please scroll down for articleabhi/papers/45-advinp.pdfdeveloped approach, using generalized...

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [Dhar, Abhishek]On: 3 December 2008Access details: Access Details: [subscription number 906332057]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Advances in PhysicsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713736250

Heat transport in low-dimensional systemsAbhishek Dhar aa Raman Research Institute, Bangalore 560080, India

Online Publication Date: 01 September 2008

To cite this Article Dhar, Abhishek(2008)'Heat transport in low-dimensional systems',Advances in Physics,57:5,457 — 537

To link to this Article: DOI: 10.1080/00018730802538522

URL: http://dx.doi.org/10.1080/00018730802538522

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713736250http://dx.doi.org/10.1080/00018730802538522http://www.informaworld.com/terms-and-conditions-of-access.pdf

Advances in Physics

Vol. 57, No. 5, September–October 2008, 457–537

Heat transport in low-dimensional systems

Abhishek Dhar*

Raman Research Institute, Bangalore 560080, India

(Received 22 August 2008; final version received 10 October 2008)

Recent results on theoretical studies of heat conduction in low-dimensionalsystems are presented. These studies are on simple, yet non-trivial, models.Most of these are classical systems, but some quantum-mechanical work is alsoreported. Much of the work has been on lattice models corresponding tophononic systems, and some on hard-particle and hard-disc systems. A recentlydeveloped approach, using generalized Langevin equations and phonon Green’sfunctions, is explained and several applications to harmonic systems are given.For interacting systems, various analytic approaches based on the Green–Kuboformula are described, and their predictions are compared with the latest resultsfrom simulation. These results indicate that for momentum-conserving systems,transport is anomalous in one and two dimensions, and the thermal conductivity� diverges with system size L as ��L�. For one-dimensional interacting systemsthere is strong numerical evidence for a universal exponent �¼ 1/3, but there is noexact proof for this so far. A brief discussion of some of the experiments on heatconduction in nanowires and nanotubes is also given.

Keywords: heat conduction; low-dimensional systems; anomalous transport

Contents page1. Introduction 4582. Methods 460

2.1. Heat bath models, definitions of current, temperature and conductivity 4602.2. Green–Kubo formula 4672.3. Non-equilibrium Green’s function method 471

3. Heat conduction in harmonic lattices 4723.1. The RLL method 4733.2. Langevin equations and Green’s function formalism 4753.3. Ordered harmonic lattices 480

3.3.1. One-dimensional case 4813.3.2. Higher dimensions 483

3.4. Disordered harmonic lattices 4843.4.1. One-dimensional disordered lattice 4853.4.2. Two-dimensional disordered harmonic lattice 489

*Email: [email protected]

ISSN 0001–8732 print/ISSN 1460–6976 online

� 2008 Taylor & FrancisDOI: 10.1080/00018730802538522

http://www.informaworld.com

Downloaded By: [Dhar, Abhishek] At: 18:24 3 December 2008

3.5. Harmonic lattices with self-consistent reservoirs 4914. Interacting systems in one dimension 494

4.1. Analytic results 4954.1.1. Hydrodynamic equations and renormalization group theory 4954.1.2. Mode coupling theory 4964.1.3. Kinetic and Peierls–Boltzmann theory 4984.1.4. Exactly solvable model 500

4.2. Results from simulation 5014.2.1. Momentum-conserving models 5014.2.2. Momentum-non-conserving models 511

5. Systems with disorder and interactions 5146. Interacting systems in two dimensions 5197. Non-interacting non-integrable systems 5258. Experiments 5269. Concluding remarks 530Acknowledgements 532References 532

1. Introduction

It is now about 200 years since Fourier first proposed the law of heat conduction thatcarries his name. Consider a macroscopic system subjected to different temperatures at itsboundaries. One assumes that it is possible to have a coarse-grained description witha clear separation between microscopic and macroscopic scales. If this is achieved, it isthen possible to define, at any spatial point x in the system and at time t, a localtemperature field T(x, t) which varies slowly both in space and time (compared withmicroscopic scales). One then expects heat currents to flow inside the system and Fourierargued that the local heat current density J(x, t) is given by

Jðx, tÞ ¼ ��rTðx, tÞ, ð1Þ

where � is the thermal conductivity of the system. If u(x, t) represents the localenergy density, then this satisfies the continuity equation @u/@tþr �J¼ 0. Using therelation @u/@T¼ c, where c is the specific heat per unit volume, leads to the heat diffusionequation:

@Tðx, tÞ@t

¼ 1cr � ½�rTðx, tÞ�: ð2Þ

Thus, Fourier’s law implies diffusive transfer of energy. In terms of a microscopic picture,this can be understood in terms of the motion of the heat carriers, i.e. molecules, electrons,lattice vibrations (phonons), etc., which suffer random collisions and hence movediffusively. Fourier’s law is a phenomological law and has been enormously successfulin providing an accurate description of heat transport phenomena as observed inexperimental systems. However, there is no rigorous derivation of this law starting from

458 A. Dhar


a microscopic Hamiltonian description and this basic problem has motivated a largenumber of studies on heat conduction in model systems. One important and somewhatsurprising conclusion that emerges from these studies is that Fourier’s law is probablynot valid in one- and two-dimensional systems, except when the system is attached toan external substrate potential. For three-dimensional systems, one expects that Fourier’slaw is true in generic models, but it is not yet known as to what are the necessaryconditions.

Since one is addressing a conceptual issue it makes sense to start by looking at thesimplest models which incorporate the important features that one believes are necessaryto see normal transport. For example, one expects that for a solid, anharmonicity anddisorder play important roles in determining heat transport properties. Thus, most of thetheoretical studies have been on these simple models, rather than on detailed modelsincluding realistic interparticle potentials, etc. The hope is that the simple models capturethe important physics, and understanding them in detail is the first step towardsunderstanding more realistic models. In this review we almost exclusively talk aboutsimple models of heat conduction in low-dimensional systems, mostly one-dimensionaland some two-dimensional systems. Also a lot of the models that have been studied arelattice models, where heat is transported by phonons, and are relevant for understandingheat conduction in electrically insulating materials. Some work on hard particle and harddisc systems is also reviewed.

There are two very good earlier review articles on this topic: those by Bonetto et al. [1]and Lepri et al. [2]. Some areas that have not been covered in much detail herecan be found in those reviews. Another good review, which also gives somehistoric perspective, is that by Jackson [3]. Apart from being an update on the olderreviews, one area which has been covered extensively in this review is the use of thenon-equilibrium Green’s function approach for harmonic systems. This approach nicelyshows the connection between results from various studies on heat transportin classical harmonic chain models, and results obtained from methods such asthe Landauer formalism, which is widely used in mesoscopic physics. As we will see,this is one of the few methods where explicit results can also be obtained for thequantum case.

The article is organized as follows. In Section 2, some basic definitionsand a description of some of the methods used in transport studies are given.In Section 3, results for the harmonic lattice are given. The non-equilibrium Green’sfunction theory is developed using the Langevin equation approach andvarious applications of this method are described. The case of interacting particles(non-harmonic interparticle interactions) in one dimension is treated in Section 4.In this section we also briefly summarize the analytic approaches, and then give resultsof the latest simulations in momentum-conserving and momentum-non-conserving one-dimensional systems. In Section 5 we look at the joint effect of disorder and interactionsin one-dimensional systems. In Section 6 results on two-dimensional interacting systemsare presented while Section 7 gives results for billiard-like systems of non-interactingparticles. Some of the recent experimental results on nanowires and nanotubes arediscussed in Section 8. Finally the conclusions of the review are summarized in Section 9and a list of some interesting open problems is provided.

Advances in Physics 459


2. Methods

The most commonly used approaches in heat transport studies have been: (i) those whichlook at the non-equilibrium steady state obtained by connecting a system to reservoirs

at different temperatures; and (ii) those based on the Green–Kubo relation between

conductivity and equilibrium correlation functions. In this section we introduce some of

the definitions and concepts necessary in using these methods (Sections 2.1 and 2.2). Apart

from these two methods, an approach that has been especially useful in understandingballistic transport in mesoscopic systems, is the non-equilibrium Green’s function method

and we describe this method in Section 2.3. Ballistic transport of electrons refers to the

case where electron–electron interactions are negligible. In the present context ballistic

transport means that phonon–phonon interactions can be neglected.

2.1. Heat bath models, definitions of current, temperature and conductivity

To study steady-state heat transport in a Hamiltonian system, one has to connect it

to heat reservoirs. In this section we first discuss some commonly used modelsof reservoirs, and give the definitions of heat current, temperature and thermal

conductivity. It turns out that there are some subtle points involved here and we try to

explain these.First let us discuss a few models of heat baths that have been used in the literature.

For simplicity we discuss here the one-dimensional case since the generalization to higher

dimensions is straightforward. We consider a classical one-dimensional system of particles

interacting through a nearest-neighbour interaction potential U and which are in an

external potential V. The Hamiltonian is thus

H ¼XNl¼1

p2l2mlþ VðxlÞ

� �þXN�1l¼1

Uðxl � xlþ1Þ ð3Þ

where fml, xl, pl ¼ ml _xlg for l¼ 1, 2, . . . ,N denotes the masses, positions and momentaof the N particles. For the moment we assume that the interparticle potentialis such that the particles do not cross each other and so their ordering on the line is

maintained.To drive a heat current in the above Hamiltonian system, one needs to connect it to

heat reservoirs. Various models of baths have been used in the literature and here wediscuss three of the most popular ones.

