stochastic dynamics

7/22/2019 Stochastic Dynamics

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Effective stochastic dynamics in deterministic

systems: model problems and applications

by

Gil Ariel

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Mathematics

New York University

September 2006

Eric Vanden-EijndenAdvisor

roduced with permission of the copyright owner. Further reproduction prohibited without permission.


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UMI Number: 3234111

Copyright 2006 by

Ariel, Gil

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Gil Ariel

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to Orit, Guy, Mia,

and who ever may join us next

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Acknowledgements

First and foremost, I would like to thank my advisor, Eric Vanden-

Eijnden. Working with Eric, who has a rare skill to find the interesting

mathem atics in physical problems has been a rewarding experience. His

guidance is evident in every aspect of this work.

I would also like to thank the faculty at the Courant institute, in partic

ular Jonathan Goodman, Robert Kohn, Charles Newman, Ragu Varadhan

and Lai-Sang Young for always being willing to explain, talk and advise.

I am also grateful to my fellow students: An drea Barreiro, M atthias

Heymann, Jose Koiller, Stan Mintchev, Junyoep Park, Haiping Shen and

Paul Wright for making my time at Courant pleasant and helping with all

those little things we know we should know, but are nevertheless not quite

sure of.

Finally, I would like to than k my family. My wife, Orit, who followed

me to New York and now to Texas, and with great love built our new home.

Our son Guy, who grew to become a little boy and provided ample enjoy

able distractions, and last but not least Mia, who came just in time to be

included.

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Contents

Dedicat ion ........................................................................................................ iv

Acknowledgements ....................................................................................... v

List of Figures ................................................................................................. ix

List of T a b le s ................................................................................................. xi

1 Introd uction 1

2 The Kac-Zw anzig m odel 13

2.1 Introduction ......................................................................................... 13

2.2 Th e Kac-Zwanzig model: Formulation

and elementary propert ies ............................................................... 15

2.3 A stro ng limit th e o re m ...................................................................... 19

2.3.1 Introduction ............................................................................. 19

2.3.2 Main results and ex am ple s .................................................. 21

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2.3.3 Proof of the th e o re m ............................................................. 27

2.3.4 Conclusion and generalizations ......................................... 49

2.4 Testing Transition State Theory on Kac-Zwanzig Model . . . . 53

2.4.1 Introduction .............................................................................. 53

2.4.2 The m od el ................................................................................. 58

2.4.3 M etastability in K ac -Z w an zig ............................................. 61

2.4.4 Numerical ex pe rim en ts .......................................................... 65

2.4.5 Tran sition rate s for the limiting dynamics ......................... 70

2.4.6 Transition state th e o ry .......................................................... 82

2.4.7 Local dynamics around the hyperbolic point:

Why does VTS T works while naive TS T does not? . . 93

2.4.8 Concluding re m ar ks ................................................................. 96

3 Accelerated simulation of a heavy particle in a gas of elastic

spheres 101

3.1 Introduction ............................................................................................. 101

3.2 The m od el ................................................................................................. 105

3.3 Limiting dynam ics ................................................................................ 109

3.3.1 Ap proximate Markov nature of colloid-gascollisions . . 112

3.3.2 The rate of momentum tr a n s fe r............................................ 114

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3.3.3 Th e limiting equations of m o t i o n .........................................115

3.3.4 Improved a c c u ra c y ................................................................... 120

3.4 Th e numerical scheme ......................................................................... 121

3.4.1 The basic algorithm ................................................................ 123

3.4.2 Higher order accuracy ............................................................. 127

3.4.3 Some technical a s p e c ts ............................................................. 128

3.4.4 Pra ctical considerations ......................................................... 130

3.5 Numerical e x p erim en ts ..........................................................................131

3.5.1 Simulation m e th o d ................................................................... 131

3.5.2 Simulation re su lts ....................................................................... 133

3.6 Conclusion ................................................................................................. 134

Bibliography 138



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List of Figures

2.1 Comparison of the rand om noise, /y(f) (solid line), and the

limiting one (f) (dashed line). The frequencies are draw n

from a one sided exponential distribution and N ~ 2000. The

bath in itial cond itions are cano nical .............................................

2.2 A semi-log plot of the distrib ution of the waiting times between

transitions, P [ t s > s], for the case 7 = 10. In figure (a), the

sharp jump near the origin is due to rapid re-crossings of the

{xo = 0} plane. In figure (b), the statistic s of trans ition times

confirms the quasi-Markov hypothesis. The slope of the curve

is 1.1-10-4, the same as the average tran sition rate. The graph

diverges from the linear fit near the origin ..................................



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2.3 A semi-log plot of the joint dis tribu tion of successive waiting

time, -P[tqV + > s], for the case 7 = 10. As expected, th e

slope of the curve is 5 10- 5 which is about half the average

transition rate ........................................................................................... 1 0 0

3.1 A car icature of a colloid moving with velocity v. The large

sphere collides with all particles whose center of mass is within

a distance of R + r from its center. Hence, the colloid collides

with all particles in a volume of vdtn(R + r) 2 ................................. 108

3.2 Th e figure depic ts a situatio n in which subsequent collisions

be tween the colloid an d a part icula r gas pa rt icle carries some

memory: a slow moving particle approaches the colloid (a).

After this collision, the particle bounces off with almost the

same speed, only in the oppo site direction (b). After a second

collision with a fast moving particle (c), the colloid catches up

with the first particle and collides again (d) ..................................... I l l

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List of Tables

2.1 Com parison between tran sition rates with two different can

didate metastable sets.............................................................................. 67

2.2 Com parison between TS T predictions and simulation results

for small and large values of 7 ................................................................90

3.1 Sim ulation results for a system of iV = 40000 gas partic les

and inverse temperature ( 3 = 1 . The table compares the av

erage kinetic energy and diffusivity of the colloid as obtained

using our newly suggested, accelerated method, with values

predicted by th e lim iting OU process................................................. 134



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Chapter 1

Introduction

Deterministic dynamical systems, in particular, Hamiltonian systems, often

display very com plicated chaotic behav ior when th e number of degrees of free

dom in the systems is large. It is not surprising th a t the evo lution of certain

observables in these systems can be approximated by a stochastic process.

Results in this direction abound in the literature [29, 36, 42, 47, 77]. A typ

ical example is the extraction of conformational dynamics of biomolecules,

which is stochastic in nature even though the underling dynamics is Hamil

ton ian [31, 61]. In this con text, th e development of simple models in which

the emergence of the stochastic behavior out of the deterministic dynamics

can be analyzed rigorously is of importance.

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In this thesis we consider two different model examples in which a single

particle is coupled to a larger, man y pa rt icle system which plays th e role of

a heat ba th. The dynamics in both cases is deterministic. As a first step we

prove th a t taking a part icula r lim it, and un der appro priate in itia l conditions,

the dynamics of the single particle of interest is given by a solution to a

stochastic equation. Next, we apply the knowledge of the limiting equation

for the study of some properties of the deterministic dynamics.

The first example, presented in Chapter 2, considers a variant of the Kac-

Zwanzig model [24, 25, 76]. In th is model a single dis tinguish ed partic le is

moving in a given external potential and is coupled to N harmonic oscilla

tors via linear springs. The system is one-dimensional. The oscillators have

random frequencies that are chosen independently from a probability distri

bu tio n. In it ia l co nd itions are chosen rando mly from an inva rian t measure.

In Section 2.3 we prove that under certain sufficient conditions on the model

pa rameters , and in any fin ite time segment, th e traje cto ry of th e partic le

converges strongly (in L 2) to the solution of an effective stochastic equa

tion. The limiting equation has the form of a generalized Langevin equation

(GLE) and satisfies a fluctuation-dissipation relation. Strong convergence is

proved by map ping the in itial co nd itions of the fin ite dimensional system to

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the probability space on which the random noise term of the GLE is defined.

Bo th canonical and micro-canonical ensembles are considered. We also prove

convergence for a more general class of initial conditions that are absolutely

continuous with respect to the above two invariant measures.

Once the form of the limiting equation is established, it is used for study

ing properties of the d ynamics a t large, but finite system size. In the second

half of the chapter, the model is used as a benchmark problem for a rigorous

analysis of transition state theory (TST) and variational TST (VTST).

