Approach to a non-equilibrium steady state

ARTICLE IN PRESS

0378-4371/$ - se

doi:10.1016/j.ph

�Correspond28040 Madrid,

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Physica A 369 (2006) 379–386

www.elsevier.com/locate/physa

Approach to a non-equilibrium steady state

Jaros"aw Piaseckia, Rodrigo Sotob,c,�

aInstitute of Theoretical Physics, University of Warsaw, Hoza 69, 00 681 Warsaw, PolandbDepartamento de Fısica, Facultad de Ciencias Fısicas y Matematicas, Universidad de Chile, Casilla 487-3, Santiago, Chile

cDepartamento de Fısica Aplicada I (Termologıa), Facultad de Ciencias Fısicas, Universidad Complutense, 28040 Madrid, Spain

Received 10 May 2005; received in revised form 17 November 2005

Available online 23 January 2006

Abstract

We consider a non-interacting one-dimensional gas accelerated by a constant and uniform external field. The energy

absorbed from the field is transferred via elastic collisions to a bath of scattering obstacles. At gas–obstacle encounters the

particles of the gas acquire a fixed kinetic energy. The approach to the resulting stationary state is studied within the

Boltzmann kinetic theory. It is shown that the long time behavior is governed by the hydrodynamic mode of diffusion

superposed on a convective flow. The diffusion coefficient is analytically calculated for any value of the field showing a

minimum at intermediate field intensities. It is checked that the properly generalized Green–Kubo formula applies in the

non-equilibrium stationary state.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Kinetic theory; Non-equilibrium; Diffusion

1. Introduction

The problem of constructing adequate statistical ensembles for non-equilibrium stationary states (hereafterdenoted by NESS) has a long history (see e.g. Refs. [1,2]). Related to it there are very interesting findingsconcerning the nature of fluctuations in the presence of spatial gradients inducing stationary flows [3]. Thecomputation of exact expressions for the probability distribution of simple NESS allows to study differentstatic properties as, for example, the long-range density correlations [4]. The recent development of the theoryof fluidized granular matter has also attracted a lot of attention to the structure of NESS (see for exampleRefs. [5–7]). Finally, there is at present an important research carried on in order to properly define theentropy production accompanying stationary dissipative currents (see e.g. Ref. [8]).

However, whereas the modes of approach of fluids to thermal equilibrium are well understood, there is stillno satisfactory theory of the dynamics of reaching NESS. In both cases the final state is stationary, but inNESS, in contradistinction to equilibrium, there persist non-vanishing fluxes sustained by the coupling to theenvironment.

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2005.12.045

ing author. Departamento de Fısica Aplicada I (Termologıa), Facultad de Ciencias Fısicas, Universidad Complutense,

Spain.

ess: [email protected] (R. Soto).

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ARTICLE IN PRESSJ. Piasecki, R. Soto / Physica A 369 (2006) 379–386380

The kinetic theory of fluids revealed the fundamental role of the separation between different time scales inthe approach to equilibrium (see e.g. the presentation of Bogoliubov’s ideas in Ref. [9]). In particular, in theevolution of the density of particles in the one-particle phase space all the degrees of freedom but thoserepresented by the densities of globally conserved quantities relax at a short time scale producing a state closeto local equilibrium. The subsequent slow evolution is essentially governed by changes of the hydrodynamicfields (an original construction of the hydrodynamic modes within the kinetic theory has been developed byResibois and De Leener [10]). One can wonder whether this kind of mechanism based on the time-scaleseparation still persists when the final state involves some dissipative stationary process related to the currentflowing through the system. This is precisely the question we want to address in this paper.

Our object here is to exploit an analytically soluble one-dimensional model discussed in Ref. [11] in order toperform a detailed study of the evolution towards a NESS. Although the system is very simple indeed, we findthat the possibility of a rigorous study of its dynamics is precious from the point of view of the search for thelaws governing the approach to NESS. The coupling to the environment is represented in the model by theaction of an external field. One of the original results of the paper is an explicit formula for the field-dependentcoefficient of diffusion governing the evolution of the only hydrodynamic mode in the approach to thestationary state.