(i) Langevin baths. These are defined by adding additional force terms in the equation of

motion of the particles in contact with baths. In the simplest form, the additional forcesconsist of a dissipative term, and a stochastic term, which is taken to be Gaussian white

noise. Thus, with Langevin reservoirs connected to particles l¼ 1 and l¼N, the equationsof motion are given by

_p1 ¼ f1 ��Lm1

p1 þ �LðtÞ,

_pl ¼ fl for l ¼ 2, 3, . . . ,N� 1,

_pN ¼ fN ��RmN

pN þ �RðtÞ,

ð4Þ

460 A. Dhar


where

fl ¼ �@H

@xl

is the usual Newtonian force on the lth particle. The noise terms given by �L,R areGaussian, with zero mean, and related to the dissipation coefficients �L,R by the usualfluctuation dissipation relations

h�LðtÞ�Lðt0Þi ¼ 2kBTL�L�ðt� t0Þh�RðtÞ�Rðt0Þi ¼ 2kBTR�R�ðt� t0Þh�LðtÞ�Rðt0Þi ¼ 0,

where TL, TR are the temperatures of the left and right reservoirs, respectively.

More general Langevin baths where the noise is correlated are described in Section 3.2.

Here, we briefly discuss one particular example of a correlated bath, namely the Rubin

model. This model is obtained by connecting our system of interest to two reservoirs which

are each described by semi-infinite harmonic oscillator chains with a Hamiltonian of the

form Hb ¼P1

l¼1 P2l =2þ

P1l¼0ðXl � Xlþ1Þ

2=2, where {Xl,Pl} denote reservoir degrees offreedom and X0¼ 0. One assumes that the reservoirs are initially in thermal equilibriumat different temperatures and are then linearly coupled, at time t¼�1, to the two endsof the system. Let us assume that the coupling of the system with left reservoir is of the

form �x1X1. Then, following the methods to be discussed in Section 3.2, one finds that theeffective equation of motion of the left-most particle is a generalized Langevin equation of

the form:

_p1 ¼ f1 þZ t�1

dt0�Lðt� t0Þx1ðt0Þ þ �LðtÞ, ð5Þ

where the fourier transform of the kernel �L(t) is given by

~�Lð!Þ ¼Z 10

dt�LðtÞei!t ¼eiq for j!j5 2,�e�� for j!j4 2,

�

and q,� are defined through cos(q)¼ 1�!2/2, cosh(�)¼!2/2� 1, respectively. The noisecorrelations are now given by

h ~�Lð!Þ ~�Lð!0Þi ¼kBTL�!

Im½ ~�Lð!Þ� �ð!þ !0Þ, ð6Þ

where ~�Lð!Þ ¼ ð1=2�ÞR1�1 dt �LðtÞei!t. Similar equations of motion are obtained for the

particle coupled to the right reservoir.

(ii) Nosé–Hoover baths. These are deterministic baths with time-reversible dynamics which

however, surprisingly, have the ability to give rise to irreversible dissipative behaviour. In

its simplest form, Nosé–Hoover baths attached to the end particles of the system described

by the Hamiltonian (3), are defined through the following equations of motion for the set



of particles:

_p1 ¼ f1 � Lp1,_pl ¼ fl for l ¼ 2, 3, . . . ,N� 1,

_pN ¼ fN � RpN,ð7Þ

where L and R are also dynamical variables which satisfy the following equations ofmotion:

_L ¼1

L

p21m1kBTL

� 1� �

_R ¼1

R

p2NmNkBTR

� 1� �

,

with L and R as parameters which control the strength of coupling to reservoirs.Note that in both models (i) and (ii) of baths, we have described situations where baths

are connected to particular particles and not located at fixed positions in space. These are

particularly suited for simulations of lattice models, where particles make small

displacements about equilibrium positions. Of course one could modify the dynamics by

saying that particles experience the bath forces (Langevin or Nosé–Hoover type) whenever

they are in a given region of space, and then these baths can be applied to fluids too.

Another dynamics where the heat bath is located at a fixed position, and is particularly

suitable for simulation of fluid systems, is the following.

(iii)Maxwell baths. Here we take particles described by the Hamiltonian (3), and moving

within a closed box extending from x¼ 0 to x¼L. The particles execute usualHamiltonian dynamics except when any of the end particles hit the walls. Thus, when

particle l¼ 1 at the left end (x¼ 0) hits the wall at temperature TL, it is reflected witha random velocity chosen from the distribution:

PðvÞ ¼ m1vkBTL

ðvÞe�m1v2=ð2kBTLÞ, ð8Þ

where (v) is the Heaviside step function. A similar rule is applied at the right end.

There are two ways of defining a current variable depending on whether one is using

a discrete or a continuum description. For lattice models, where every particle moves

around specified lattice points, the discrete definition is appropriate. In a fluid system,

where the motion of particles is unrestricted, one has to use the continuum definition. For

the one-dimensional hard particle gas, the ordering of particles is maintained, and in fact

both definitions have been used in simulations to calculate the steady-state current. We

show here explicitly that they are equivalent. Let us first discuss the discrete definition of

heat current.For the Langevin and Nosé–Hoover baths, we note that the equation of motion has the

form _pl ¼ fl þ �l,1fL þ �l,NfR where fL and fR are forces from the bath. The instantaneousrate at which work is done by the left and right reservoirs on the system are, respectively,

given by

j1,L ¼ fLv1

462 A. Dhar


and

jN,R ¼ fRvN,

and these give the instantaneous energy currents from the reservoirs into the system.

To define the local energy current inside the wire we first define the local energy density

associated with the lth particle (or energy at the lattice site l) as follows:

�1 ¼p212m1þ Vðx1Þ þ

1

2Uðx1 � x2Þ,

�l ¼p2l2mlþ VðxlÞ þ

1

2½Uðxl�1 � xlÞ þUðxl � xlþ1Þ�, for l ¼ 2, 3, . . . ,N� 1,

�N ¼p2N2mNþ VðxN Þ þ

1

2UðxN�1 � xN Þ:

ð9Þ

Taking a time derivative of these equations, and after some straightforward manipula-

tions, we obtain the discrete continuity equations given by

_�1 ¼ �j2,1 þ j1,L,_�l ¼ �jlþ1,l þ jl,l�1 for l ¼ 2, 3, . . . ,N� 1,

_�N ¼ jN,R þ jN,N�1,ð10Þ

with

jl,l�1 ¼1

2ðvl�1 þ vlÞfl,l�1 ð11Þ

and where

fl,lþ1 ¼ �flþ1,l ¼ �@xlUðxl � xlþ1Þ

is the force that the (lþ 1)th particle exerts on the lth particle and vl ¼ _xl. From the aboveequations one can identify jl,l�1 to be the energy current from site l� 1 to l. The steady-state average of this current can be written in a slightly different form which has a clearer

physical meaning. We denote the steady-state average of any physical quantity A by hAi.Using the fact that hdU(xl�1� xl)/dti¼ 0 it follows that hvl�1 fl,l�1i¼ hvl fl,l�1i and, hence,

hjl,l�1i ¼1

2ðvl þ vl�1Þfl,l�1

� �¼ hvl fl,l�1i, ð12Þ

and this has the simple interpretation as the average rate at which the (l� 1)th particle doeswork on the lth particle. In the steady state, from (10), we obtain the equality of current

flowing between any neighbouring pair of particles:

J ¼ hj1,Li ¼ hj2,1i ¼ hj3,2i ¼ � � � ¼ hjN,N�1i ¼ �hjN,Ri, ð13Þ

where we have used the notation J for the steady-state energy current per bond.

In simulations one can use the above definition, which involves no approximations, and

a good check of convergence to steady state is to verify the above equality on all bonds.

In the case where interaction is in the form of hard-particle collisions we can write the



expression for the steady-state current in a different form. Replacing the steady-state

average by a time average we obtain

hjl,l�1i ¼ hvl fl,l�1i ¼ lim�!1

1

�

Z �0

dt vlðtÞfl,l�1ðtÞ ¼ lim�!1

1

�

Xtc

�Kl,l�1, ð14Þ

where tc denotes time instances at which particles l and (l� 1) collide and �Kl,l�1 is thechange in energy of the lth particle as a result of the collision.

Next we discuss the continuum definition of current which is more appropriate for

fluids but is of general validity. We discuss it for the case of Maxwell boundary conditions

(BCs) with the system confined in a box of length L. Let us define the local energy density

at position x and at time t as

�ðx, tÞ ¼XNl¼1

�l�½x� xlðtÞ�, ð15Þ

where "l is as defined in (9). Taking a time derivative we obtain the required continuumcontinuity equation in the form (for 05 x5L):

@�ðx, tÞ@tþ @jðx, tÞ

@x¼ j1,L�ðxÞ þ jN,R�ðx� LÞ ð16Þ

where

jðx, tÞ ¼ jKðx, tÞ þ jIðx, tÞ

with

jKðx, tÞ ¼XNl¼1

�lðtÞvlðtÞ�½x� xlðtÞ�

and

jIðx, tÞ ¼XN�1l¼2ð jlþ1,l � jl,l�1Þ½x� xlðtÞ� þ j2,1½x� x1ðtÞ� � jN,N�1½x� xNðtÞ�:

Here, jl,l�1, j1,L, jN,R are as defined earlier in the discrete case, and we have written the

current as a sum of two parts, jK and jI, whose physical meaning we now discuss. To see

this, consider a particle configuration with x1, x2, . . . , xk5 x5 xkþ1, xkþ2, . . . , xN. Thenwe obtain

jIðx, tÞ ¼ jkþ1,k,

which is thus simply the rate at which the particles on the left of x do work on the particles

on the right. Hence, we can interpret jI(x, t) as the contribution to the current density

coming from interparticle interactions. The other part jK(x, t) arises from the physical flow

of particles carrying energy across the point x. Note, however, that even in the absence of

any net convection particle flow, both jK and jI can contribute to the energy flow. In fact,

for point particles interacting purely by hard elastic collisions jkþ1,k is zero whenever the

kth and (kþ 1)th particles are on the two sides of the point x and, hence, jI is exactly zero.