In the particular case in which the distinguished particle moves in a

double-well potential the dynamics is bistable. Amid the complexity of

individual trajectories, it is found that that most trajectories remain con

fined for very long periods of time in well separated regions of phase-space

whose boundary are loosely determined by the double-well potential, and

only switch from one region to ano ther occasionally. The confinement is due

to the presence of dynam ical bottlenecks between these regions. The system

is then said to display m etastability, and th e regions in which the trajec tories

remain confined are referred to as meta stable sets. Example of systems dis

playing m eta stability abound in natu re , with examples arising from physics

(phase transitions), chemistry (chemical reactions, conformation changes of

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bio-molecules), biology (switches in popula tio n dynamics), and many others.

In these systems, it is worth verifying whether the dynamics can be approx

imated by a Markov chain over the state-space of the metastable sets with

appropriate ra te constants. The m ain question of interest then becomes the

determination of these rate constants. This question is nontrivial because it

amounts to understanding the pathways by which the transitions between

the metastable sets occur. These pathways are usually non-trivial and com

plicated .

It is here that our knowledge of the limiting dynamics can be applied

by ca lculat ing th e ex ac t transit io n ra te s for th e lim iting equation. This is

done by spectral decomposition of the backward operator characterizing the

limiting stochastic equation. Eigenvalues and eigenfunctions are found using

the matched asymptotics method [60].

One of the earliest attempts to determine transition pathways and rate

con stan ts is tran sitio n sta te theo ry (TST ) [22, 35, 75]. Under the sole as

sumption that the dynamics of the system is ergodic with respect to some

known equilibrium distribution, TST gives the exact average frequency at

which trajectories cross a given hypersurface or hyperplane which separates

two metastable sets of interest. This average frequency can be used as a

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first approximation for the frequency of transition between the metastable

sets. Unfortunately, it was recognized early on th a t this approx ima tion can

be qui te po or , for not every crossing of th e dividing surface correspond s to

a transition between the m etastable sets. Indeed, the trajectories can cross

the dividing surface many times in the course of one transition . As a result,

the TST prediction for the frequency of transition always overestimates the

ac tua l frequency, sometimes grossly so. One way to minimize this problem is

to use the freedom in the choice of dividing surface. The best prediction for

the frequency from TST is the one corresponding to the dividing surface with

minimum crossing frequency. In variational transitio n state theory (VTST )

[35, 51, 65, 70] one minimizes the TST rate over some class of candidate

separating hyperplanes.

Unfortunately, VTST (just like TST) is an uncontrolled approximation,

for it only provides an upper bound on the transition frequency between

the m etastab le sets. In general, one does not know how sharp th is boun d is.

Other assumptions beyond TST and VTST are usually difficult to assess too.

Are successive transitions between the metastable sets well approximated by

Poisson events (i.e. statistically indepen dent and with exponentially dis

tributed waiting times) as required for the approximation of the dynamics

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by a Markov chain to hold? How does th is pro perty depend s on th e defin ition

of the m etastable sets? Etc.

The questions described above are first studied here through computer

simulations. M etastable sets are identified and it is shown tha t the dynam ics

is well app rox imated by a two-s tate Markov process over these sets. We also

examine the various assumptions required for a consistent use of TST in this

system.

We then turn to evaluate the transition rate using TST and VTST and

compare them to both numerical results and to transition rates obtained

from the limiting equations. We find th a t the VTST approxim ation for

the transition rate is exact, rather than just an upper bound, which is not

generally the case. We explain why the m ethod works for the c urrent model,

draw necessary conditions for the applicability of VT ST in the way used here,

and discuss the nature of spurious recrossings of the TST plane in light of

our findings.

The motivation for studying TST in the context of the Kac-Zwanzig

model is its simplicity, which allows a detailed , rigorous analysis. We have

shown that in this model, VTST given the exact rates of crossing between

the two me tastable sets. Unfortunately, th e general case is not as simple. In

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part icular, th e only reason why hy perplanes, ra th er th an more complicated

hypersurfaces were sufficient is the quadratic nature of the interaction with

the bath . This is obviously an artifact of this particu lar model. Because

the transition region is not localized at a saddle point, the VTST dividing

surface is in general a more complicated surface than the lift-up in phase

space of a hyperplane in configu ration space. It has been proved [64] tha t

VTST is always exact when minimizing over hypersurfaces in phase space.

The question of whether minimization over hypersurfaces in configuration

space (position only, not momentum) is sufficient is still open, and is of

much practical importance.

The theoretica l pa rt of proving stong convergence to the limiting equa tion

poses some inte rest ing op en problems and generalizations as well. The ra te

at which the solution of the finite system converges to the solution of the

limiting one depends exponentially on the time length considered. Hence,

it seems like stron g convergence is too strong . It would be interesting

to prove a weak type of convergence (in dis tribu tion) for all times. Such a

statement is not trivial since there is no clear separation between the time

scale for the dynamics of the distinguished particle and that of the bath.

The second example considered in this thesis, presented in Chapter 3,

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studies a system of elastic spheres. The dy namical theory of Brownian motion

has been the subject of extensive research both from the theoretical and

the num erical points of view. A particularly simplified model is th a t of

spheres interacting only through elastic collisions. In this model, a large

heavy spherical particle (colloid) is placed in a gas of smaller lighter ones.

Due to the collective effect of numerous collisions between colloid and gas,

the d isplacement an d velocity of the colloid seem erratic an d stochastic. It

has been proved [15, 34, 62, 63] that in the limit of an infinitely massive

colloid, and under specific scaling of time and space, the dynamics of the

colloid can be described by an Ornstein-Uhlenbeck (OU) process

t

dx(y) = v( t )dt

( i . i )

dv(t) = av{t)dt + V D d W ( t ) ,K

where x( t) and v(t) denote the position and velocity of the colloid, respec

tively, W(t) denotes the Wiener process, a > 0 is the viscosity and D > 0 is

the diffusion coefficient. Althoug h the limiting eq uation is known, the rate

of convergence is still undeterm ined. As a result, comparison between the

dynamics predicted by the limiting OU process and numerical simulations is

of importance.

Different attempts for simulating hard sphere systems with disparate

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masses can be found in the litera ture [2, 39], all of which amou nt to advancing

the system collision by collision [1, 27]. Th e main draw back of this metho d is

that due to the disparity between the masses of the two constituents, it takes

an extremely large number of colloid-gas collisions to produce a significant

change in the colloid position or velocity. In add ition, one also has to resolve

all collisions between gas partic les themselves. A simple estimation shows

that the required simulation time is proportional to the ratio between the

colloid and gas particle masses.

In the second chapter we present an alternative method for simulating

the system in three dimensions. Although it is stochastic in natu re, we

prove th a t th is method ap proximates th e dynamics of th e colloid with in a

calculated error. The efficiency of the me thod is indepen dent of the ra tio

be tween th e two masses, which const itutes a big improvem ent over th e full

simulation method . The a lgorithm is based on the heterogeneous multiscale

meth od [16, 18] and is co nstructed along the lines of [18, 23, 69]. Th e basic

idea is to calculate the drift, av, and diffusion terms, DdW/dt , on the fly

by simulating th e gas for a sh ort tim e segment At . Using these values the

colloid is advanced by a big time step As according to the limiting equation

(1 . 1 ).

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The key property of the system used in this Chapter is the gap between

the timescale on which the gas and the colloid evolve. Because the colloid

is so much heavier than gas particles, the gas will reach equilibrium before

x(s) and v(s) change considerably. This mo tivates an approximation of the

stochastic differential equation 1.1 by a forward Euler scheme with step size

As

(x i+i = xt+ ViAs

( 1.2 )

vi+i = Vi - aViAs + y/DAs{,i,

where are indepen dent random variables with normal distribu tion in M3.

The idea behind th e algorithm is to evaluate a and D in each step by making

a short simulation of the gas with fixed v V{. Each Euler step consists of

three parts:

1. Simulate the gas for a time segment of length A t while keeping the

velocity of the colloid is fixed.

2. Use the statistics o btained from pa rt 1 for evaluating a and D.

3. Move the colloid according to the forward Euler approximated equa

tions of motion (1 .2 ).