In Section 2, we define the system and its dynamics. The following analysis of the approach to the stationarystate is presented in Section 3. Finally, Section 4 contains the concluding comments.

2. One-dimensional system and its dynamics

We consider non-interacting point particles of mass m moving in one dimension with acceleration a underthe influence of an external constant and uniform field. The particles are surrounded by a bath of pointobstacles of mass m that move with velocities �U , both directions being equally probable. At elastic collisionsbetween the particles and the obstacles, a simple interchange of velocities is taking place. The scatteringobstacles are uniformly distributed with a number density n.

The statistical state of the particles is described by the probability density f ðr; v; tÞ for finding a particle attime t at point r with velocity v. We assume that the evolution of the density f ðr; v; tÞ is governed by the linearBoltzmann equation [11]

qqtþ v

qqrþ a

qqv

� �f ðr; v; tÞ ¼ n

Zdcjv� cj f ðr; c; tÞfðvÞ � f ðr; v; tÞfðcÞ½ �, (1)

putting

fðvÞ ¼ 12dðvþUÞ þ dðv�UÞ½ � (2)

for the velocity distribution of the obstacles. When writing Eq. (1) it has been supposed that the particlesalways encountered obstacles with velocities distributed according to (2). The possibility of recollisions withobstacles perturbed by previous collisions with the particles is not taken into account by the Boltzmanncollision term in (1).

In dimensionless variables w ¼ v=U , x ¼ nr, and t ¼ Unt Eq. (1) takes the form

qqtþ w

qqxþ �

qqwþ

1

2ðjwþ 1j þ jw� 1jÞ

� �F ðx;w; tÞ ¼

1

2½dðw� 1Þm�ðx; tÞ þ dðwþ 1Þmþðx; tÞ�, (3)

where F ðx;w; tÞ ¼ f ðx=n;Uw; t=nUÞU=n is the dimensionless probability density, and

m�ðx; tÞ ¼Z

dwjw� 1jF ðx;w; tÞ. (4)

In Eq. (3), there appears the dimensionless intensity of the field � ¼ a=nU2.Provided that appropriate boundary conditions are supplied at �1, Eq. (3) has a non-equilibrium

stationary solution representing a homogeneous NESS whose velocity distribution F 0ðw; �Þ, that can be chosento be normalized as

Rdw F 0ðw; �Þ ¼ 1, has been derived and analyzed in Ref. [11]. The NESS is characterized

ARTICLE IN PRESSJ. Piasecki, R. Soto / Physica A 369 (2006) 379–386 381

by a constant current VNESSð�Þ ¼Rdw wF 0ðw; �Þ that shows a linear response, VNESSð�Þ��, in the weak field

limit j�j51 and a non-linear response, VNESSð�Þ�sgnð�Þffiffiffiffiffij�jp

, for strong fields j�jb1.Whereas the paper [11] focused on the properties of the stationary distribution, our aim here is to analyze

the approach to the asymptotic homogeneous NESS starting from an arbitrary inhomogeneous initialcondition. This requires solving the time-dependent Boltzmann equation. The way towards the construction ofthe solution has been already found in Ref. [11]. Using the Fourier–Laplace transformation

~F ðk;w; zÞ ¼

Z 10

dtZ

dx e�zt�ikxF ðx;w; tÞ (5)

one finds from (3) the integral equation

~F ðk;w; zÞ ¼ ~Hðk;w; zÞ þ1

2j�jexp �

wðwÞ�� ik

w2 � 1

2�

� �

� exp1� zðw� 1Þ

�

� �y½�ðw� 1Þ� ~m�ðk; zÞ þ exp �

1þ zðwþ 1Þ

�

� �y½�ðwþ 1Þ� ~mþðk; zÞ

� �, ð6Þ

where

wðwÞ ¼ ðwþ 1Þjwþ 1j þ ðw� 1Þjw� 1j½ �=4, (7)

which obey the relation wðwÞ ¼ �wð�wÞ,

~m�ðk; zÞ ¼Z

dwjw� 1j ~F ðk;w; zÞ, (8)

and

~Hðk;w; zÞ ¼

Z 10

e�zt�ikxHðx;w; tÞ (9)

relates to the initial condition F ðx;w; 0Þ through

Hðx;w; tÞ ¼ exp½ðwðw� �tÞ � wðwÞÞ=��F ðx� wtþ �t2=2;w� �t; 0Þ. (10)