464 A. Dhar


The only contribution to the energy current then comes from the part jK and thus for the

steady-state current we obtain

hjðx, tÞi ¼XNl¼1

mlv2l

2�ðx� xlÞ

� �: ð17Þ

In simulations either of expressions (14) or (17) can be used to evaluate the steady-state

current and will give identical results. For hard-particle simulations one often uses

a simulation which updates between collisions and in this case it is more efficient to

evaluate the current using (14). We now show that, in the non-equilibrium steady state, the

average current from the discrete and continuum definitions are the same, i.e.

hjl,l�1i ¼ hjðx, tÞi ¼ J: ð18Þ

Note that the steady-state current is independent of l or x. To show this we first define the

total current as

J ðtÞ ¼Z L0

dx jðx, tÞ ¼Xl¼1,N

�lvl �X

l¼2,N�1xlð jlþ1,l � jl,l�1Þ � x1j2,1 þ xNjN,N�1

¼Xl¼1,N

�lvl þX

l¼1,N�1ðxlþ1 � xlÞjlþ1, l:

ð19Þ

Taking the steady-state average of the above and using the fact that h�lvli ¼ �h _�lxli ¼hð jlþ1,l � jl,l�1Þxli, where j1,0¼ j1,L, jNþ1,N¼�jN,R we obtain

hJ i ¼ �hx1j1L þ xNjNRi:

Since the Maxwell baths are located at x¼ 0 and x¼L, the above then gives hJ i¼Lhj(x, t)i¼�LhjNRi and, hence, from (13), we obtain hj(x, t)i¼ J¼hjl,l�1i, which provesthe equivalence of the two definitions.

The extensions of the current definitions, both the discrete and continuum versions, to

higher dimensions is straightforward. Here, for reference, we outline the derivation for the

continuum case since it is not easy to see a discussion of this in the literature. Consider

a system in d-dimensions with Hamiltonian given by:

H ¼Xl

p2l2mlþ VðxlÞ

� �þ 12

Xl6¼n

Xn

UðrlnÞ, ð20Þ

where xl ¼ ðx1l , x2l , . . . , xdl Þ and pl ¼ ð p1l , p2l , . . . , pdl Þ are the vectors denoting the positionand momentum of the lth particle and rln¼ jxl� xnj. The particles are assumed to be insidea hypercubic box of volume Ld. As before we define the local energy density as

�ðx, tÞ ¼Xl

�ðx� xlÞ�l

where

�l ¼p2l2mlþ VðxlÞ þ

1

2

Xn 6¼l

UðrlnÞ:



Taking a derivative with respect to time (and suppressing the source terms arising from the

baths) gives

@�ðx, tÞ@t¼ �

Xd�¼1

�@

@x�

Xl

�ðx� xlÞ �lv�l�þXl

�ðx� xlÞ _�l

¼ �X�

@

@x�½jK� þ jI��,

ð21Þ

where

j�Kðx, tÞ ¼Xi

�ðx� xlÞ�lv�l

and

j�I ðx, tÞ ¼ �Xl

Xn6¼l

ðx� � x�l ÞY� 6¼�

�ðx� � x�l Þjl,n

where

jl,n ¼1

2

X�

ðv�l þ v�nÞf�l,n,

and f �l,n ¼ �@Uðrl,nÞ=@x�l is the force, in the �th direction, on the lth particle due tothe nth particle. We have defined jl,n as the current, from particle n to particle l, analogously

to the discrete one-dimensional current. The part j�I gives the energy flow as a result of

physical motion of particles across x�. The part j�I also has a simple physical interpretation,

as in the one-dimensional case. First note that we need to sum over only those n for

which x�n 5 x�. Then the formula basically gives us the net rate, at which work is done,

by particles on the left of x�, on the particles to the right. This is thus the rate at which energy

flows from left to right. By integrating the current density over the full volume of the system,

we obtain the total current:

J �ðtÞ ¼Xl

�lv�l þ

1

2

Xl 6¼n

Xn

ðx�l � x�nÞjl,n: ð22Þ

Thus, we obtain an expression similar to that in one-dimensional given by (19).

In simulations making non-equilibrium measurements, any of the various definitions

for current can be used to find the steady-state current. However, it is not clear

whether other quantities, such as correlation functions obtained from the discrete

and continuum definitions, will be the same.The local temperature can also be defined using either a discrete approach (giving Tl)

or a continuum approach (giving T(x, t)). In the steady state these are respectively given

by (in the one-dimensional case)

kBTl ¼p2lml

� �,

kBTðxÞ ¼hP

lð p2l =mlÞ�ðx� xlÞihP

l �ðx� xlÞi:

ð23Þ

466 A. Dhar


Again it is not obvious that these two definitions will always agree. Lattice simulations

usually use the discrete definition while hard-particle simulations use the continuum

definition.The precise definition of thermal conductivity would be

� ¼ limL!1

lim�T!0

JL

�T, ð24Þ

where �T¼TL�TR. In general � would depend on temperature T. The finiteness of �(T ),along with Fourier’s law, implies that even for arbitrary fixed values of TL, TR the current J

would scale as�L�1 (orN�1).What is of real interest is this scaling property of Jwith systemsize. We are typically interested in the large N behaviour of the conductivity defined as

�N ¼JN

�T, ð25Þ

which will usually be denoted by �. For large N, systems with normal diffusive transportgive a finite � while anomalous transport refers to the scaling

� � N�, � 6¼ 0, ð26Þ

and the value of the heat conduction exponent � is one of the main objects of interest. Forthe current, this implies that J�N��1.

The only examples where the steady-state current can be evaluated analytically, and

exact results are available for the exponent �, correspond to harmonic lattices, using veryspecific methods that are discussed in Section 3.

Coupling to baths and contact resistance.

In the various models of heat baths that we have discussed, the efficiency with which

heat exchange takes place between reservoirs and the system depends on the strength of

coupling constants. For example, for the Langevin and Nosé–Hoover baths, the

parameters � and respectively determine the strength of coupling (for the Maxwell bathone could introduce a parameter which gives the probability that after a collision the

particle’s speed changes and this can be used to tune the coupling between system and

reservoir). From simulations it is found that typically there is an optimum value of the

coupling parameter for which energy exchange takes place most efficiently, and at

this value one obtains the maximum current for a given system and fixed bath

temperatures. For too high or too small values of the coupling strength the current

is small. The coupling to the bath can be thought of as giving rise to a contact resistance.

An effect of this resistance is to give rise to boundary jumps in the temperature profile

measured in simulations. One expects that these jumps will be present as long as the

contact resistance is comparable to the systems resistance. We will later see that in order

to be sure that one is measuring the true resistance of the system, it is necessary to be in

parameter regimes where the contact resistances can be neglected.

2.2. Green–Kubo formula

The Green–Kubo formula provides a relation between transport coefficients, such as the

thermal conductivity � or the electrical conductivity , and equilibrium time correlation



functions of the corresponding current. For the thermal conductivity in a classical one-dimensional system, the Green–Kubo formula gives

� ¼ 1kBT2

lim�!1

limL!1

1

L

Z �0

dt hJ ð0ÞJ ðtÞi, ð27Þ

where J is the total current as defined in (19) and h. . .i denotes an average over initialconditions chosen from either a micro-canonical ensemble or a canonical ensemble attemperature T. Two important points to be remembered with regard to use of the aboveGreen–Kubo formula are the following.

(i) It is often necessary to subtract a convective part from the current definitionor, alternatively, in the microcanonical case one can work with initial conditions chosensuch that the centre of mass velocity is zero (see also the discussions in [4,5]). Tounderstand this point, let us consider the case with V(x)¼ 0. Then for a closed system thatis not in contact with reservoirs, we expect the time average of the total current to vanish.However, this is true only if we are in the centre of mass frame. If the centre of mass ismoving with velocity v, then the average velocity of any particle hvli¼ v. Transforming tothe moving frame let us write vl¼ v0lþ v. Then the average total current in the rest frame isgiven by (in d-dimensions)

hJ �i ¼ M2v2 þ

Xl

h�0li" #

v� þXv

Xl

Xhmlv

0�l v0�l i þ

1

2

Xl, nl 6¼n

hðx�l � x�nÞf �l,ni

264

375v�

¼ ðEþ PVÞ v�, ð28Þ

where M¼P

ml and E is the average total energy of the system as measured in the restframe and V¼Ld. In deriving the above result we have used the standard expression forequilibrium stress tensor given by

V�� ¼Xhmlv

0�l v0�l i þ

1

2

Xl6¼nhðx�l � x�nÞf �l,ni, ð29Þ

and assumed an isotropic medium. Thus, in general, to obtain the true energy current in anarbitrary equilibrium ensemble one should use the expression:

J �c ¼ J � � ðEþ PVÞv�: ð30Þ

The corresponding one-dimensional form should be used to replace J in (27).(ii) The second point to note is that in (27) the order of limits L!1 and then �!1

has to be strictly maintained. In fact, for a system of particles inside a finite box of length Lit can be shown exactly that: Z 1

0

dt hJ cð0ÞJ cðtÞi ¼ 0: ð31Þ

To prove this, let us consider a microcanonical ensemble (with hJ i¼ 0, so that J c¼J ), inwhich case from (19) we obtain

J ðtÞ ¼ ddt

XNl¼1

�lðtÞxlðtÞ" #

: ð32Þ

468 A. Dhar


Multiplying both sides of the above equation by J (0), integrating over t and noting thatboth the boundary terms on the right-hand side vanish, we obtain the required result

in (31). With the correct order of limits in (27), one can calculate the correlation functions

with arbitrary BCs and apply the formula to obtain the response of an open system with

reservoirs at the ends.

Derivation of the Green–Kubo formula for thermal conductivity.

There have been a number of derivations of this formula by various authors including

Green, Kubo, Mori, McLennan, Kadanoff and Martin, Luttinger and Visscher [4–12].

