planck equation [Eq. (D.3)] can be written as a conti-nuity equation:

∂tPt(ϕ) = −(δϕ|Jt(ϕ)), (D.6)

where the transient flux is given by

|Jt(ϕ)) = |F ′(ϕ))Pt(ϕ)− D(ϕ)|δϕPt(ϕ)), (D.7)

with the effective drift vector defined as

|F ′(ϕ)) = |F (ϕ))−[(δϕ|D(ϕ)

]ᵀ. (D.8)

The notation ᵀ represents the operation of transpose, which is necessary to ensurethat this term is a ket as the other two terms in this equation. Note that we donot need to use the operation of conjugate transpose since we have assumed theHilbert space to be real. Equation (D.7) can be reformulated into the transientdynamical decomposition equation:

|F ′(ϕ)) = −D(ϕ)|δϕS(ϕ)) + |Vt(ϕ)), (D.9)

191

Non-equilibrium Thermodynamics of Spatially InhomogeneousSystems

The set of non-equilibrium thermodynamic equations is:

U = Sed − Sex

S = Spd − Sfl

A = Sed − Sna

Spd = Sad + Sna

Sfl = Sad + Sex

(D.18)

with the sign properties Sad ≥ 0, Sna ≥ 0, Spd ≥ 0 and the definitions U =⟨− lnPs⟩, S = ⟨− lnPt⟩, A = ⟨ln[Pt/Ps]⟩. For systems with one effective statetransition mechanism, the expressions of these thermodynamic quantities in theabstract representation are given by:

U = ⟨∂tU⟩+ ⟨(Vt|δϕU)⟩ (D.19)S = ⟨(Vt|δϕS)⟩ (D.20)A = ⟨∂tU⟩+ ⟨(Vt|δϕA)⟩ (D.21)

Sed = ⟨∂tU⟩ (D.22)Sex = −⟨(Vt|δϕU)⟩ (D.23)

Spd =⟨(Vt|D−1|Vt)

⟩(D.24)

Sfl =⟨(Vt|D−1|F ′)

⟩(D.25)

Sad =⟨(Vt|D−1|Vs)

⟩=⟨(Vs|D−1|Vs)

⟩(D.26)

Sna =⟨(Vt|D−1|Vr)

⟩= −⟨(Vt|δϕA)⟩

=⟨(Vr|D−1|Vr)

⟩= −⟨(Vr|δϕA)⟩ =

⟨(δϕA|D|δϕA)

⟩(D.27)

For systems with multiple state transition mechanisms, the expressions of the ther-modynamic quantities in the abstract representation are given by:

U = ⟨∂tU⟩+ ⟨(Vt|δϕU)⟩ = ⟨∂tU⟩+∑m

⟨(V

(m)t |δϕU)

⟩(D.28)

S = ⟨(Vt|δϕS)⟩ =∑m

⟨(V

(m)t |δϕS)

⟩(D.29)

193

A = ⟨∂tU⟩+ ⟨(Vt|δϕA)⟩ = ⟨∂tU⟩+∑m

⟨(V

(m)t |δϕA)

⟩(D.30)

Sed = ⟨∂tU⟩ (D.31)

Sex = −⟨(Vt|δϕU)⟩ = −∑m

⟨(V

(m)t |δϕU)

⟩(D.32)

Spd =∑m

⟨(V

(m)t |[D(m)]−1|V (m)

t )⟩

(D.33)

Sfl =∑m

⟨(V

(m)t |[D(m)]−1|F ′(m))

⟩(D.34)

Sad =∑m

⟨(V

(m)t |[D(m)]−1|V (m)

s )⟩=⟨(V (m)

s |[D(m)]−1|V (m)s )

⟩(D.35)

Sna =⟨(Vt|D−1|Vr)

⟩=∑m

⟨(V

(m)t |[D(m)]−1|V (m)

r )⟩

= −⟨(Vt|δϕA)⟩ = −∑m

⟨(V

(m)t |δϕA)

⟩=

⟨(Vr|D−1|Vr)

⟩=∑m

⟨(V (m)

r |[D(m)]−1|V (m)r )

⟩= −⟨(Vr|δϕA)⟩ = −

∑m

⟨(V (m)

r |δϕA)⟩

=⟨(δϕA|D|δϕA)

⟩=∑m

⟨(δϕA|D(m)|δϕA)

⟩(D.36)

D.2 Representation TransformationWe discuss how to transform the abstract representation into the space config-

uration representation and into the orthonormal basis representation |en) as wellas how to relate these two different representations. The mathematical formalismto be presented in the following has a close connection with the representationtransformation theory in quantum mechanics; we refer readers to textbooks ofquantum mechanics for more perspectives [159]. We will use examples from theabove subsections as illustrations.