Denoting the ratio between the colloid and gas particle masses by E 1, it is

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shown that ifA tis 0 (1 ), i.e., independ ent of E, while As is proportio nal to E,

then the variance in the error introduced by this approximation is bounded.

Hence, the simulation is accelerated by a factor E. This is the m ain result of

Chapter 3.

The numerical experiments presented here are benchmark examples, sim

ple enough for demonstra ting th e ad vantag es of th e method. First , we calcu

lated the viscosity and diffusion coefficients of the colloid in the case E = 100.

Th e results deviate substan tially from values obtained by Du rr et. al. [15]

in the limit E >oo are found. The source of the errors are co rrections due

to correlations and the finite size of gas particles, not taken into account by

the limiting equation. A second example, in which E 104, demonstrates

that when E is large enough, the dynamics of the colloid is well described by

the limiting stochastic equation.

The new simulation method suggests numerous applications and general

izations in which the colloid dynamics is more complicated, and the limiting

equation is more difficult to ob tain or may no t be known. For instance, th e

case in which the colloid is not a sphere but an ellipsoid poses an enormous

com putation al challenge. The an gular mom entum of the colloid has to be

taken into account and calculating collision times is much more complicated.

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It is also interesting to investigate more thoroughly a system in which the

gas is in a dense regime. In this case the correlation function C{t\ 12) does

not decay exponentially, and a larger time step A t may be required.

Another interesting generalization is to the case of two or more colloids.

When the separation between two colloids is a few times the diameter of gas

pa rticles , th e collective effect of colloid-gas collisions is to pu sh th e colloids

closer toge ther [7, 45]. In other words, th e gas induces an effective close

range attra ction . Because the rang e of attra ctio n depend s on the size of gas

particles, it vanishes in th e lim it E > oo. This implies th a t th e lim iting

equations of motion of the two colloids is the same as for a single one (except

for elastic collisions between th e colloids). However, with a fixed bu t large E,

the dynamics is completely different. We expect to find th a t und er app ropri

ate scaling, the two colloids become effectively trapped in a metastable state

keeping them close for a long time. Hence, the limiting rate for the dynamics

may be the escape rate o ut of this state. This rate does not depend on E

expo nentially since fluctu ation s do not vanish even in the limit E oo.

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Chapter 2

The Kac-Zwanzig model

2.1 Introduction

In this Chapter we consider a variant of the Kac-Zwanzig model [24, 25, 76].

In this model a single distinguished particle is moving in a given external

potential and is coupled to N free one dimensional free particles via lin

ear springs. The pa rticles oscillate around th e distinguished one and have

random frequencies that are chosen independently from a probability distri

bution. In itia l co nd itions are chosen rand omly from an inva rian t measure.

In Section 2.2 we introduce the model in detail and derive the Hamiltonian

equation of motion.

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The re st of the chap ter is devided into two topics. In the first, Section 2.3,

we prove that with appropriate initial conditions, and in any finite time

segment, the trajectory of the particle converges strongly (in L2) to the

solution of an effective stochastic equation. The limiting equa tion has the

form of a generalized Langevin equation (GLE) and satisfies a fluctuation-

dissipation relation. Strong convergence is proved by map ping the initial

conditions of the finite dimensional system to the probability space on which

the rando m noise term of the GLE is defined. Bo th canonical and micro-

canonical ensembles are considered. We also prove convergence for a more

general class of initial conditions that are absolutely continuous with respect

to the above two invariant measures.

The second topic is an application of the limiting equation of motion for

the distiguished particle in a particu lar choice of model param eters. In Sec

tion 2.4 we use the Kaz-Zwanzig model as a b enchm ark problem for a rigorous

analysis of transition state theory (TST) and variational TST (VTST).

In a par ticula r case, in which the distinguished particle moves in a double

well potential the dynamics is bistable. We begin by studying metastab ility

in Kac-Zwanzig throu gh num erical simulations. In particula r, we show th at

by an appro priate choice of metastable sets th e dynamics of th e pa rtic le may

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be ap proximately described by a two state s Markov process. We th en use

our knowledge of the limiting equation to calculate transition rates for the

limiting equations. This is done by spectral decom position of the backward

ope rator characterizing the limiting stochastic equation. Eigenvalues and

eigenfunctions are found using matched a symptotics [60]. The numerical

and limiting results will serve in our analysis of the various assumptions

underlying TST , VSTS and th e predictions of these theories. We find th at

the VTST approximation for the transition rate is exact, rather than just an

upp er bound , which is not generally the case. We explain why the m ethod

works for the current model, draw necessary conditions for the applicability

of VT ST in the way used here, and discuss the n atu re of spurious recrossings

of the TST plane in light of our findings.

2.2 The Kac-Zwanzig model: Formulation

and elementary properties

The Kac-Zwanzig model is a system describing the evolution of a distin

guished particle with unit mass and position x 0, moving in an external po

tential V (x q) [24, 25, 58, 76]. Th e pa rticle is coupled by a harm onic p otentia l

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to a bath of N particles of mass m* > 0 with positions Zj, i = 1 , . . ., N. The

system is described by the Hamiltonian

# ( x , p) = i p i + v ( x 0) + 2 ^ - + Y l ^ X i ~ x ^ ^ t2-1)i = l 1 t = l

where, for short hand we use the vector notation x = (x0, X \ , - , x ^ ) , P =

(po.Pi, ,Pn)i and Pi is the momentum associated with X{. The coupling

constant between the distinguished particle and each oscillator in the bath

is 7/ N > 0. Th e scaling with N emphasizes the fact that the interaction is

weak. The equations of mo tion derived from this Ham iltonian are:

N

Xq = - V ' ( x q) - ~ - X i )

- 1 (2 .2)

X i = U J i { x0 - X i )

where

J - z k - (2'3)

We also assume that the frequencies {(^1) 1=1,.,.,jv are independent and iden

tically distrib uted (i.i.d.) rando m variables whose distribu tion is absolutely

continuous to the Lebesgue measure on M.

Here we proceed informally and postpone the mathematically rigorous

analysis to Section 2.3. Similar arguments can be found in [1 1 , 29, 32, 58].

In the derivation presented here we assume that initial conditions for the

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bath are dis tr ib uted according to th e canonica l ensemble, i.e., a t t = 0 bath

pa rticles are dis tr ib uted according to th eir equilib rium Gibbs measure which

is given by

Xi (0) = A f ( x 0(Q), N/ /3at i)

(2.4)

P i ( 0 ) = N ( Q , m i / P ) ,

where a 2)denotes the Gaussian distribution w ith mean m and variance


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In the limit N > oo, th e strong law of large numbers implies th a t for any

fixed t, R N(t) will converge to its average R(t)

R( t) = lim RnU) = 7 lim -J- V '' cosuit = yElcoswf]. (2.9)N00 N>OON

In order to ev aluate the rate of convergence, we calculate the second moment

of Rn. Breaking all double sums into the diagonal and off-diagonal pa rts

yields

E[RN( ti)RN(t2)] E[.Rjv(i)]E[i?/v(f2)] = ^ ( i v ) (2-10)

We conclude that the limiting equation describing the dynamics of the dis

tinguished particle has the form of a Generalized Langevin equation

rf 1x 0+ V'{ xQ) + 7 J R( t - T)x0(T)dT = (2 .1 1 )

where ^(t) is a G aussian process with zero mean a nd covariance function R( t) .

The equality between the memory kernel R(t) and the covariance function

of the random noise is a fluctuation-dissipation relation.

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2.3 A strong limit theorem

2.3.1 Introduction

Deterministic dynamical systems with a large number of degrees of freedom

typically display chaotic behavior and it is not surprising that the evolution

of certain observables in these systems can be approximated by a stochastic

process. Results in th is direction abound in th e li tera tu re . Most of these

results, however, are of weak convergence-type, i.e. it is shown th a t the

evolution of the observables tends to that of a stochastic process in some dis

tribu tiona l sense. It is more surprising th a t th e evolution of some observables

in deterministic dynamical systems converges pathwise to that of a stochas

tic process, since this requires relating the probability space on which this

stochastic process is defined to the pa ram eters in the system. The present

chapter offers a result in this direction.