None of these derivations are rigorous and all require certain assumptions. Luttinger’s

derivation was an attempt at a mechanical derivation and involves introducing

a fictitious ‘gravitational field’ which couples to the energy density and drives an

energy current. However, one now has to relate the response of the system to the field

and its response to imposed temperature gradients. This requires additional inputs

such as use of the Einstein relation relating the diffusion coefficient (or thermal

conductivity) to the response to the gravitational field. We believe that this derivation,

as is also the case for most other derivations, implicitly assumes local thermal

equilibrium. Although these derivations are not rigorous, they are quite plausible, and

it is likely that the assumptions made are satisfied in a large number of cases of

practical interest. Thus, the wide use of the Green–Kubo formula in calculating thermal

conductivity and transport properties of different systems is possibly justified in many

situations.Here we give an outline of a non-mechanical derivation of the Green–Kubo formula,

one in which the assumptions can be somewhat clearly stated and their physical basis

understood. The assumptions we make here are as follows.

(a) The non-equilibrium state with energy flowing in the system can be described

by coarse-grained variables and the condition of local thermal equilibrium is

satisfied.(b) There is no particle flow, and energy current is equal to the heat current.

The energy current satisfies Fourier’s law which we write in the form J(x, t)¼�D@u(x, t)/@x, where D¼ �/c, c is the specific heat capacity and uðx, tÞ ¼��ðx, tÞ, Jðx, tÞ ¼ �jðx, tÞ are macroscopic variables obtained by a coarse-graining(indicated by bars) of the microscopic fields.

(c) Finally, we assume that the regression of equilibrium energy fluctuations occurs in

the same way as the non-equilibrium flow of energy.

We consider a macroscopic system of size L. Fluctuations in energy density in

equilibrium can be described by the correlation function S(x, t)¼h�(x, t)�(0, 0)i� h�(x, t)i�h�(0, 0)i. Assumption (c) above means that the decay of these fluctuations is determined bythe heat diffusion equation and given by

@Sðx, tÞ@t

¼ D @2Sðx, tÞ@x2

for t4 0,

where we have also assumed temperature fluctuations to be small enough so that the

temperature dependence of D can be neglected. From time reversal invariance we have



S(x, t)¼S(x,�t). Using this and the above equation we obtain

~Sðk,!Þ ¼Z 1�1

dt Ŝðk, tÞe�i!t ¼ 2Dk2Ŝðk, t ¼ 0Þ

D2k4 þ !2 ,

where Ŝðk, tÞ ¼R1�1 dxSðx, tÞeikx. Now, from equilibrium statistical mechanics we have

Ŝ(k¼ 0, t¼ 0)¼ ckBT2 and using this in the above equation we obtain

� ¼ cD ¼ 12kBT2

lim!!0

limk!0

!2

k2~Sðk,!Þ: ð33Þ

One can relate the energy correlator to the current correlator using the continuity equation

@�(x, t)/@tþ @j(x, t)/@x¼ 0 and this gives

h~jðk,!Þ~jðk0,!0Þi ¼ ð2�Þ2 !2

k2~Sðk,!Þ�ðkþ k0Þ�ð!þ !0Þ,

where ~jðk,!Þ ¼R1�1 dx dt jðx, tÞeiðkx�!tÞ. Integrating the above equation over k0 givesZ 1

�1dx

Z 1�1

dt hjðx, tÞjð0, 0Þieiðkx�!tÞ ¼ !2

k2~Sðk,!Þ: ð34Þ

From (33) and (34) we then obtain

� ¼ lim!!0

limk!0

1

2kBT2

Z 1�1

dx

Z 1�1

dt hjðx, tÞjð0, 0Þieiðkx�!tÞ: ð35Þ

Finally, using time-reversal and translational invariance and interpreting the !! 0, k! 0limits in the alternative way (as �!1,L!1) we recover the Green–Kubo formulain (27).

Limitations on use of the Green–Kubo formula.

There are several situations where the Green–Kubo formula in (27) is not applicable.

For example, for the small structures that are studied in mesoscopic physics, the

thermodynamic limit is meaningless, and one is interested in the conductance of a specific

finite system. Second, in many low-dimensional systems, heat transport is anomalous and

the thermal conductivity diverges. In such cases it is impossible to take the limits as in (27);

one is there interested in the thermal conductance as a function of L instead of an

L-independent thermal conductivity. The usual procedure that has been followed in the

heat conduction literature is to put a cut-off at tc�L, in the upper limit in the Green–Kubo integral [2]. The argument is that for a finite system connected to reservoirs, sound

waves travelling to the boundaries at a finite speed, say v, lead to a decay of correlations in

a time �L/v. However, there is no rigorous justification of this assumption. A related caseis that of integrable systems, where the infinite time limit of the correlation function in (27)

is non-zero.Another way of using the Green–Kubo formula for finite systems is to include the

infinite reservoirs also while applying the formula and this was done, for example, by Allen

and Ford [13] for heat transport and by Fisher and Lee [14] for electron transport. Both of

470 A. Dhar


these cases are for non-interacting systems and the final expression for conductance(which is more relevant than conductivity in such systems) is basically what one also

obtains from the Landauer formalism [15] or the non-equilibrium Green’s functionapproach (see Section 2.3).

It has been shown that Green–Kubo-like expressions for the linear response heatcurrent for finite open systems can be derived rigorously by using the steady-state

fluctuation theorem [16–22]. This has been done for lattice models coupled to stochasticMarkovian baths and the expression for linear response conductance of a one-dimensionalchain is given as

lim�T!0

J

�T¼ 1

kBT2

Z 10

hjlð0ÞjlðtÞi, ð36Þ

where jl is the discrete current defined in Section 2.1. Some of the important differencesbetween this formula and the usual Green–Kubo formula are worth keeping in mind:

(i) the dynamics of the system here is non-Hamiltonian since they are for a system coupledto reservoirs, (ii) one does not need to take the limit N!1 first, the formula being validfor a finite system, (iii) the discrete bond current appears here, unlike the continuum

current in the usual Green–Kubo formula. Recently, an exact linear response result similarto (36), for the conductance of a finite open system has been derived using a differentapproach [23]. This has been done for quite general classical Hamiltonian systems and for

a number of implementations of heat baths.It appears that the linear response formula given by (36) is the correct one to use to

evaluate the conductance in systems where there is a problem with the usual Green–Kuboformula, e.g. in finite systems or low-dimensional systems showing anomalous transport(because of slow decay of h J (0)J (t)i). We note here the important point that the current–current correlation can have very different scaling properties, for a purely Hamiltoniandynamics, as compared with a heat bath dynamics. This has been seen in simulations byDeutsche and Narayan [24] for the random collision model (defined in Section 4.2.1).

Of course this makes things somewhat complicated since the usefulness of the Green–Kubo formalism arises from the fact that it allows a calculation of transport propertiesfrom equilibrium properties of the system, and without any reference to heat baths, etc.

In fact, as we will see in Section 4.1, all of the analytic results for heat conductionin interacting one-dimensional systems rely on (27) and involve a calculation of thecurrent–current correlator for a closed system. Some of the simulation results discussed

later suggest that, for interacting (non-linearly) systems, in the limit of large system size theheat current is independent of the details of the heat baths. This means that, in the linearresponse regime and the limit of large system sizes, a description which does not take into

account bath properties may still be possible.

2.3. Non-equilibrium Green’s function method

The non-equilibrium Green’s function (NEGF) method is a method, first invented in thecontext of electron transport, to calculate the steady-state properties of a finite system

connected to reservoirs which are themselves modelled by non-interacting Hamilitonianswith infinite degrees of freedom [25–27]. Using the Keldysh formalism, one can obtainformal expressions for the current and other observables such as the electron density, in



models of electrons such as those described by tight-binding-type Hamiltonians. Recently,

this approach has been applied both to phonon [28,29] and photon [32,33] transport.The main idea in the formalism is as follows. One starts with an initial density matrix

describing the decoupled system, and two infinite reservoirs which are in thermal

equilibrium, at different temperatures and chemical potentials. The system and reservoirs

are then coupled together and the density matrix is evolved with the full Hamiltonian for

an infinite time so that one eventually reaches a non-equilibrium steady state. Various

quantities of interest such as currents and local densities, etc., can be obtained using thesteady-state density matrix and can be written in terms of the so-called Keldysh Green’s

functions. An alternative and equivalent formulation is the Langevin approach where,

instead of dealing with the density matrix, or in the classical case with the phase space

density, one works with equations of motion of the phase space variables of the full

Hamiltonian (system plus reservoirs). Again the reservoirs, which are initially in thermal

equilibrium, are coupled to the system in the remote past. It is then possible to integrate

out the reservoir degrees of freedom, and these give rise to generalized Langevin terms in

the equation of motion. For non-interacting systems, one can show that it is possible to

recover all of the results of NEGF exactly, both for electrons and phonons. Here we

discuss this approach for the case of phonons and describe the main results that have been

obtained (Section 3.2).For non-interacting systems, the formal expressions for the current obtained from the

NEGF approach is in terms of transmission coefficients of the heat carriers (electrons,

phonons or photons) across the system, with appropriate weight factors corresponding tothe population of modes in the reservoirs. These expressions are basically what one also

obtains from the Landauer formalism [15]. We note that in the Landauer approach one

simply thinks of transport as a transmission problem and the current across the system is

obtained directly using this picture. In the simplest set-up one thinks of one-dimensional

reservoirs (leads) filled with non-interacting electrons at different chemical potentials. On

connecting the system in between the reservoirs electrons are transmitted through the

system from one reservoir to the other. The net current in the system is then the sum of the

currents from left-moving and right-moving electron states from the two reservoirs.

3. Heat conduction in harmonic lattices

The harmonic crystal is a good starting point for understanding heat transport in solids.

Indeed in the equilibrium case we know that studying the harmonic crystal already gives

a good understanding of, for example, the specific heat of an (electrically) insulating solid.