194

From the abstract representation to the space configuration rep-resentation

A concrete representation is characterized by a basis of the Hilbert space. Thespace configuration representation is characterized by the space configuration ba-sis, which is written in the abstract representation as |a, x), with a discrete vec-tor index a and a continuous space index x. If the state of the system in the spaceconfiguration representation ϕ(x) has only one component (i.e., the vector ϕ is onedimensional), then the discrete index a will not be there and the space configura-tion basis is simply |x). Strictly speaking, |a, x) is not in the Hilbert space sinceits norm is divergent due to x being a continuous index. But such states can beaccommodated in an extended space of the Hilbert space. We shall not go into thetechnical details of this issue here. The orthonormality condition (normalized toDirac delta function) and the completeness condition for the space configurationbasis |a, x) are, respectively,

(a, x|b, x ′) = δabδ(x− x ′);

∫dx∑a

|a, x)(a, x| = I . (D.37)

The completeness condition is also called resolution of identity when viewedbackward, where the identity operator I is resolved into a sum and/or integralin the basis:

I =

∫dx∑a

|a, x)(a, x|, (D.38)

which is a very useful equation in facilitating representation transformations.A state of the system in the space configuration representation ϕ(x) is ob-

tained by projecting its abstract representation |ϕ) onto the space configurationbase vector (a, x|:

ϕa(x) = (a, x|ϕ). (D.39)

The space configuration representation of a linear operator as an integral kernelB(x, x ′) is obtained from its abstract representation B by projecting it onto thespace configuration base vectors on both sides:

Bab(x, x ′) = (a, x|B|b, x ′). (D.40)

The space configuration representation of other expressions and equations in theabstract representation can be obtained by projecting onto the basis |a, x) andinserting resolution of identity in appropriate places. For example, the space con-figuration representation of the equation |φ) = B|ϕ) is derived by projecting the

195

resentation given in the main text:

∂

∂tPt[ϕ] = −

∫dx∑a

δ

δϕa(x)

(F a(x)[ϕ]Pt[ϕ]

)+

∫∫dxdx ′

∑ab

δ2

δϕa(x)δϕb(x ′)

(Dab(x, x ′)[ϕ]Pt[ϕ]

).(D.42)

The space configuration representation of other expressions and equations given inthe main text can be obtained from the abstract representation in a similar fashion.

From the abstract representation to the orthonormal basis rep-resentation

Then we consider an orthonormal basis |en), with the index n discrete andgoing into infinity. One aspect in which this basis differs from the space configu-ration basis is that there is no continuous index. The orthonormality (normalizedto the Kronecker delta function) and completeness conditions of the basis |en)are

(en|em) = δnm;∑n

|en)(en| = I . (D.43)

The latter can be regarded as resolution of identity in the basis |en):

I =∑n

|en)(en|. (D.44)

A state |ϕ) in the basis |en) is represented by an infinite dimensional vectorϕn, with the component given by

ϕn = (en|ϕ). (D.45)

A linear operator B in the basis |en) is represented by an infinite dimensionalsquare matrix [Bmn], with the matrix element given by

Bmn = (em|B|en). (D.46)

The expressions and equations in the representation |en) can be obtained fromthe abstract representation by projecting onto the basis |en) and inserting reso-lution of identity Eq. (D.44) where appropriate. The equation |φ) = B|ϕ) in thebasis of |en) is derived as follows:

φm = (em|φ) = (em|B|ϕ) =∑n

(em|B|en)(en|ϕ) =∑n

Bmnϕn. (D.47)

197

The relation in Eq. (D.53) can be derived in another way. ϕn can be seen as afunctional of ϕ(x), given by:

ϕn = (en|ϕ) =∫

dx ′∑b

(en|b, x ′)(b, x ′|ϕ) =∫

dx ′∑b

ebn(x′)ϕb(x ′).