Consider

N

m0x0+ f ( x Q)+ rriiUjf(xo - Xi) = 0

< R is a C 1 function, and {u i ,mi} i=i...at

are positive param eters. Eq uation (2.12) is the well-known system introduced

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by Ford, Kac, Mazur and Zwanzig [24, 25, 76] as a toy model to investigate

several issues in nonequilibrium sta tistic al mechanics. Kac-Zwanzig model

has been intensely investigated, both in the physics and mathematical liter

atures [11, 24, 25, 29, 32, 38, 40, 41, 42, 57, 58, 74, 76]. In pa rt icular it is

known that, with appropriate choices of /(), {w,, n and initial con

ditions, in the limit as N oo the traje cto ry of the distinguished particle,

{x0 (t),x 0(t )} , can be approximated by the solution of a integro-differential

equation with random coefficients, namely

m 0Xo + f { X Q) + J R ( t - T ) X 0(T)dT = (2.13)where 0 > 0 is a parameter, R : [0, oo) t>R is a ce rta in memory kernel and

: [0, oo) t>R is a Gaussian ran dom fu nct ion w ith mean zero and covariance

/?(). Th e typical results, however, are weak convergence theorem s. The

pu rpose of th is cha pter is to offer a strong er convergence re su lt, namely the

pa thwise convergence of {xo(t), x 0(t )} toward { X 0(t),Ao(t)} as N>oo.

The dynamics of a Hamiltonian system is given by a system of determin

istic differential equations. In many particle systems, solving these equations

is either impossible or imprac tical, bo th an alytically and numerically. How

ever, it is often the case that one is not interested in the dynamics of the

full system, b ut ra the r only in a small part of it. The rest of the system is

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referred to as a bath . It is then reasonable to look for effective equations

that describe the behavior of the smaller, interesting part, in the presence of

the ba th. We prove tha t if the b ath initial conditions are distributed accord

ing to the microcanonical invariant measure (and assuming some sufficient

conditions on the model parameters), then the solution of the equations of

motion for a finite N converges strongly (in L 2), in the limit N >oo, to the

solution of an effective stochastic integro-differential equation describing the

dynamics of the distinguished particle.

2.3.2 M ain results and exam ples

We study a variation of the well-known Ford, Kac, Mazur an d Zwanzig model

[24, 25, 76], in which a single - distinguished - one dimensiona l partic le is

coupled to a ba th ofNone dimensional free particles via harmo nic po tentials.

Similar models were previously studied from several different aspects [11, 24,

25, 29, 32, 38, 40, 41, 42, 57, 58, 74, 76]. As already n oted above, the model

is described by the Hamiltonian

where xo, Po and m 0 denote the position, momentum and mass of the dis

tinguished particle we are interested in describing, and ay, px and m, denote

^ { x i - x o ) 2, (2.14)

t=i

N

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the position, momentum and mass of the i-th bath oscillator, i = 1 . . . N .

For shorthand we use the notation x = (xq,Xi, . . . , xn ) and similarly for

writing vectors in MN+1. The disting uished particle is placed in an extern al

potential V{xq). Some restrictions on this potential that guaranty existence

of solutions are specified later. The coupling between th e pa rticle and each

oscillator is taken as harmonic, with spring constant 7/ N > 0. Th e scaling

with N emphasizes the fact that the interaction between the particle and

each one of the oscillators is weak. The equa tions of mo tion derived from

this Hamiltonian are (2.12) where

For fixed N , the Hamiltonian dynamics preserves the total energy and the

dynamics is restricted to an energy shell {H = E}. Since the to tal energy is

an extensive variable, we take E = Eq+ N//3 , where Eq= p%/(2mo) + V(x q)

is the initial energy of the distinguished particle in the absence of the bath.

The parameter 0 plays the role of an inverse tempera ture. We therefore make

the following assumptions regarding initial conditions:

Assumption la: The initial conditions of the distinguished particle

are fixed to (xo,Po).

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and for the bath,

Assumption lb: The initial conditions of the bath distributed ac

cording to the microcanonical equilibrium measure.

For fixed N this measure is given by

dtiNtE(x,P) = | ^ )r (2'16)

where | | is the s tanda rd Euc lidean norm in R2iV+2, dcr(x,p) denotes a surface

element (Lebesgue measure) on H (x , p) = E, and Z E is a normalization

constant.

Louivilles Theorem implies th at the Ham iltonian dynamics preserves vol

ume in phase space. Hence, the equilibrium distributio n of the m icrocanon

ical ensemble is invariant under the dynamics (2 .1 2 ).

We also make the following assumptions regarding the model parameters:

Assum ption 2: The bat h frequencies, {wj}i=i...Ar are independen t, iden

tically distrib uted (i.i.d.) rando m variables, with probability density

function (PDF) p(u>) with respect to (w.r.t.) the Lebesgue measure

on [0, oo). In add ition , p(u>) is strictly positive in its support and the

number of spectral gaps is finite, i.e., the support is a finite union of

segments.

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The assumption on the finite number of spectral gaps can be relaxed at the

cost of a slower convergence rate. Th is generalization is discussed in Section

2.3.3. In order to avoid confusion, exp ecta tions w.r.t. initial cond itions are

denoted Eo[-], while expec tations w.r.t. the frequencies are deno ted =

J[-]p(uj)duj. Assumption 2 can be relaxed to admit a more general case in

which both the coupling coefficient 7 and the PDF p(cu) depend on N , but

converge to a b ound ed integrable function. This general set up is discussed

in Section 2.3.4.

Our goal is to derive a stochastic equation that approximates the dy

namics of the distinguished particle for large N . The effective equation has

the form of a generalized Langevin equation (2.13), where R(t) = Ew[cos ut\

is a memo ry kernel and f() is a rando m noise. It is a Gaussian process

with zero mean and covariance given by a fluctuation-dissipation relation

Eo,u[(ti)(t2)] = R ( h 2)- If V'{%q) is uniformly Lipshit, then the solution

of (2.13) is well defined.

Denote by (xN(t),pN (t)) the full solution of the 2N+ 2 equations of mo

tion (2 .1 2 ), and by (xq (t),p^ ()) its projec tion on the coordinates describing

the d istinguished particle. Th e solution of the effective stochastic equation

(2.13) is denoted (X(t), P(t)). We prove the following strong convergence

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theorem which is the main result of this chapter

Theorem: Suppose assumptions 1 and 2 hold. Then, fo r any T ). For instance, if the tail of p(uj) is polynomial, C u~ q ) < D u ~ q, then I = (q l ) / (3 q1). I f the tail of p(u>) is exponential or

better, then Z= 1/3 . In the case of macrocanonical bath init ial condit ions,

the random noise (t) is a Gaussian process with zero mean and covariance

m t i ) s ( t2 ) = R { t i - t 2).

Note th a t th is rate is only an upper bo un d. Improved bo un ds may be ob

tain ed for specific cases. Th e pro of of strong convergence requires identifi

cation of a map between the probability space on which the random initial

conditions are defined, and the probability space on which the effective ran

dom noise is defined. Path -wise convergence of Xq (t) to xq(Z) is obtained

using the Gronwall inequality.

In particular, the theorem implies that the probability that trajectories

of the distinguished particle given by the finite system (2 .1 2 ) are far from

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that of the effective one (2.13) can be made arbitrarily small, i.e.,

lim PN oo

sup (t) X()| > eo< t < T

= 0 . (2.18)

Our results improve on previous works in several aspects. Fir st, we con

sider a wide range of model parameters rather than a particular choice of

PDF p(u>). Second, our initial conditions are drawn from the microcanon-

ical invariant measure, and not the canonical one. Since the H amiltonian

dynamics conserves energy, the microcanonical ensemble is the appropriate

one for this model. Finally, we prove a stron g type of convergence rath er

than convergence in distribution or probability. The proof is constructive in

the sense that it shows explicitly how the random initial conditions blend

into the noise. W ith arb itrarily large probability, every choice of (x(0), p(0))

that satisfies #(x (0), p(0)) = Eq + N/ (3 corresponds to a continuous func

tion () such that the solution of ( 2 .1 2 ) with initial conditions (x(0),p(0))

approximates the solution of (2.13) with this particular ().