In the non-equilibrium case, the problem of heat conduction in a classical one-dimensional

harmonic crystal was studied for the first time by Rieder, Lebowitz and Lieb (RLL) [34].

They considered the case of stochastic Markovian baths and were able to obtain the steady

state exactly. The main results of this paper were: (i) the temperature in the bulk of the

system was constant and equal to the mean of the two bath temperatures; (ii) the heat

current approaches a constant value for large system sizes and an exact expression for this

was obtained. These results can be understood physically when one realizes that in theordered crystal, heat is carried by freely propagating phonons. RLL considered the case

where only nearest-neighbour interparticle interactions were present. Nakazawa [35]

extended these results to the case with a constant on-site harmonic potential at all sites and

472 A. Dhar


also to higher dimensions. The approach followed in both the RLL and Nakazawa papers

was to obtain the exact non-equilibrium stationary state measure which, for this quadratic

problem, is a Gaussian distribution. A complete solution for the correlation matrix was

obtained and from this one could obtain both the steady-state temperature profile and the

heat current.In Section 3.1 we briefly describe the RLL formalism. In Section 3.2 we describe

a different and more powerful formalism. This is the Langevin equation and NEGF

method and various applications of this will be discussed in Sections 3.3, 3.4 and 3.5.

3.1. The RLL method

Let us consider a classical harmonic system of N particles with displacements about the

equilibrium positions described by the vector X¼ {x1, x2, . . . , xN}T, where T denotestranspose. The particles can have arbitrary masses and we define a diagonal matrix M̂

whose diagonal elements Mll¼ml, for l¼ 1, 2, . . . ,N, give the masses of the particles.The momenta of the particles are given by the vector P ¼ ðM _XÞ ¼ fp1, p2, . . . , pNgT. Weconsider the following harmonic Hamiltonian for the system:

H ¼ 12PTM̂�1Pþ 1

2XT�̂X, ð37Þ

where �̂ is the force matrix. Let us consider the general case where the lth particle is

coupled to a white noise heat reservoir at temperature TBl with a coupling constant � l.The equations of motion are given by

_xl ¼plml

_pl ¼ �Xn

�lnxn ��lml

pl þ �l for l ¼ 1, 2, . . . ,Nð38Þ

with the noise terms satisfying the usual fluctuation dissipation relations

h�lðtÞ�nðt0Þi ¼ 2�lkBTBl �l,n�ðt� t0Þ: ð39Þ

Defining new variables

q ¼ fq1, q2, . . . , q2NgT ¼ fx1, x2, . . . , xN, p1, p2, . . . , pNgT

we can rewrite (38) and (39) in the form

_q ¼ �Âqþ �,h�ðtÞ�Tðt0Þi ¼ D̂�ðt� t0Þ,

ð40Þ

where the 2N-dimensional vector �T¼ (0, 0, . . . , 0, �1, �2, . . . , �N) and the 2N� 2N matricesÂ, D̂ are given by

Â ¼ 0 �M̂�1

�̂ M̂�1�̂

!, D̂ ¼

0 0

0 Ê

� �



and �̂ and Ê are N�N matrices with elements �ln¼ � l�l,n, Eln ¼ 2kBTBl �l�ln, respectively.In the steady state, time averages of total time derivatives vanish, hence we have hd(qqT)/dti¼ 0. From (40) we obtain

d

dtðqqTÞ

� �¼ hð�Âqþ �ÞqT þ qð�qTÂT þ �TÞi

¼ �Â � B̂� B̂ � ÂT þ h�qT þ q�Ti ¼ 0ð41Þ

where B̂ is the correlation matrix hqqTi. To find the average of the term involving noise wefirst write the formal solution of (40):

qðtÞ ¼ Ĝðt� t0Þqðt0Þ þZ tt0

dt0 Ĝðt� t0Þ�ðt0Þ where ĜðtÞ ¼ e�Ât: ð42Þ

Setting t0!�1 and assuming Ĝ(1)¼ 0 (so that a unique steady state exists), we getqðtÞ ¼

R t�1 dt

0 Ĝðt� t0Þ�ðt0Þ and, hence,

hq�Ti ¼Z t�1

dt0 Ĝðt� t0Þh�ðt0Þ�TðtÞi

¼Z t�1

dt0 Ĝðt� t0ÞD̂�ðt� t0Þ ¼ 12D̂,

ð43Þ

where we have used the noise correlations given by (39), and the fact that Ĝ(0)¼ Î, a unitmatrix. Using (43) in (41), we finally obtain

Â � B̂þ B̂ � ÂT ¼ D̂: ð44Þ

The solution of this equation gives the steady-state correlation matrix B̂ which completely

determines the steady state since we are dealing with a Gaussian process. In fact, the steady

state is given by the Gaussian distribution

PðfqlgÞ ¼ ð2�Þ�N Det½B̂��1=2e�12 q

TB̂�1q: ð45Þ

Some of the components of the matrix Equation (44) have simple physical interpretations.

To see this we first write B̂ in the form

B̂ ¼B̂x B̂xp

B̂xpT

B̂p

!,

where B̂x, B̂p and B̂xp are N�N matrices with elements ðB̂xÞln ¼ hxlxni, ðB̂pÞln ¼ hplpni andðB̂xpÞln ¼ hxlpni. From (44) we then obtain the set of equations

M̂�1B̂Txp þ B̂xpM̂�1 ¼ 0, ð46Þ

B̂x�̂T � M̂�1B̂p þ B̂xpM̂�1�̂ ¼ 0, ð47Þ

M̂�1�̂B̂p þ B̂p�̂M̂�1 þ �̂B̂xp þ B̂Txp�̂ ¼ Ê: ð48Þ

474 A. Dhar


From (46) we obtain the identity hxlpn/mni¼�hxnpl/mli. Thus, hxlpli¼ 0. The diagonalterms in (47) give X

n

hxlð�lnxnÞi �1

mlhp2l i þ hxlpli

�lml¼ 0,

) xl@H

@xl

� �¼ p

2l

ml

� �¼ kBTl,

where we have defined the local temperature kBTl ¼ hp2l =mli. This equation has the formof the ‘equipartition’ theorem of equilibrium physics. It is, in fact, valid quite generallyfor any Hamiltonian system at all bulk points and simply follows from the fact that h(d/dt)xlpli ¼ 0. Finally, let us look at the diagonal elements of (48). This gives the equationX

n

ð�lnxnÞplml

� �¼ �l

mlkBðTBl � TlÞ, ð49Þ

which again has a simple interpretation. The right-hand side can be seen to be equal toh(�� lpl/mlþ �l)pl/mli which is simply the work done on the lth particle by the heat bathattached to it (and is thus the heat input at this site). On the left-hand side h(��lnxn)pl/mliis the rate at which the nth particle does work on the lth particle and is therefore the energycurrent from site n to site l. The left-hand side is thus is just the sum of the outgoing energycurrents from the lth site to all of the sites connected to it by �̂. Thus, we can interpret (49)as an energy-conservation equation. The energy current between two sites is given by

Jn!l ¼ � ð�lnxnÞplml

� �: ð50Þ

Our main interests are in computing the temperature profile and the energy current in thesteady state. This requires the solution of (44) and this is quite difficult and has beenachieved only in a few special cases. For the one-dimensional ordered harmonic lattice,RLL were able to solve the equation exactly and obtain both the temperature profile andthe current. The extension of their solution to higher-dimensional lattices is straightfor-ward and was done by Nakazawa. More recently Bonetto et al. [36] have used thisapproach to solve the case with self-consistent reservoirs attached at all of the bulk sites ofa ordered harmonic lattice.

A numerical solution of (44) requires inversion of a N(2Nþ 1)�N(2Nþ 1) matrixwhich restricts one to rather small system sizes. The RLL approach is somewhat restrictivesince it is not easily generalizable to other kinds of heat baths or to the quantum case.In addition, except for the ordered lattice, it is difficult to obtain useful analytic resultsfrom this approach. In the next section we discuss a different approach which is bothanalytically more tractable, as well as numerically more powerful.

3.2. Langevin equations and Green’s function formalism

This approach involves a direct solution of generalized Langevin equations. Using thissolution one can evaluate various quantities of interest such as steady-state current andtemperature profiles. Compact formal expressions for various quantities of interest can beobtained and, as pointed out earlier, it turns out that these are identical to results obtained



by the NEGF method described in Section 2.3. The method can be developed for quantum

mechanical systems, in which case we deal with quantum Langevin equations (QLE), and

we see that the classical results follow in the high-temperature limit. In all applications we

restrict ourselves to this approach which we henceforth refer to as the Langevin equations

and Green’s function (LEGF) method. As we will see, for the ordered case to be discussed

in Section 3.3, one can recover the standard classical results as well as extend them to the

quantum domain using the LEGF method. For the disordered case (Section 3.4), one can

also make significant progress. Another important model that has been well studied in the

context of harmonic systems is the case where self-consistent heat reservoirs are attached

at all sites of the lattice. In Section 3.5 we review results for this case also obtained from the

LEGF approach. All examples in this section deal with the case where particle

displacements are taken to be scalars. The generalization to vector displacements is

straightforward.The LEGF formalism has been developed in [37–41] and relies on the approach first

proposed by Ford, Kac and Mazur [42] of modelling a heat bath by an infinite set of

oscillators in thermal equilibrium. Here we outline the basic steps as given in [41]. One again

considers a harmonic system which is coupled to reservoirs which are of a more general

form than the white noise reservoirs studied in the last section. The reservoirs are now

themselves taken to be a collection of harmonic oscillators, whose number will be eventually

taken to be infinite. As we will see, this is equivalent to considering generalized Langevin

equations where the noise is still Gaussian but in general can be correlated. We present

the discussion for the quantum case and obtain the classical result as a limiting case.We consider here the case of two reservoirs, labelled as L (for left) and R (for right),

which are at two different temperatures. It is easy to generalize to the case where there are

more than two reservoirs. For the system let X¼ {x1, x2, . . . ,xN}T now be the set ofHeisenberg operators corresponding to the displacements (assumed to be scalars) of the N

particles, about equilibrium lattice positions. Similarly let XL and XR refer to position

operators of the particles in the left and right baths, respectively. The left reservoir has NLparticles and the right has NR particles. Also let P, PL, PR be the corresponding

momentum variables satisfying the usual commutation relations with the position

operators (i.e. [xl, pn]¼ ih��l,n, etc.). The Hamiltonian of the entire system and reservoirsis taken to be