(D.55)Using the chain rule of differentiation, we have:

δ

δϕa(x)=

∑n

δϕn

δϕa(x)

∂

∂ϕn

=∑n

[δ

δϕa(x)

∫dx ′

∑b

ebn(x′)ϕb(x ′)

]∂

∂ϕn

=∑n

[∫dx ′

∑b

ebn(x′)δϕb(x ′)

δϕa(x)

]∂

∂ϕn

=∑n

[∫dx ′

∑b

ebn(x′)δabδ(x− x ′)

]∂

∂ϕn

=∑n

ean(x)∂

∂ϕn

. (D.56)

It is the same result as in Eq. (D.53), showing that the definitions (a, x|δϕ) =δ/δϕa(x) and (en|δϕ) = ∂/∂ϕn are consistent with each other. Plugging Eqs.(D.49) and (D.51) into Eq. (D.42) and using Eq. (D.54), it is easy to see that thefunctional Fokker-Planck equation in the space configuration representation inEq. (D.42) is transformed into its representation in the basis |en) in Eq. (D.48)directly, showing their equivalence explicitly. Reciprocally, we can transform thefunctional Fokker-Planck equation in the basis |en) into its space configurationrepresentation by plugging Eqs. (D.50) and (D.52) into Eq. (D.48) and using Eq.(D.53) to obtain Eq. (D.42).

Next we touch upon the integration measure on the Hilbert space in differentrepresentations. Consider integrating a (general) function f(ϕ) over the Hilbertspace, which is formally represented as

∫D(ϕ)f(ϕ) in the abstract representa-

tion. When working in specific representations (e.g., the space configuration rep-resentation |a, x) or the representation of |en)), the integral takes on differentforms but the result should be the same, i.e.,∫

D(ϕ)f(ϕ) =

∫D[ϕ(x)

]f[ϕ(x)

]=

∫D(ϕn)f(ϕn). (D.57)

200

If the value of the function f(ϕ) is the same in different representations and onlyϕ is replaced by its specific form in the corresponding representation, that is,

f(ϕ) = f[ϕ(x)

]= f(ϕn), (D.58)

then we also have the equality of the integration measures in different representa-tions:

D(ϕ) = D[ϕ(x)

]= D(ϕn), (D.59)

where in these equations ϕ, ϕ(x) and ϕn are related to each other by Eq. (D.55)(or Eq. (D.49) with F replaced by ϕ). Practically, the representation |en)seems to be easier to work with as it makes the generalization from finite di-mensional spaces to infinite dimensional spaces more direct and explicit. Thuswe can first work things out in the representation |en) and then transform intoother representations. For example, if we formally define the integration mea-sure in the representation |en) as D(ϕn) = Πn

∫dϕn, then the integration

measure in the space configuration representation can be defined indirectly asD[ϕ(x)] = D(ϕn) = Πn

∫dϕn, where ϕn is regarded as a functional of ϕ(x)

given by Eq. (D.55). This might avoid some issues involved in defining D[ϕ(x)] inthe space configuration representation directly by discretizing the physical spaceand then taking the continuum limit.

We mention that the entire formalism developed here in this appendix canalso accommodate the treatment of spatially homogeneous systems with finitedegrees of freedom. In that case, the state space is assumed to be a Euclideanspace, which is a finite-dimensional real Hilbert space. The state of the systemq = (q1, ..., qi, ..., qn) can be expressed in the Dirac bra-ket notation as qi = (ei|q),where |q) is the abstract representation of the state and |ei) is the base vector ofthe state space onto which it projects. The orthonormality and completeness con-ditions for the basis |ei) are formally the same as those in Eq. (D.43), with therestriction that the index i runs from 1 and n rather than going into infinity, sincethe state space is now finite-dimensional. The results for spatially homogeneoussystems can thus be recovered from the abstract representation. Therefore, thisformalism provides a unified language for the treatment of both spatially homo-geneous and inhomogeneous systems.

201

Appendix E

Ornstein-Uhlenbeck Processes forSpatially Inhomogeneous Systems

This appendix is supplementary to Sec. 3.3.5 of the main text. We give theabstract representation of the dynamical equations and the space configurationrepresentation of the thermodynamic expressions of OU processes of spatiallyinhomogeneous systems.