It is interesting to note that due to pathwise convergence, the result of

the Theorem holds for all initial distributions for the bath that are absolutely

continuous with respect to the microcanonical one. Any initial distribution

on and { p ,} ^ , will induce a probability distribution on the noise

w(0- This, in turn, induces a probab ility distribu tion on the limiting process

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(t) = lini/v-xxj w(0- Th e limiting noise will generally no t be Gau ssian, and

will not satisfy a fluctuation-dissipation relation.

As an example, take P (u ) to be the one sided Lorentz distribution

P M = 2, ^ > 0 (2.19)7T OL U)

and p(u>)= 0 otherwise, where a > 0. In a previous pape r [5] we stud ied this

model both formally and numerically in the case when V(x0) is a double-well

potential an d th e dynamics is metastab le. In Section 2.3.3 we prove th a t th e

effective equation for x 0(t) is

m 0X 0 + V ' ( X 0) + 7 j \ ~ a't- T' X 0( r)dr = ~ ^ t ) , (2.20)

and (t) is an Ornstein-Uhlenbeck (OU) process at equilibrium with zero

drift and an exponentially decaying covariance with rate a.

2 .3 .3 Pr oof o f the theorem

Although the appropriate invariant measure for the Hamiltonian dynamics

(2 .1 2 ) is the microcano nical one, it is technically simpler to use the cano ni

cal invariant measure. Under the canonical ensemble, initial conditions are

indepen dent G aussian random variables. The ir distribution is given by the

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Gibbs measure

Xi = N ( x 0,N /pyi ) ,

( 2 .21 )

Pi = N(0,rrii /f3) , i = l . . . N ,

where N( fi, 1 / a 2) denotes the G aussian distribution with mean n and vari

ance a 2. Most auth ors studying such models consider only the canonical

ensemble [29, 32, 38, 40, 41, 42, 57, 58, 74]. We will first prove the Theo

rem for the case of canonical initial conditions and then prove that the same

result holds for the microcanonical one.

Canonical initial conditions

We begin with a few prelimina ry calculations. Using either variation of pa

rameters or the Laplace transform, (2.12) can be solved for x ^ .. . x N. Sub

stituting into the equation for x 0(t) and integrating by parts, the equation

for Xq (t) can be written as

(2 .22)

where

N

(2.23)

and /v(t) is given by

(2.24)

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Note th a t th e bath in itia l co nd itions appear only in ^ . For th is reason we

will refer to /v(t) as a ran dom noise term. Changing variables into dimen-

sionless, centered coordinates

IQ'Y

hi = V77 ~~

9i = ~ P*(0)>V rr i i

(2.25)

the noise term r(t) can be written as

I N= J ^ 2 (hi cosujit + gi sin LUit). (2.26)

1=1

The T heorem is proved in three steps. The first considers convergence

of the memory kernel i?/v(f). For fixed time t, the random variable cosu t is

bo unded. Hence, th e law of large nu mbers implies th a t Rn(I) converges to

its average for almost all u>

N

lim f?Ar(t) = lim -7- cosujt =7 ECi)cosu t R( t), (2.27)N-+O O N OO N /

1= 1

In Lemma 1 we prove that Rn(I) converges to R(t) strongly. The second

step considers the noise. In Lem ma 2 we prove th a t jv() converges strongly

to a limiting Gaussian process, (), with zero mean and covariance R( t) .

The final step considers the position and momentum of the distinguished

pa rticle. Stron g convergence of x ff(t) to x 0(t) and of (t) to p0(t) is proved

in Theorem 1. All steps consider a finite time interval 0 < t < T, T


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Initial conditions of the b ath have a Gibbs distribution. Hence, hi and i, and the definition of R(t),

(2.27), yields (2.28). Th e strong convergence is also an imm ediate conse

quence of Birkoffs ergodic theorem in L 2 [73].

A similar calculation shows that for large N,

sup E J i ? ( t ) - i ? ( t ) |4 < I ^ . (2.30)0 < t < T (V

This is used in the proof of Theorem 1.

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L e m m a 2 : For all T < oo, there exists a constant C independent of T , N

and p(u>) such that,

C T 2

sup E j ( f ) - 6 v ( * ) | 2 ^ (231)0 < t < T A

where (f) is a Gaussian process with zero mean and covariance R( t) . The

rate of convergence, I, is determined by the tail of the distribution ofp(u>) as

detailed before.

Proof: For fixed N and t, jv() is a linear combination of independent Gaus

sian rando m variables and is therefore Gaussian itself. Hence, () is also

Gaussian for fixed t. As a result, all marginals of the form ( (f i) ,. .. ,(&))

are Gaussian vectors, which implies th a t () is a Gaussian process. It is

interesting to note that (2.31) implies that jv() converges in distribution

to (). The ra te of this weaker convergence is always N ~ 1/2 and does not

depend on the tail of p(u>).

In order to prove strong convergence, we must identify how the random

initial conditions hi and gi blend into the limiting process. In other words,

both th e sequence of processes i( t) , 2^) , and the limiting process (t)

have to be defined on the same pro bability space. To achieve this we write

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the limiting process as

m = \ / 7 / p l/ 2(ui)[costutdhu+ sin u>tdgj\ , (2.32)Jo

where and gw are independent copies of the standard Brownian motion

(BM) on [0, oo). To see this is indeed (), we use the Ito iso metry to show

that the average of this process is zero and the covariance is

We now wish to define the initial conditions, hi and gx,on the same probabil

ity space of hMand gw. The difficulty is in finding a scaling tha t relates th e

random variables hi and ^ to parts of h^ and g^. We begin with a few defini

tions. Let a be a permutation of {1,.. . , N } that rearranges the frequencies

in increasing order, i.e., ui^i) ). For short

hand we denote p x = ipf(u>a(i)). We take the maps to b e increasing in the

sense that


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Th e rand om variables are defined similarly using gw instead of h^. The

motivation for this definition is that for any family of such maps, hi and gl

are independen t random v ariables with normal distribution. This is easily

verified using the Ito isometries. We now look for the scaling maps th a t

would yield strong convergence, and show that for all 0 < t < T,

for some I > 0. The constant C depends only on the density function p(u>).

A necessary condition is to have convergence at t 0. Th is simple case will

provide us with th e requ ired scaling. Substi tu ting in the representation s for

at and ,

where x a {u) denotes the indicator function of the set A evaluated at u>.

Using the Ito isometries yields

(2.35)

Eo.0,16v(0) - mi2=27 - 27 L V Pi-Pi-1- (2-37)

It is clear that we want to find a map ip such that

(2.38)

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Squaring this expression and using Jense ns inequality yields an up per bound:

^(^Ev^wn) < 1. (2.39)

Equality holds if all the terms Pj P,_i are equal, i.e., Pj = i / N . Hence,

strong convergence is obtained if

( d ' K w ) = P - 1 ( - 0 = P - \ ( 2 . 40)

rpr l K( i) = V N / p l,2{ u) dh u, (2.41)

Jp- \

where P 1denotes the inverse of the distribution function Pw{u < z). There

fore, initial conditions take the simple representation

rp,r

and similarly for gt .

For t > 0, we have, substituting in the representations for f, N, hi an d

9i

Eo.c m ) - ZN{t)I2 = 27 (1 - S N) (2.42)

with

x

5 iv = E 0iW / p 1 2(x) cos(xt)dhx + / p 1 2(x) sin (xt)dgxL v R Jr

N ( rpi X P E 1V ] / p 1/2(x)dhx cos(uaii)t) + / p 1/2(x )dg x sm(u


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Note th a t averaging with respec t to in itia l co nd itions implies averaging over

the two BM hx and gx, while averaging with respect to u implies averaging

over mu ltiple samples of Clearly, we need to show th a t Sn > 1 .

Using the Ito isometries yields

N f P r 1SV = y ~ ' E u, / p( x) cos( (x - u a(i))t)dx. (2.44)

i = i J p r - \

Bounding the cosine by 1 yields an up per bo und , Sn < 1. To get a lower

boun d, we show th a t for most i, (x oJa{i)) is small, and the cosine is almost

one. The reason why (x u>a(i)) is not small for all i is due to the tail of

the density p(uj). We therefore break the sum into two parts: up to kN and

above. For i = 1 . . . k^ , we use co sx > 1 x 2/2. Let u>k be such that

Pw[u> < u>k] kN/ N . For i = (kn + 1) N , we bou nd the cosine by 1.