H ¼ HS þHL þHR þHIL þHIR ð51Þ

where

HS ¼1

2PTM̂�1Pþ 1

2XT�̂X,

HL ¼1

2PTLM̂

�1L PL þ

1

2XTL�̂LXL,

HR ¼1

2PTRM̂

�1R PR þ

1

2XTR�̂RXR,

HIL ¼ XTV̂LXL, HIR ¼ XTV̂RXR,

476 A. Dhar


where M, ML, MR are real diagonal matrices representing masses of the particles in the

system, left and right reservoirs, respectively. The quadratic potential energies are given by

the real symmetric matrices �̂, �̂L, �̂R while V̂L and V̂R denote the interaction between the

system and the two reservoirs, respectively. It is assumed that at time t¼ t0, the system andreservoirs are decoupled and the reservoirs are in thermal equilibrium at temperatures TLand TR, respectively.

The Heisenberg equations of motion for the system (for t4 t0) are

M̂ €X ¼ ��̂X� V̂LXL � V̂RXR, ð52Þ

and the equations of motion for the two reservoirs are

M̂L €XL ¼ ��̂LXL � V̂TLX, ð53Þ

M̂R €XR ¼ ��̂RXR � V̂TRX: ð54Þ

One first solves the reservoir equations by considering them as linear inhomogeneous

equations. Thus, for the left reservoir the general solution to (53) is (for t4 t0)

XLðtÞ ¼ f̂þL ðt� t0ÞM̂LXLðt0Þ þ ĝþL ðt� t0ÞM̂L _XLðt0Þ

�Z tt0

dt0 ĝþL ðt� t0ÞV̂TLXðt0Þ, ð55Þ

with

f̂þL ðtÞ ¼ ÛL cos ð�̂LtÞÛTLðtÞ, ĝþL ðtÞ ¼ ÛLsin ð�̂LtÞ

�̂LÛTLðtÞ,

where (t) is the Heaviside function, and ÛL, �̂L are the normal mode eigenvector andeigenvalue matrices, respectively, corresponding to the left reservoir Hamiltonian HL and

which satisfy the equations

ÛTL�̂LÛL ¼ �̂2L, ÛTLM̂LÛL ¼ Î:

A similar solution is obtained for the right reservoir. Plugging these solutions back into the

equation of motion for the system, (52), one obtains the following effective equations of

motion for the system:

M̂ €X ¼ ��̂Xþ �L þZ tt0

dt0 �̂Lðt� t0ÞXðt0Þ þ �R þZ tt0

dt0 �̂Rðt� t0ÞXðt0Þ, ð56Þ

where

�̂LðtÞ ¼ V̂LĝþL ðtÞV̂TL, �̂RðtÞ ¼ V̂RĝþRðtÞV̂TRand

�L ¼ �V̂L½f̂þL ðt� t0ÞM̂LXLðt0Þ þ ĝþL ðt� t0ÞM̂L _XLðt0Þ�,�R ¼ �V̂R½f̂þRðt� t0ÞM̂RXRðt0Þ þ ĝþRðt� t0ÞM̂R _XRðt0Þ�:



This equation has the form of a generalized quantum Langevin equation. The properties

of the noise terms �L and �R are determined using the condition that, at time t0, thetwo isolated reservoirs are described by equilibrium phonon distribution functions.

At time t0, the left reservoir is in equilibrium at temperature TL and the population of the

normal modes (of the isolated left reservoir) is given by the distribution function

fbð!,TLÞ ¼ 1=½e�h!=kBTL � 1�. One then obtains the following correlations for the leftreservoir noise:

h�LðtÞ�TLðt0Þi

¼ V̂LÛL cos �̂Lðt� t0Þ�h

2�̂Lcoth

�h�̂L2kBTL

!� i sin �̂Lðt� t0Þ

�h

2�̂L

" #ÛTLV̂

TL,

ð57Þ

and a similar expression for the right reservoir. The limits of infinite reservoir sizes

(NL,NR!1) and t0!�1 are now taken. One can then solve (56) by taking Fouriertransforms. Let us define the Fourier transforms

~Xð!Þ ¼ 12�

Z 1�1

dt XðtÞei!t,

~�L,Rð!Þ ¼1

2�

Z 1�1

dt �L,RðtÞei!t,

ĝþL,Rð!Þ ¼Z 1�1

dt ĝþL,RðtÞei!t:

One then obtains the following stationary solution to the equations of motion (56):

XðtÞ ¼Z 1�1

d! ~Xð!Þe�i!t, ð58Þ

with

~Xð!Þ ¼ Ĝþð!Þ½ ~�Lð!Þ þ ~�Rð!Þ�,

where

Ĝþð!Þ ¼ 1½�!2M̂þ �̂� �̂þL ð!Þ � �̂þRð!Þ�

and

�̂þL ð!Þ ¼ V̂LĝþL ð!ÞV̂TL, �̂þRð!Þ ¼ V̂RĝþRð!ÞV̂TR:

For the reservoirs one obtains [using Equation (55), etc.]

�V̂L ~XLð!Þ ¼ ~�Lð!Þ þ �̂þL ~Xð!Þ,�V̂R ~XRð!Þ ¼ ~�Rð!Þ þ �̂þR ~Xð!Þ: ð59Þ

478 A. Dhar


The noise correlations, in the frequency domain, can be obtained from (57) and we obtain(for the left reservoir):

h ~�Lð!Þ ~�TLð!0Þi ¼ �ð!þ !0Þ�̂Lð!Þ�h

�½1þ fbð!,TLÞ� ð60Þ

where

�̂Lð!Þ ¼ Im½�̂þL ð!Þ�which is a fluctuation–dissipation relation. This also leads to the more commonly usedcorrelation:

1

2h ~�Lð!Þ ~�TLð!0Þ þ ~�Lð!0Þ ~�TLð!Þi ¼ �ð!þ !0Þ�̂Lð!Þ

�h

2�coth

�h!

2kBTL

� �: ð61Þ

Similar relations hold for the noise from the right reservoir. The identification of Ĝþð!Þ asa phonon Green function, with �̂þL,Rð!Þ as self-energy contributions coming from thebaths, is the main step that enables a comparison of results derived by the LEGF approachwith those obtained from the NEGF method. This has been demonstrated in [41].

3.2.1. Steady-state properties

The simplest way to evaluate the steady-state current is to evaluate the followingexpectation value for current from left reservoir into the system:

J ¼ �h _XTV̂LXLi: ð62Þ

This is just the rate at which the left reservoir does work on the wire. Using the solutionin (58), (59) and (60) one obtains, after some manipulations,

J ¼ 14�

Z 1�1

d! T ð!Þ�h!½f ð!,TLÞ � f ð!,TRÞ� ð63Þ

where

T ð!Þ ¼ 4Tr½Ĝþð!Þ�̂Lð!ÞĜ�ð!Þ�̂Rð!Þ�,

and Ĝ�ð!Þ ¼ Ĝþyð!Þ. This expression for the current is of identical form as the NEGFexpression for electron current (see, for example, [25–27,47]) and has also been derived forphonons using the NEGF approach in [28,29]. In fact, this expression was first proposedby Angelescu et al. [30] and by Rego and Kirczenow [31] for a one-dimensional channeland they obtained this using the Landauer approach. Their result was obtained moresystematically later by Blencowe [43]. Note that (63) can also be written as an integral overonly positive frequencies using the fact that the integrand is an even function of !.The factor T (!) is the transmission coefficient of phonons at frequency ! through thesystem, from the left to right reservoir. The usual Landauer result for a one-dimensionalchannel precisely corresponds to Rubin’s model of bath, to be discussed in Section 3.4.1.

For small temperature differences �T¼TL�TR�T, where T¼ (TLþTR)/2, i.e. inthe linear response regime the above expression reduces to

J ¼ �T4�

Z 1�1

d! T ð!Þ�h! @f ð!,T Þ@T

: ð64Þ



The classical limit is obtained by taking the high-temperature limit h�!/kBT! 0. This gives

J ¼ kB�T4�

Z 1�1

d! T ð!Þ: ð65Þ

One can similarly evaluate various other quantities such as velocity–velocity correlationsand position–velocity correlations. The expressions for these in the classical case are,respectively,

K ¼ h _X _XTi

¼ kBTL�

Z 1�1

d!!Gþð!Þ�Lð!ÞG�ð!Þ þkBTR�

Z 1�1

d!!Gþð!Þ�Rð!ÞG�ð!Þ,

C ¼ hX _XTi

¼ ikBTL�

Z 1�1

d!Gþð!Þ�Lð!ÞG�ð!Þ þikBTR�

Z 1�1

d!Gþð!Þ�Rð!ÞG�ð!Þ:

The correlation functions K can be used to define the local energy density which can inturn be used to define the temperature profile in the non-equilibrium steady state of thewire. Also we note that the correlations C give the local heat current density. Sometimes itis more convenient to evaluate the total steady-state current from this expression ratherthan that in (63).

For one-dimensional wires the above results can be shown [44] to lead to expressionsfor current and temperature profiles obtained in earlier studies of heat conduction indisordered harmonic chains [44–46].