E.1 Dynamical Equations in the Abstract Represen-tation

We list in the following the compact form of some dynamical equations andexpressions for OU processes of spatially inhomogeneous systems in the abstractrepresentation using Dirac bra-ket notations. The functional Langevin equationfor OU processes reads as follows in Dirac notation:

d

dt|ϕ) = −γ|ϕ) + |ξ(t)), (E.1)

where −γ|ϕ) is the deterministic driving force |F (ϕ)) and |ξ(t)) is the stochasticdriving force with the following Gaussian white noise statistical properties

⟨|ξ(t))⟩ = 0, ⟨|ξ(t))(ξ(t′)|⟩ = 2Dδ(t− t′), (E.2)

where the diffusion operator D does not depend on ϕ. The corresponding func-tional Fokker-Planck equation is

d

dtPt(ϕ) = (δϕ|γ

[|ϕ)Pt(ϕ)

]+ (δϕ|D|δϕ)Pt(ϕ). (E.3)

202

Compare with the general equation given in Eq. (D.3). The fact that γ and D donot depend on ϕ allowed the form of the equation to be simplified a little bit. Thetransient probability functional is

Pt(ϕ) =1√

det(2πσ)exp

[−1

2(ϕ−K|σ−1|ϕ−K)

]. (E.4)

The stationary probability functional is

Ps(ϕ) =1√

det(2π ˆσ)exp

[−1

2(ϕ|ˆσ

−1|ϕ)]. (E.5)

The equations determining |K) and σ are:

d

dt|K) = −γ|K) (E.6)

d

dtσ = −γσ − γᵀσ + 2D. (E.7)

The equation determining ˆσ is

γ ˆσ + γᵀ ˆσ = 2D. (E.8)

The potential field landscapes are given by

U(ϕ) =1

2(ϕ|ˆσ

−1|ϕ) + 1

2tr(ln(2π ˆσ)

)(E.9)

S(ϕ) =1

2(ϕ−K|σ−1|ϕ−K) +

1

2tr (ln(2πσ)) (E.10)

A(ϕ) =1

2(ϕ|ˆσ

−1|ϕ)− 1

2(ϕ−K|σ−1|ϕ−K) +

1

2tr(ln ˆσ − ln σ

).(E.11)

Their functional gradients are

|δϕU(ϕ)) = ˆσ−1|ϕ) (E.12)

|δϕS(ϕ)) = σ−1|ϕ−K) (E.13)

|δϕA(ϕ)) = ˆσ−1|ϕ)− σ−1|ϕ−K). (E.14)

The flux velocity fields are given by

|Vs(ϕ)) = −γ|ϕ) + D ˆσ−1|ϕ) (E.15)

|Vt(ϕ)) = −γ|ϕ) + Dσ−1|ϕ−K) (E.16)

|Vr(ϕ)) = Dσ−1|ϕ−K)− D ˆσ−1|ϕ). (E.17)

203

E.2 Thermodynamic Expressions in the Space Con-figuration Representation

In the following, we give the explicit expressions of some thermodynamicquantities for OU processes of spatially inhomogeneous systems in the space con-figuration representation in the functional language. For comparison, we alsowrite down the compact form of these thermodynamic quantities in the abstractrepresentation using the Dirac notation. The expression in the space configurationrepresentation can be obtained by inserting, into appropriate places of the abstractrepresentation, resolution of identity in the space configuration basis:

I =

∫dx∑a

|a, x)(a, x|. (E.18)

For example, the rate of change of the transient entropy S of OU processes istransformed from the abstract representation into the space configuration repre-sentation as follows:


)(E.19)

= tr

((∫dx∑a

|a, x)(a, x|

)σ−1

(∫dx ′

∑b

|b, x ′)(b, x ′|

)D

−

(∫dx∑a

|a, x)(a, x|

)γ

)=

∫∫dxdx ′

∑ab

tr(|a, x)(a, x|σ−1|b, x ′)(b, x ′|D

)−

∫dx∑a

tr (|a, x)(a, x|γ)

=

∫∫dxdx ′

∑ab

(a, x|σ−1|b, x ′)(b, x ′|D|a, x)−∫

dx∑a

(a, x|γ|a, x)

=

∫∫dxdx ′

∑ab

[σ−1]ab

(x, x ′)Dba(x ′, x)−∫

dx∑a

γaa(x, x), (E.20)

where we have used the formula tr(|ϕ)(φ|B) = (φ|B|ϕ). The expression of otherthermodynamic quantities in the space configuration representation can be derived

204

similarly by inserting the identity in Eq. (E.18) and using that trace formula. Wejust list the results without going through the derivation again.