Hence,

fcjv pP~l N rW 1I , p{x ) [ ^ ~ i.x - ^ a ( i ) f T2/ ^ ] d x - / p{x)dx

1 = 1 J p i - l i = k N + l ^ p i - \

(2.45)

p - 1

Noting th a t f p p(x)dx 1 /N yields

\SN - 1| < ~ P^ E^ X ~ u (i))2(lx+ 2 ( l (2-46)

Th e main difficulty is in evaluating the exp ecta tion E w(x tu^j))2. Noting

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that for x G [P ^ ),Pi *]

K , ( x ~ ^ ) 2 < 2 ( Pr ' - ^ - l ) 2 + 2EW( P - 1 - coa { i ) ) \ (2.47)

we proceed in several steps:

L e m m a 2 .1 : For all T < oo, there exists a constant C such that for all

i < kf j, and except fo r a possible fini te number of indices,

We begin with the second term of (2.47), which has a form similar to the

setu p of the Kolmogorov-Smirnov statistics. Bre iman [10] gives a proof of

the following Theorem:

Let U \ , . . . ,Un denote n independent samples from a random variable, un i

forml y dis tributed on [0,1]. Let P CT(i ), . . . , Ua(n) denote the same set of sam

ples arranged in increasing order. Then, the random variable

L > n = Vnm ax \Uaii) - i / n \ ,i < n

has a limiting distribution with finite variance.

This implies that,

(2.48)

Proof:

m ax E ( U. (2.49)

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Noting tha t P(u>lT ')) is uniformally distributed [0,1], i / n = P(Pi : )

P (v) > 0 for all u>< u>k and for k 1 , . . . , we have

e u p - 1 - u a{i))2 0j)(w ) ) ]2

Since p{u ) is strictly positive in its support, the minimum is bounded and

E ( / r UVW)2 < (2.51)

If there are sp ectral gaps then (2.51) fails whenever two subsequent samples

u>cr(i) and u>tr(i+i) are in different conectivity classes of the s upp ort ofp(u>). In

the latter case, which may happen in a finite set of indices, we have

(jEu)( i^ _1 - u a{i))2 < + largest gap. (2.52)

Substituting (2.51) and (2.52) into (2.46) yields,

+T ?


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for some I2> 0 that depend on l\.

Proof:

We prove Lemma 2.2 for different types of distributions p(u), and show

that the rate of convergence of t) to (t) depends on the tail of p(uj).

Suppose that the tail of the density function p(oj) decreases polynomially,

i.e.,

C u~ q < p{uj) < D u~ q, (2.55)

for some q > 1 and C, D > 0. We use C and D to denote generic constants

whose values may vary between expressions. Den ote l - k N/ N = N ~ l, I > 0.

Since p{u>) is strictly positive, then for large enough N we have,

Cu k q )COk] = / p(u})du> < D qdu = ----- 7^l~q.

J^k Jwk Q~ k

(2.57)

Using the opposite bound yields

C N h/(q-i) < U k


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Hence, for i < k^,

. r 'N\j = ( ' p(u>)dw > C(P- ' - P ~\ y i > C ( p r ' - p r_ \) N - W i -'), (2.59)v Jpr~\

which implies that

|P f1 - < N hqn* - l)- 1. (2.60)

Similar evaluations can be done with different tails. For instance, if p(u>)

decreases like e~ax, e~ax or has finite sup port th an th e rate is the sup over

all qs, I2= h 1 .

Combining the above bounds we find that with a w~ g tail, the optimal

rate is obtained with

With an exponential tail or better,

l = i . (2.62)

This concludes the proof of Lemma 2.

Figure 2.1 compares the random noise at() for N = 2000 obtained by

drawing random initial conditions hi and gi, with (i) obtained by sampling

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two BMs hw and th a t satisfy the con straints (2.41). The sample is pre

pared in th e following way: we first draw th e frequencies, {uJi}i=i,..N, and

then initial conditions hi and from the canonical distribution. For the fre

quencies we use the one sided exponential distribution p(u>) = e~w for u > 0 ,

and p(u) = 0 otherwise. We can then comp ute /v(t). The exponen tial distri

bu tion is convenient since th e an ti-de riva tiv e of e~u^2cosuit, which appears

in (2.41), is easily calcula ted. We the n app rox imate two BMs hu and that

satisfy the constraints (2.41) by a linear interpolation between the points

P _1(i/./V), i = 0 . . . N . This partition becomes finer as N >oo. Hence, the

approx imation converges to BM by the invariance principle. The limiting

noise (t) is then obtained through its representation given by (2.32). Since

the approximation is piece-wise linear, the integration w.r.t. dh^ and dg^

can be preformed analytically.

Theorem 1 : For all T < oo, there exists a constant C(T), independent of

N such that,

sup E0,w0< t < T

|xK(t ) - X ( t ) I2+ 1 ^ ( 0 - F ( ( )|2] < ^ 2 , (2.63)

where the rate of convergence, I, is the same as in Lemma 2.

This implies that Xq (t), the solution of the full equations of motion (2.12),

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2.5

0.5

- 0.5

- 1.520

t

Figure 2.1: Com parison of the rando m noise, jv(f) (solid line), and the

limiting one (t) (dashed line). The frequencies are drawn from a one sided

exponential distribution and N = 2000. The ba th initial conditions are

canonical.

converges strongly to X ( t ) , the solution of the limiting stochastic equation

(2.13).

Proof: Compare the solutions of the limiting equation for (X ,P):

X = P / m 0

t (2 -64 )

P = - f - V ( X ) - ^ f R ( t - T ) P ( T ) d T + - ; ( ( t ) ,m0 mlQJo v/3m0

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(2.65)

with the one obtained for finite N:

%o = P o / m o

Po = - V ' ( x Z ) - f R N(t - r )p g ( r )d r + N(t).m 0 rag J0 V/5m0

For shorthand, we drop the subscript zero from mo,XQ and Pq for the rest

of the section. We wish to show th at

sup E 0,a, [X (t) - ^ { t ) ] 2+ [P(t) ~ pN(t)]2 oo, one has to

consider the convergence of R ^ ( t ) , iv(t) and (x q (t ) ,p(t )) . However, since

jRiv(t) does not depend on initial cond itions, th e only difference between the

microcanonical ensemble and the canonical one considered in Section 2.3.3 is

the convergence of the noise, #() Once this is established, convergence of

Xq (t) follows by the exac t same arg um ent d etailed in the proof of Theorem 1,

(2.79)

The normalization Zis obtained by taking /( h i) = 1. S etting /( h i) = e lthl

yields the characteristic function of the marginal h\

(2.80)

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Section 2.3.3. In this Section we use the s up er sc rip tsm and c to distinguish

between micro- and canonical initia l co nd itions or noise.

In the case of Gaussian initial conditions, %(t) is trivially a Gaussian

process for any N. Hence, the sequence %(t) converges to a Gaussian pro

cess, c(f). Th e situa tion is not as simple with th e microcanonica l ensemble.

We recall the definition of jv, (2.26):

1 NG W = - j = ^ 2 y / T i ( h ^ c o s u } i t + g ^ s i n u i t) . (2.81)

' i = l

For fixed t, it is easy to see th at the characte ristic function of converge,

as IV > oo to the sam e charac teristic function in th e can onical case, c().

This is because contributions of order iV- 1 do not sum up to 0(1) due to

the N ~ 1/2 prefactor in (t). Hence, () converge in distr ibut ion to the

same limit c(f). The rat e is prop ortiona l to N ~ ^ 2. Th e same holds for

any linear com bination of marginals a t different times. This implies tha t the

whole process $() converges in distrib utio n to (), which is Gaussian. For

this reason, from now on we can drop the ensemble label from the limiting

noise ().

In order to prove strong convergence, we need to map the probability

space of the initial conditions hand gto the one on which () was defined:

the two BM hu and gu. The idea is to draw independent initial conditions

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according to the Gibbs distribution and th en scale them to the desired energy.

The dependence between different variables comes from the scaling factor.