In our derivation of the LEGF results we have implicitly assumed that a unique steadystate will be reached. One of the necessary conditions for this is that no modes outside thebath spectrum are generated for the combined model of system and baths. These modes,

when they exist, are localized near the system and any initial excitation of the mode isunable to decay. This has been demonstrated and discussed in detail in the electroniccontext [47].

We note that unlike other approaches such as the Green–Kubo formalism andBoltzmann equation approach, the Langevin equation approach explicitly includes thereservoirs. The Langevin equation is physically appealing since it gives a nice picture of thereservoirs as sources of noise and dissipation. Also just as the Landauer formalism andNEGF have been extremely useful in understanding electron transport in mesoscopicsystems it is likely that a similar description will be useful for the case of heat transportin (electrically) insulating nanotubes, nanowires, etc. The LEGF approach has someadvantages over NEGF. For example, in the classical case, it is easy to write Langevin

equations for non-linear systems and study them numerically. Unfortunately, in thequantum case, one does not yet know how to achieve this, and understanding steady-statetransport in interacting quantum systems is an important open problem.

3.3. Ordered harmonic lattices

As mentioned above, heat conduction in the ordered harmonic chain was first studied inthe RLL paper and its higher-dimensional generalization was obtained by Nakazawa.The approach followed in both the RLL and Nakazawa papers was to obtain the exact

480 A. Dhar


non-equilibrium stationary-state measure which, for this quadratic problem, is a Gaussiandistribution. A complete solution for the correlation matrix was obtained and from thisone could obtain both the steady-state temperature profile and the heat current.

Here we follow [48] to show how the LEGF method, discussed in the previous section,can be used to calculate the heat current in ordered harmonic lattices connected to Ohmicreservoirs (for a classical system this is white noise Langevin dynamics). We will see how

exact expressions for the asymptotic current (N!1) can be obtained from this approach.We also briefly discuss the model in the quantum regime and extensions to higherdimensions.

3.3.1. One-dimensional case

The model considered in [48] is a slightly generalized version of the models studied by RLLand Nakazawa. An external potential is present at all sites and the pinning strength at theboundary sites are taken to be different from those at the bulk sites. Thus, both the RLLand Nakazawa results can be obtained as limiting cases. Also it seems that this model moreclosely mimics the experimental situation. In experiments the boundary sites would beinteracting with fixed reservoirs, and the coupling to those can be modified by an effectivespring constant that is expected to be different from the interparticle spring constant in the

bulk. We also note here that the constant on-site potential present along the wire relates toexperimental situations such as that of heat transport in a nanowire attached to a substrateor, in the two-dimensional case, a monolayer film on a substrate. Another example wouldbe the heat current contribution from the optical modes of a polar crystal.

Consider N particles of equal masses m connected to each other by harmonic springs ofequal spring constants k. The particles are also pinned by on-site quadratic potentials withstrengths ko at all sites except the boundary sites where the pinning strengths are koþ k0.The Hamiltonian is thus:

H ¼XNl¼1

p2l2mþ 12kox

2l

� �þXN�1l¼1

1

2kðxlþ1 � xlÞ2 þ

1

2k0ðx21 þ x2NÞ, ð66Þ

where xl denotes the displacement of the lth particle from its equilibrium position.The particles 1 and N at the two ends are connected to heat baths at temperature TL andTR, respectively, assumed to be modelled by Langevin equations corresponding to Ohmic

baths (�(!)¼ i�!). In the classical case the steady-state heat current from left to rightreservoir can be obtained from (65) and given by [44,49]

J ¼ kBðTL � TRÞ4�

Z 1�1

d! T Nð!Þ, ð67Þ

where

T Nð!Þ ¼ 4�2ð!ÞjĜ1Nð!Þj2, Ĝð!Þ ¼Ẑ�1

k

and

Ẑ ¼ ½�m!2Îþ �̂� �̂ð!Þ�

k,



where Î is a unit matrix, �̂ is the force matrix corresponding to the

Hamiltonian in (66). The N�N matrix �̂ has mostly zero elements except for�11¼�NN¼ i�(!) where �(!)¼ �!. The matrix Ẑ is tri-diagonal matrix withZ11¼ZNN¼ (kþ koþ k0�m!2�i�!)/k, all other diagonal elements equal to 2þ ko/k�m!2/k and all off-diagonal elements equal to �1. Then it can be shown easilythat jG1N(!)j ¼ 1/(kj�Nj) where �N is the determinant of the matrix Ẑ. This can beobtained exactly. For large N, only phonons within the spectral band of the system can

transmit, and the integral over ! in (67) can be converted to an integral over q to give

J ¼ 2�2kBðTL � TRÞ

k2�

Z �0

dqd!

dq

!

2q

j�Nj2, ð68Þ

with m!2q ¼ ko þ 2k½1� cos ðqÞ�. Now using the result

limN!1

Z �0

dqg1ðqÞ

1þ g2ðqÞ sinNq¼Z �0

dqg1ðqÞ

½1� g22ðqÞ�1=2

, ð69Þ

where g1(q) and g2(q) are any two well-behaved functions, one can show that in the limit

N!1, (68) gives

J ¼ �k2kBðTL � TRÞ

m�2ð��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 ��2

pÞ, ð70Þ

where

� ¼ 2kðk� k0Þ þ k02 þ ðko þ 2kÞ�2

mand � ¼ 2kðk� k0Þ þ 2k�

2

m:

Two different special cases lead to the RLL and Nakazawa results. First in the case

of fixed ends and without on-site potentials, i.e. k0 ¼ k and ko¼ 0, we recover the RLLresult [34]:

JRLL ¼ kkBðTL � TRÞ2�

1þ �2� �2

ffiffiffiffiffiffiffiffiffiffiffi1þ 4

�

r" #where � ¼ mk

�2: ð71Þ

In the other case of free ends, i.e. k0 ¼ 0, one obtains the Nakazawa result [35]:

JN ¼ k�kBðTL � TRÞ2ðmkþ �2Þ 1þ

�

2� �

2

ffiffiffiffiffiffiffiffiffiffiffi1þ 4

�

r" #where � ¼ ko�

2

kðmkþ �2Þ : ð72Þ

In the quantum case, in the linear response regime, (64) and similar manipulations made

above for the N!1 limit lead to the following final expression for the current:

J ¼ �k2�h2ðTL � TRÞ4�kBmT2

Z �0

dqsin2 q

�� cos q!2q cosech

2 �h!q2kBT

� �, ð73Þ

where !2q ¼ ½ko þ 2kð1� cos qÞ�=m. While one cannot perform this integral exactly,numerically it is easy to obtain the integral for given parameter values. It is interesting

to examine the temperature dependence of the conductance J/�T. In the classical case this

is independent of the temperature while one finds that at low temperatures the quantum

482 A. Dhar


result is completely different. For three different cases one finds, in the low-temperature

(T� h�(k/m)1/2/kB) regime, the following behaviour:

J �

T3 for k0 ¼ k, ko ¼ 0,T for k0 ¼ 0, ko ¼ 0,

e��h!o=ðkBT Þ

T1=2for k0 ¼ 0, ko 6¼ 0,

8>><>>: ð74Þ

where !o¼ (ko/m)1/2. In studies trying to understand experimental work on nanosystems(see Section 8), the temperature dependence of the conductance is usually derived from the

Landauer formula, which corresponds to the Rubin model of the bath. The temperature

dependence will then be different from the above results.

3.3.2. Higher dimensions

As shown by Nakazawa [35] the problem of heat conduction in ordered harmonic

lattices in more than one dimension can be reduced to an effectively one-dimensional

problem. We briefly give the arguments here and also give the quantum generalization.Let us consider a d-dimensional hypercubic lattice with lattice sites labelled by

the vector n¼ {n�}, �¼ 1, 2, . . . , d, where each n� takes values from 1 to L�. The totalnumber of lattice sites is thus N¼L1 L2. . . Ld. We assume that heat conduction takes placein the �¼ d direction. Periodic BCs are imposed in the remaining d� 1 transversedirections. The Hamiltonian is described by a scalar displacement Xn and, as in the one-

dimensional case, we consider nearest-neighbour harmonic interactions with a spring

constant k and harmonic on-site pinning at all sites with spring constant ko. All boundary

particles at nd¼ 1 and nd¼Ld are additionally pinned by harmonic springs with stiffnessk0 and follow Langevin dynamics corresponding to baths at temperatures TL and TR,respectively.

Let us write n¼ (nt, nd) where nt¼ (n1, n2, . . . , nd�1). Also let q¼ (q1, q2, . . . , qd�1) withq�¼ 2�s/L� where s goes from 1 to L�. Then defining variables

Xnd ðqÞ ¼1

L1=21 L1=22 . . .L

1=2d�1

Xnt

Xnt, ndeiq:nt , ð75Þ

one finds that, for each fixed q, Xnd(q) (nd¼ 1, 2, . . . ,Ld) satisfy Langevin equationscorresponding to the one-dimensional Hamiltonian in (66) with the on-site spring constant

ko replaced by

�ðqÞ ¼ ko þ 2k�d� 1�

X�¼1,d�1

cos ðq�Þ�: ð76Þ

For Ld!1, the heat current J(q) for each mode with given q is then simply given by (70)with ko replaced by �q. In the quantum mechanical case we use (73). The heat current perbond is then given by

J ¼ 1L1L2 . . .Ld�1

Xq

JðqÞ: ð77Þ



Note that the result holds for finite lengths in the transverse direction. For infinite

transverse lengths we obtain J ¼R� � �R 2�0 dq JðqÞ=ð2�Þ

d�1.

3.4. Disordered harmonic lattices

From the previous section we see that the heat current in an ordered harmonic lattice

is independent of system size (for large systems) and, hence, transport is ballistic.