U = tr

(1

2(σ + |K)(K| − ˆσ)

d

dtˆσ−1

− ˆσ−1D ˆσ

−1(σ + |K)(K|) + γ

)(E.21)

=

∫∫dxdx ′

∑ab

1

2

[σab(x, x ′) +Ka(x)Kb(x ′)− σab(x, x ′)

]× d

dt

[σ−1]ba

(x ′, x)

−∫∫∫∫

dxdx ′dx ′′dx ′′′∑abcd

[σ−1]ab

(x, x ′)Dbc(x ′, x ′′)[σ−1]cd

(x ′′, x ′′′)

×(σda(x ′′′, x) +Kd(x ′′′)Ka(x)

)+

∫dx∑a

γaa(x, x) (E.22)

A = tr

(1

2(σ + |K)(K| − ˆσ)

d

dtˆσ−1

− ˆσ−1D ˆσ

−1(σ + |K)(K|)

− σ−1D + 2γ)

(E.23)

=

∫∫dxdx ′

∑ab

1

2


]× d

dt

[σ−1]ba

(x ′, x)

−∫∫∫∫

dxdx ′dx ′′dx ′′′∑abcd

[σ−1]ab

(x, x ′)Dbc(x ′, x ′′)[σ−1]cd

(x ′′, x ′′′)

×(σda(x ′′′, x) +Kd(x ′′′)Ka(x)

)−∫∫

dxdx ′∑ab

[σ−1]ab

(x, x ′)

× Dba(x ′, x) + 2

∫dx∑a

γaa(x, x) (E.24)

Sed = tr

(1

2

(σ + |K)(K| − ˆσ

) d

dtˆσ−1)

(E.25)

=

∫∫dxdx ′

∑ab

1

2


]× d

dt

[σ−1]ba

(x ′, x) (E.26)

205


−1(σ + |K)(K|)− γ

)(E.27)

=

∫∫∫∫dxdx ′dx ′′dx ′′′

∑abcd

[σ−1]ab

(x, x ′)Dbc(x ′, x ′′)

×[σ−1]cd

(x ′′, x ′′′)(σda(x ′′′, x) +Kd(x ′′′)Ka(x)

)−

∫dx∑a

γaa(x, x) (E.28)


)(E.29)

=

∫∫∫∫dxdx ′dx ′′dx ′′′

∑abcd

γba(x ′, x)[D−1

]bc(x ′, x ′′)γcd(x ′′, x ′′′)

×(σda(x ′′′, x) +Kd(x ′′′)Ka(x)

)−∫

dx∑a

γaa(x, x) (E.30)


)(E.31)

=

∫∫∫∫dxdx ′dx ′′dx ′′′

∑abcd

γba(x ′, x)[D−1

]bc(x ′, x ′′)γcd(x ′′, x ′′′)

×(σda(x ′′′, x) +Kd(x ′′′)Ka(x)

)+

∫∫dxdx ′

∑ab

[σ−1]ab

(x, x ′)

× Dba(x ′, x)− 2

∫dx∑a

γaa(x, x) (E.32)

Sad = tr((


−1)(σ + |K)(K|)

)(E.33)

=

∫∫∫∫dxdx ′dx ′′dx ′′′

∑abcd

γba(x ′, x)

[D−1

]bc(x ′, x ′′)γcd(x ′′, x ′′′)

−[σ−1]ab

(x, x ′)Dbc(x ′, x ′′)[σ−1]cd

(x ′′, x ′′′)

×(σda(x ′′′, x) +Kd(x ′′′)Ka(x)

)(E.34)

206


−1(σ + |K)(K|) + σ−1D − 2γ

)(E.35)

=

∫∫∫∫dxdx ′dx ′′dx ′′′

∑abcd

[σ−1]ab

(x, x ′)Dbc(x ′, x ′′)

×[σ−1]cd

(x ′′, x ′′′)(σda(x ′′′, x) +Kd(x ′′′)Ka(x)

)+

∫∫dxdx ′

∑ab

[σ−1]ab

(x, x ′)Dba(x ′, x)

− 2

∫dx∑a

γaa(x, x) (E.36)

207

Appendix F

Spatial Stochastic Neuronal Model

This appendix is supplementary to Sec. 3.3.6 of the main text. In the followingwe calculate the potential field landscapes, their functional gradients, and the fluxvelocity fields of the spatial stochastic neuronal model in the space configurationrepresentation in the basis of en(x).