Define

K ^ h y r ; g? = r f / r , ( 2 .8 2 )

where

= v / 2H / N =

\ 1 = 1Then, with hc{ and g\ given by (2.41), h' f and ghave the same distribution

as the microcanonical initial conditions hand g. We can therefore drop

the tilde notation. Substituting into (2.81) yields

( 0 = i& W - (2.84)

An elementary calculation shows that, in the limit N > oo, the average of

r te nds to 1 while the variance is 0 (N ~ 1). Hence, $() convergence to (t)

strongly. The dependence of the ra te on N is the same as in Sections 2.3.3.

To sum up, we have proved the following Theorem:

Theorem 2: For all T < oo, there exists a constant C(T), independent of

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N such that,

sup E 0)U0 < t < T

X, \

7 t = 7 If the product 'y(N,u>)pn{uj) converge in Li(ui) to a limiting

function p(u>), then the cosine transform ofp(uj) is bounded and continuous.

We denote R(t) = J p(u>) cosuitdu). Note tha t p(u/) may not be normalized.

Assuming that 7 (N,lo)pn(u>) converge also in L 2 (w), and that 7 ( N , u ) < C N

for some constant C > 0, it is easily shown that R E (t) converges to R(t) in

L 2 (u>). The proof is the same as in Lemma 1, however, the convergence rate

may be smaller. Once strong convergence of the covariance function is estab

lished, stron g convergence of the noise w(t ) and the trajectory ( x ^(t),p$ (t))

follows. For instance , S tu ar t and W arren [58] suggest the following example

Pn {u ) = t ^ X [ o , j v ] ( w )

N (2 .86 )

7(NM=7r a z + ur

where a, 7 > 0 and 0 < a


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(2.19).

If we remove the requirement for L\ (u>) convergence, then the model ad

mits a m uch larger variety of limiting processes. For instance, if y (N , cu)p^r(u)

is not uniformly bounded, then the covariance function R(t) is not necessar

ily bounded and continuous. Taking p(u>) = N a / n / ( N 2a -f- u 2) and 7j = N a,

yields, for any 0 < a < 1/2, i?jv(t) = N ae~N


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fortunately, it is often these long time scales, or even the asy mp totic behavior

of the pa rticle th a t are interesting. An example of this is a case in which

the external potential V(x0) is metastable [5]. However, it seems reasonable

that the process (x q,Pq) should converge to the limiting one weakly or in

distribution uniformly for all times T > 0. This is because the invariant

measure of the finite system should converge to the invariant measure of the

limiting one. This problem is beyond the scope of the present m anuscript

and will be trea ted in a later publication. A proof for a specific case similar

to (2.86) was given in [42].

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2.4 Testing Transition State Theory on Kac-

Zwanzig Model

2.4.1 Introduction

Deterministic dynamical systems, in particular, Hamiltonian systems, often

display very complicated chaotic behavior when the number of degrees of

freedom in the systems is large. Amid the complexity of individual traje c

tories, it is sometimes the case that most trajectories remain confined for

very long periods of time in well separated regions of phase-space and only

switch from one region to ano ther occasionally. The confinement is due to

the presence of dynam ical bottlenecks between these regions. The system is

then said to display metastability, and the regions in which the trajectories

remain confined are referred to as me tastable sets. Example of systems dis

playing m eta stabil ity abound in natu re , with exam ples ar ising from physics

(phase transitions), chemistry (chemical reactions, conformation changes of

bio-molecules), biology (switches in po pula tion dynamics), and many others.

In these systems, it is worth verifying whether the dynamics can be approx

imated by a Markov chain over the state-space of the metastable sets with

approp riate rate constants. The m ain question of interest then becomes the

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determ ination of these rate con stants. This question is nontrivial because it

amounts to understanding the pathways by which the transitions between

the m etastab le sets occur. These pathways are usually non-trivial and com

plicated.

One of the earliest attempts to determine transition pathways and rate

con stants is tran sition s tate theory (TST ) [22, 35, 75]. Under the sole as

sumption that the dynamics of the system is ergodic with respect to some

known equilibrium distribution, TST gives the exact average frequency at

which trajectories cross a given hypersurface or hyperplane which separates

two m etasta ble sets of interest. This average frequency can be used as a

first approximation for the frequency of transition between the metastable

sets. Unfortunately, it was recognized early on tha t this approxim ation can

be qu ite po or , for not every crossing of th e dividing surface corresponds to

a transition between the meta stable sets. Indeed, the trajectorie s can cross

the dividing surface many times in the course of one transition. As a result,

the TST prediction for the frequency of transition always overestimate the

actua l frequency, sometimes grossly so. One way to minimize this problem

is to use the freedom in the choice of dividing surface. The best prediction

for the frequency from TST is the one corresponding to the dividing surface

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with minimum crossing frequency. This idea is at th e core of the so-called

variational transition state theory (VTST) [35, 51, 65, 70], which aims at

identifying the dividing surface with minimum crossing rate which is the

lift-up in phase space of a surface defined in configuration space.

Unfortunately, VTST (just like TST) is an uncontrolled approximation,

for it only provides an upper bound on the transition frequency between

the m etastab le sets. In general, one does not know how sharp this b oun d is.

Other assumptions beyond TST and VTST are usually difficult to assess too.

Are successive transitions between the metastable sets well approximated by

Poisson events (i.e. statistically indepen dent and with exponentially dis

tributed waiting times) as required for the approximation of the dynamics

by a Markov chain to hold? How does th is pro perty dep en ds on th e definition

of the metastable sets? Etc.

In this section we study a benchmark problem, which is simple enough

so that many of the assumptions and approximations underlying TST and

VT ST can be examined. Bu t the m odel is also complex enough to display a

wide variety of phenomena common to many dynamical systems exhibiting

me tastability. The problem we consider is a variant of a model originally pro

posed by Ford, Kac, and Mazur [24, 25] and Zwanzig [76]. It was revisited

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in the context of transition rates in [13, 30, 52, 53, 54, 55] and more re

cently, from a more analytical point of view in [11, 29, 32, 36, 40, 41, 42, 58].

The Kac-Zwanzig model is a Hamiltonian system describing the evolution

of a distinguished particle moving in a double-well external potential and

coupled to a bath of N free particles via linear springs. The dynamics in

this model is deterministic. However with app rop riate choice of the pa ram e

ters, the evolution of the distinguished particle can be captured by a closed

stochastic differential equation in the limit of infinite bath, N >oo. W ith

the double-well external potential, the model displays metastability over two

sets (bistability). Tran sition rate con stants between these sets can be com

puted exac tly from th e effective stocha stic dynamics in th e lim it N oo.

The values for these exact transition rate constants can then be compared

to the predictions of TST and V TST. This comparison is the m ain objective

of this paper. In particular , we will show tha t the application of TS T w ith a

naive (but natural) choice of dividing surface based only on the position of

the distinguished particle leads to a wrong prediction for the transition rate

constants. This is because the naive dividing surface is crossed many times

in the course of each transition between the two me tastable sets. However,

we will also show that if one optimizes over the dividing surface following

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VT ST, these many spurious crossings can be eliminated completely. Hence,

the correct transition rate constants for the model can be computed within

VTST. The optimal dividing surface which allows one to do so is then a

surface whose normal spans all the configurational degrees of freedom in the

system and n ot only the one associated with the distinguished particle. We

shall try to explain why this is the case and when a similar success of VTST

can be expected in other more realistic systems.

In [52], Poliak et al. app rox imate a generalized Langevin equation , which

has the same form of the limiting equation in our case, by the Hamiltonian

dynamics of the Kac-Zwanzig model, and then use TST for obtaining the

escape rates. In [54, 55], they ob tain th e same rate s from the limiting equa

tion by a generalization of Kram ers meth od [28]. Although our results are

similar, th e po int of view is different. Here, the Kac-Zwanzig model is used

as a platform for analyzing the predictions of TST and VTST and testing

the und erlying assumptions of these theories. Our results are also all de

rived from basic principles, and rely on the only (uncontrolled) assumption

of ergodicity.

The rem ainder of this section is organ ized as follows. In Section 2.4.2

we present the model and derive some of its basic properties. Section 2.4.4

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details numerical experiments, discusses some of the properties of the sys

tem conjectured in Section 2.4.2 and presents results for the transition rate.

In Section 2.4.5 we derive the effective stochastic differential equation that

describes the dynamics of the distinguished particle in the limit TV > oo.