Of course, this is expected since there is no mechanism for scattering of the heat carriers,

namely the phonons. Two ways of introducing scattering of phonons are by introducing

disorder into the system or by including anharmonicity which would cause phonon–

phonon interactions. In this section we consider the effect of disorder on heat conduction

in a harmonic system.Disorder can be introduced in various ways, for example by making the masses

of the particles random as would be the case in a isotopically disordered solid

or by making the spring constants random. Here, we discuss the case of mass disorder

only since the most important features do not seem to vary much with the type of

disorder one is considering. Specifically, we consider harmonic systems where the

mass of each particle is an independent random variable chosen from some fixed

distribution.It can be expected that heat conduction in disordered harmonic systems will be

strongly affected by the physics of Anderson localization. In fact, the problem of

finding the normal modes of the harmonic lattice can be directly mapped to that

of finding the eigenstates of an electron in a disordered potential (for example, in a tight-

binding model) and so we expect the same kind of physics as in electron localization.

In the electron case the effect of localization is strongest in one dimension, where it

can be proved rigorously that all eigenstates are exponentially localized, hence the current

decays exponentially with system size and the system is an insulator. This is believed

to be true in two dimensions also. In the phonon case the picture is much the same

except that, in the absence of an external potential, the translational invariance

of the problem leads to the fact that low-frequency modes are not localized and are

effective in transporting energy. Another important difference between the electron

and phonon problems is that electron transport is dominated by electrons near the

Fermi level while in the case of phonons, all frequencies participate in transport.

These two differences lead to the fact that the disordered harmonic crystal in

one and two dimensions is not a heat insulator, unlike its electronic counterpart.

Here we present results using the LEGF approach to determine the system size

dependence of the current in one-dimensional mass-disordered chains. We note that

basically this same approach (NEGF) is also popular in the electron case and is widely

used in mesoscopic physics. Also earlier treatments by, for example, Casher and Lebowitz

[49] and by Rubin and Greer [45] of the disordered harmonic chain, can be viewed as

special examples of the LEGF approach. We also discuss results of simulations for the

two-dimensional case.Our main conclusions here will be that Fourier’s law is not valid in a disordered

harmonic crystal in one and two dimensions, the current decays as a power law

with system size and the exponent � is sensitive to BCs and spectral properties of theheat baths.

484 A. Dhar


3.4.1. One-dimensional disordered lattice

We first briefly review earlier work on this problem [2]. The thermal conductivity ofdisordered harmonic lattices was first investigated by Allen and Ford [13] who, using theKubo formalism, obtained an exact expression for the thermal conductivity of a finitechain attached to infinite reservoirs. From this expression they concluded, erroneously aswe now know, that the thermal conductivity remains finite in the limit of infinite systemsize. Simulations of the disordered lattice connected to white noise reservoirs were carriedout by Payton et al. [50]. They were restricted to small system sizes (N� 400) and alsoobtained a finite thermal conductivity.

Possibly the first paper to note anomalous transport was that by Matsuda and Ishii(MI) [51]. In an important work on the localization of normal modes in the disorderedharmonic chain, MI showed that all high-frequency modes were exponentially localized.However, for small ! the localization length in an infinite sample was shown to vary as!�2, hence normal modes with frequency !9!d have localization length greater than N,and cannot be considered as localized. For a harmonic chain of length N, given the averagemass m¼hmli, the variance 2¼h(ml�m)2i and interparticle spring constant k, it wasshown that

!d �km

N2

� �1=2ð78Þ

They also evaluated expressions for thermal conductivity of a finite disordered chainconnected to two different bath models, namely:

. model (a), white noise baths; and

. model (b), baths modelled by semi-infinite ordered harmonic chains (Rubin’smodel of bath).

In the following, we also consider these two models of baths. For model (a) MI usedfixed BCs (boundary particles in an external potential) and the limit of weak coupling tobaths, while for case (b) they considered free BCs (boundary particles not pinned) and thiswas treated using the Green–Kubo formalism given by Allen and Ford [13]. They found�¼ 1/2 in both cases, a conclusion which we will see is incorrect. The other two importanttheoretical papers on heat conduction in the disordered chain were those by Rubin andGreer [45] who considered model (b) and of Casher and Lebowitz [49] who used model(a) for baths. A lower bound [J]� 1/N1/2 was obtained for the disorder averaged current [J]in [45] who also gave numerical evidence for an exponent �¼ 1/2. This was later provedrigorously by Verheggen [52]. On the other hand, for model (a), Casher and Lebowitz [49]found a rigorous bound [J]� 1/N3/2 and simulations by Rich and Visscher [53] with thesame baths supported the corresponding exponent �¼�1/2. The work in [44] gavea unified treatment of the problem of heat conduction in disordered harmonic chainsconnected to baths modelled by generalized Langevin equations and showed that models(a) and (b) were two special cases. An efficient numerical scheme was proposed and used toobtain the exponent � and it was established that �¼�1/2 for model (a) (with fixed BCs)and �¼ 1/2 for model (b) (with free BCs). It was also pointed out that, in general, � wasdependent on the spectral properties of the baths. We briefly describe this formulation [54]here and see how one can understand the effect of BCs on heat transport in the disorderedchain. We consider both the white noise (model (a)) and Rubin baths (model (b)). One of



the main conclusions is that the difference in exponents obtained for these two cases arises

from use of different BCs, rather than because of differences in the spectral properties of

the baths.The Hamiltonian of the mass-disordered chain is given by

H ¼XNl¼1

p2l2mlþXN�1l¼1

1

2kðxlþ1 � xlÞ2 þ

1

2k0ðx21 þ x2NÞ: ð79Þ

The random masses {ml} are chosen from, say, a uniform distribution between (m��) to(mþ�). The strength of on-site potentials at the boundaries is k0. The particles at two endsare connected to heat baths, at temperature TL and TR, and modelled by generalized

Langevin equations. The steady-state classical heat current through the chain is given by

J ¼ kBðTL � TRÞ4�

Z 1�1

d! T Nð!Þ, ð80Þ

where

T Nð!Þ ¼ 4�2ð!ÞjĜ1Nð!Þj2, Ĝð!Þ ¼Ẑ�1

k

and

Ẑ ¼ ½�!2M̂þ �̂� �̂ð!Þ�

k,

where M̂ and �̂ are, respectively, the mass and force matrices corresponding to (79).

As shown in [44], the non-zero elements of the diagonal matrix �̂ð!Þ for models (a) and (b)are given by

�11ð!Þ ¼ �NNð!Þ ¼ �ð!Þ ¼�i�! model ðaÞ,kf1�m!2=2k� i!ðm=kÞ1=2½1�m!2=ð4kÞ�1=2g model ðbÞ,

(

ð81Þ

where � is the coupling strength with the white noise baths, while in the case of Rubin’sbaths it has been assumed that the Rubin bath has spring constant k and equal masses m.

As noted above, T N(!) is the transmission coefficient of phonons through the disorderedchain. To extract the asymptotic N dependence of the disorder averaged current [J] one

needs to determine the Green’s function element G1N(!). It is convenient to writethe matrix elements Z11¼�m1!2/kþ 1þ k0/k��/k¼�m1!2/kþ 2��0, where �0 ¼�/k�k0/kþ 1 and similarly ZNN¼�mN!2/kþ 2��0. Following the techniques used in [44,49]one obtains

jG1Nð!Þj2 ¼ k�2j�Nð!Þj�2 ð82Þ

with

�Nð!Þ ¼ D1,N ��0ðD2,N þD1,N�1Þ þ�02D2,N�1,

486 A. Dhar


where �N(!) is the determinant of Ẑ and the matrix elements Dl,m are given by thefollowing product of (2� 2) random matrices T̂l:

D̂ ¼D1,N �D1,N�1D2,N �D2,N�1

� �¼ T̂1T̂2 . . . T̂N, ð83Þ

where

T̂l ¼2�ml!2=k �1

1 0

� �:

We note that the information about bath properties and BCs are now contained entirely in

�0(!) while D̂ contains the system properties. It is known that jDl,mj � ecN!2

(see [51]),

where c is a constant, and so we need to look only at the low-frequency (!9 1/N1/2) formof �0. Let us now discuss the various cases. For model (a) free BCs correspond to k0 ¼ 0and so �0 ¼ 1� i�!/k while for model (b) free boundaries correspond to k0 ¼ k and thisgives, at low frequencies, �0 ¼ 1� i(m/k)1/2!. Other choices of k0 correspond to pinnedboundary sites with an on-site potential kox

2/2 where ko¼ k0 for model (a) and ko¼ k0 � kfor model (b). The main difference, from the unpinned case, is that now Re[�0] 6¼ 1.The arguments of [44] then immediately give �¼ 1/2 for free BCs and �¼�1/2 for fixedBCs for both bath models. In fact, for the choice of parameters �¼ (mk)1/2, the imaginarypart of �0 is the same for both baths and, hence, we expect, for large system sizes, that theactual values of the current will be the same for both bath models. This can be seen in

Figure 1 where the system size dependence of the current for the various cases is shown.

The current was evaluated numerically using (80) and averaging over many realizations.

Note that for free BCs, the exponent �¼ 1/2 settles to its asymptotic value at relativelysmall values (N� 103) while, with pinning, one needs to examine much longer chains(N� 105).

100 1000 10000 1e+05N

1e-06

0.0001

0.01

[J]

Model (a) free BCModel (b) free BCModel (a) fixed BCModel (b) fixed BC

Figure 1. Plot of [J] versus N for free and fixed BCs. Results are given for both models (a) and (b)of baths. The two straight lines correspond to the asymptotic expressions given in (85) and (86)and have slopes �1/2 and �3/2. Parameters used were m¼ 1, �¼ 0.5, k¼ 1, �¼ 1, TL¼ 2, TR¼ 1and ko¼ 1. Reprinted with permission from [54]. Copyright � (2008) by the American PhysicalSociety.



These results clearly show that, for both models (

please scroll down for articleabhi/papers/45-advinp.pdfdeveloped approach, using generalized...

Documents