Note that the abstract representation of en(x) using the Dirac notation is |en).en(x) can be obtained from its abstract representation |en) by projecting it onto thespace configuration basis |x), i.e., en(x) = (x|en). Therefore, the basis en(x)can also be abstractly represented as |en). We list the expressions of the vectorcomponent and matrix elements of the involved states and linear operators in thebasis |en), which was derived in the main text and will be used here:

γmn = (em|γ|en) = γnδmn, Dmn = (em|D|en) =δmn

2,

Kn = (en|K) = e−γntfn, σmn = (em|σ|en) =1− e−2γnt

2γnδmn,

σmn = (em|ˆσ|en) =δmn

2γn.

(F.1)We start from the expressions of the potential and flux velocity fields for OU

processes in the abstract representation given by Eqs. (E.9)-(E.17). Then we gointo the representation of |en) by inserting resolution of identity in the basis of|en):

∑n |en)(en| = I , into places where appropriate. The stationary potential

208

The relative potential field landscape is thus given by

A[ϕ] = U [ϕ]− S[ϕ]

=∞∑n=0

[(n2 + 1)ϕ2

n −(n2 + 1)

1− e−2(n2+1)t

(ϕn − fne

−(n2+1)t)2

− 1

2ln(1− e−2(n2+1)t

)].

(F.4)

Next we calculate their functional gradients. Starting from the abstract rep-resentation of the functional gradient of the stationary potential field landscapegiven in Eq. (E.12), we calculate its expression in the space configuration repre-sentation, but resolved in the basis of en(x):

δU

δϕ(x)= (x|δϕU(ϕ))

= (x|ˆσ−1|ϕ)

=∞∑n=0

(x|en)(en|ˆσ−1|ϕ)

=∞∑n=0

en(x)ϕn

σnn

=∞∑n=0

2(n2 + 1)ϕn en(x), (F.5)

where we have used en(x) = (x|en). Similarly, using Eq. (E.13) we calculate thefunctional gradient of the transient potential field landscape:

δS

δϕ(x)= (x|σ−1|ϕ−K)

=∞∑n=0

(x|en)(en|σ−1|ϕ−K)

=∞∑n=0

en(x)ϕn −Kn

σnn

=∞∑n=0

2(n2 + 1)

1− e−2(n2+1)t(ϕn − fne

−(n2+1)t) en(x). (F.6)

210

Thus the functional gradient of the relative potential field landscape is given by

δA

δϕ(x)=

δU

δϕ(x)− δS

δϕ(x)

=∞∑n=0

[2(n2 + 1)ϕn −

2(n2 + 1)

1− e−2(n2+1)t(ϕn − fne

−(n2+1)t)

]en(x). (F.7)

Then we calculate the flux velocity fields. Using Eq. (E.15), the stationaryflux velocity field is given by:

Vs(x)[ϕ] = (x|Vs(ϕ))

= −(x|γ|ϕ) + (x|D ˆσ−1|ϕ)

= −∞∑n=0

(x|en)(en|γ|ϕ) +∞∑n=0

(x|en)(en|D ˆσ−1|ϕ)

=∞∑n=0

en(x)

[−γnnϕn +

Dnn

σnn

ϕn

]=

∞∑n=0

[−(n2 + 1)ϕn + (n2 + 1)ϕn

]en(x) = 0. (F.8)

Similarly, using Eq. (E.16), we have the transient flux velocity field given by:

Vt(x)[ϕ] = −(x|γ|ϕ) + (x|Dσ−1|ϕ−K)

= −∞∑n=0

(x|en)(en|γ|ϕ) +∞∑n=0

(x|en)(en|Dσ−1|ϕ−K)

=∞∑n=0

en(x)

[−γnnϕn +

Dnn

σnn

(ϕn −Kn)

]=

∞∑n=0

[−(n2 + 1)ϕn +

(n2 + 1)

1− e−(n2+1)t(ϕn − fne

−(n2+1)t)

]en(x). (F.9)

Therefore, the relative flux velocity field is given by:

Vr(x)[ϕ] = Vt(x)[ϕ]− Vs(x)[ϕ]

=∞∑n=0

[−(n2 + 1)ϕn +

(n2 + 1)

1− e−(n2+1)t(ϕn − fne

−(n2+1)t)

]en(x). (F.10)

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potential and flux field landscape theory of spatially wei...

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