We then calculate trans ition ra tes for the limiting dynamics. This is done by

spectral decomposition of the backward operator of the stochastic equations.

Eigenvalues and eigenfunctions are obtaine d using matched asym ptotics. In

Section 2.4.6 we develop TST and VTST. We find predictions for the tran

sition rates from these theories. Finally, in Section 2.4.8 we summarize our

findings and discuss possible generalizations.

2.4 .2 Th e m odel

In order to simplify the discussion, for the rest of this section we will restrict

our analysis to the following particular choice of probabilty density for the

frequencies

p(u) = * 1 y r ~ 2 if w > o

tt 0,2 + ^ 2 (2.87)

0 otherwise

where o,* > 0 is a parameter playing the role of a characteristic frequency.

Unless sta ted otherwise, we will take o,* = 1. Notice th a t all the moments

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of (2.87) are infinite, i.e. u>* ^ E ul = oo, where E denotes expectation with

respect to (2.87). Hence, using the no tation of the previous sections

R(t) = qEfcoswf] = 7 e- ^. (2.88)

In section 2.4.5, we proved that if we choose initial conditions (x(0), p(0))

so that

E = N/P, (2.89)

for some P > 0 playing the role of an inverse temperature, then in the

limit N >oo, the evolution of the distinguished particle is described by the

stochastic equations

f t iX o + V ' { x o ) + 7 j e~(t~T)x0(T)dr= (2-90)

where (t) is a G aussian process with zero mean a nd covariance function

7 e- ^. Hence, the noise is an Ornstein-Uhlenbeck process at equilibrium

which solves the stochastic differential equation

d= - d t + x/27 dWt , C(O) = A7(0,1 ), (2.91)

where Wt isa stan da rd Brow nian motion. Using (2.90) and (2.91), the lim-

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iting equation (2.90) can be written as [29, 58]:

xq = Po

* Po = s / i s - V \ x q) (2.92)

s = - s - ^ p Q+y/W~mt

Hence, with this particu lar choice of model parameters th e limiting dynamics

satisfy a stochastic differential equation.

In Section 2.4.5 we will find the following form of (2.92) useful:

Ldt

( \Xq

Po

\ S /

= -K X 7H (x 0, po, s) + x / ^ W t.

Here H ( x q , P q , s ) = V(x0)+ \ p l+ | s 2,

/ \0

(2.93)

(2.94)

V 1 /

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and K= K s+ K A, with

/0 0 0

K s = aaT = 0 0 0

V0 0 1

/(2.95)

0 - 1 0

K A = 1 0 -V 7

2.4.3 M etastabi l ity in Kac-Zwanzig

In order to study metastability, the distinguished particle is placed in a

double-well poten tial. Unless state d o ther wise, for the rest of this Section

we take

When the inverse temperature (3 and the size of the bath N are large

enough and satisfy 1


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instance be taken as

S - ( N , P , 5 ) = {(x, p) :H(x , p) = N/(3,x0 < 0, and H 0{x0,po) < 5}(2.97)

S+(N,0 ,6) = {(x,p) :H (x, p) = N/(3,x0 > 0, and H 0{ x0,Po) oo. (2.98)

Here (Xe is the microcanonical distribution on H(x , p) = E,

(2.99)

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element (Lebesgue measure) on H(x , p) = E, and 2 ( E ) is a normalization

constan t. The ergodic assumption can not be proved rigorously for the po

tential in (2.96), but it will be corroborated by the numerical experiments

presen ted in section 2.4.41.

The key property implying metastability over the sets in (2.97) is that

for every 5 E (0,1], and similarly for the integral over S -( N , (3,6). Note that

the o rder in which the limits are taken m atters. The validity of (2.100) can

be checked by direct ca lculation. Indeed, performing first th e in tegrat ion

over x \ , . . . , xn and p i , . . . , p we have

where Z 0(N/ /3) = f Ho


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stant. In the limit as Noo, this is

lim [ dtiE=N/0(X;P) = Z q 1 [ e~/3Hodx0dp0 (2 .1 0 2 )N ^ J S + ( N , 0 , 8 ) J H 0< 6

where Zq = f R2 e~^Hodx0dp0. The B oltzmann-Gibbs probability density func

tion ZQ1e~/3H is in fact the marginal density for the position and momentum

of the distinguished particle in the limit of infinite bath, N oo. For every

5G (0 , 1], the minimum at (x0,po) = (1 , 0 ) is the only minimum that belongs

to the domain where H 0 < 8. Therefore, (2.100) follows from (2 .1 0 2 ) by

simple evaluation of this integral by Laplace method. It is im po rtan t to note

that S are cylindrical sets in R27V+ 2 and n ot small neighborhoods around of

the energy minima. Due to the high dimensionality of the model, the mass

of the equilibrium measure is not concentrated in a small volume of phase

space since the Laplace approximation does not hold when N > (3.

Equation (2.100) implies that, when 1 -C j3 N , any generic trajectory

solution of (2.2) spends most of its time in either S+(N, (3,8) or S- (N , (3,8).

However, under the ergodicity assumption, this trajectory must switch be

tween S+(N,(3,8) and S-(N,(3,8) infinitely often. W hat are the rate con

stants of these transitions ? How do they depend on 81 Are they statistically

independent, with transitio n events Poisson distributed? In othe r words, can

the dynamics in (2.2) be reduced to a Markov process over S+(N,(3,8) and

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S-(N, /3 ,5) for some suitable choice of 6? These are the questions which we

shall investigate in the remainder of this paper, first via a series of numer

ical experiments with (2.2) (section 2.4.4), then using the limiting equation

in (2.92) (section 2.4.5), and finally within TST and VTST (section 2.4.6).

2.4 .4 N um erical experim ents

In this section, we perform a series of numerical experiments with (2.2) to

investigate when the dynamics can be app roxim ated by a Markov process over

the two m etastab le sets in (2.97). The questions we are especially interested

in are:

1 . W hat are the rate of the transition? How do they depend on the

para m ete r 7 (interaction strength with the bath) in the model? How

do they depend on the choice of 5 in (2.97)?

2. Are successive transitions to a good approximation statistically inde

pendent? Are th e transit io n tim es in th e sets (2.97) Poisson dis tr ibuted

with intensity equal to the rate of transition? How do these properties

depend on 5?

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The second question is especially important since it determines when the

dynamics in (2.2) can be approximated by a Markov process, and how the

metastable sets have to be chosen in this case to get the correct transition

rate constants to use in the chain.

For these experiments, we will take N = 1000 and /? = 7. We will

also consider two different values of 7: 7 = 1 and 7 = 10. In (2.2), the

equations of motion of the bath are integrated explicitly, while the equations

of motion describing the distinguished particle are integrated numerically

using the Verlet algo rithm [71]. Each time step is made reversible by a

Tro tter expansion of the time evolution ope rator [27, 67]. In all the results

reported below we use the double-well potential (2.96), but the integration

scheme was also checked using the harmonic potential, V(xq) ^Xq, for

which (2.2) can be integra ted analytically. Initial conditions are chosen once

from the microcanonical invariant distribution on the energy shell E = N/ /?.

The integration is performed up to time T = 2 106. T he param eters N , T

and the step size are chosen so that further increase (in Nand T) or decrease

(in step size) does not change the average rates considerably. Transition ra tes

are obtained by counting the number of times the trajectory switches between

S + and S - . Table 1 deta ils resu lts obtained for


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7 5 = 1 (5 ==0 .1

1 4.1 1 0 4 3.7 1 0 ~ 4

1 0 3.5 1 0 ~ 4 1 .1 1 0 ~ 4

Table 2.1: Com parison between tran sition ra tes with two different candida te

metastable sets.

5 = 0.1 is arb itrary . Any choice of 0.05 <


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between successive transi tio ns, we ex pect th a t it will have an expo nen tia l

distribution,

P[tS> s ] = e ~ k s , (2.103)

for some rate constant k,which is also the average transitio n rate. Figure 2.2

depicts the distribution of the waiting times between transitions, P [ts > s],

on a semi-log plot for the case 7 = 10. For 5 = 1 , the graph is not linear

near the origin. However, for 5 = 0.1, the linear fit is very good, indicating

th a t transitio ns events have

stochastic dynamics

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