Eq. (6) is implicit expressing ~F ðk;w; zÞ in terms of ~m�ðk; zÞ that are linear functionals of ~F ðk;w; zÞ. However,the solution can be made explicit in a straightforward way. Indeed, multiplying Eq. (6) by jw� 1j andintegrating with respect to w one obtains a closed system of linear equations for ~m� in the form

Mðk; z; �Þ~m�ðk; zÞ

~mþðk; zÞ

" #¼

h�ðk; zÞ

hþðk; zÞ

" #. (11)

Here,

h�ðk; zÞ ¼

Zdwjw� 1j ~Hðk;w; zÞ (12)

and Mðk; z; �Þ is a two-by-two matrix analytic in z given by

Mðk; z; �Þ ¼M11ðk; z; �Þ; M12ðk; z; �Þ

M21ðk; z; �Þ; M22ðk; z; �Þ

!, (13)

with

M11 ¼1

2j�j

Zdwjw� 1j exp �

wðwÞ�� ik

w2 � 1

2�

� �exp

1� zðw� 1Þ

�

� �y½�ðw� 1Þ�,

M12 ¼1

2j�j

Zdwjw� 1j exp �

wðwÞ�� ik

w2 � 1

2�

� �exp �

1þ zðwþ 1Þ

�

� �y½�ðwþ 1Þ�,


M21 ¼1

2j�j

Zdwjwþ 1j exp �

wðwÞ�� ik

w2 � 1

2�

� �exp

1� zðw� 1Þ

�

� �y½�ðw� 1Þ�,

M22 ¼1

2j�j

Zdwjwþ 1j exp �

wðwÞ�� ik

w2 � 1

2�

� �exp �

1þ zðwþ 1Þ

�

� �y½�ðwþ 1Þ�.

When �40, the matrix Mðk; z; �Þ takes the form

Mðk; z; �Þ ¼1� ðA� BÞ eð2zþ1Þ=2�; C �Dþ ðB� AÞ e�ð2zþ3Þ=2�

�ðAþ BÞ eð2zþ1Þ=2�; 1� ðAþ BÞ e�ð2zþ3Þ=2� � C �D

!, (14)

where

Aðk; z; �Þ ¼1

2�

Z 11

dw w exp �1

2�½w2ð1þ ikÞ þ 2wz� ik�

� �, (15)

Bðk; z; �Þ ¼1

2�

Z 11

dw exp �1

2�½w2ð1þ ikÞ þ 2wz� ik�

� �, (16)

Cðk; z; �Þ ¼1

2�

Z 1

�1

dw w exp �1

2�½ikw2

þ 2ðwþ 1Þðzþ 1Þ � ik�

� �, (17)

Dðk; z; �Þ ¼1

2�

Z 1

�1

dw exp �1

2�½ikw2

þ 2ðwþ 1Þðzþ 1Þ � ik�

� �. (18)

The corresponding formulae for �o0 follow from the important symmetry relations

M11ðk; z; �Þ ¼M22ð�k; z;��Þ, (19)

M12ðk; z; �Þ ¼M21ð�k; z;��Þ. (20)

3. Modes of approach to the non-equilibrium steady state

The linear equation (11) contains all information about the modes by which the system approaches theNESS. Whether these modes behave for long wavelengths as hydrodynamic modes is the question we wantto study now. To this end we use the explicit solution of the Boltzmann equation. The structure of (6) and(11) implies that ~F ðk;w; zÞ is an analytic function in the complex plane except at points z where the inversematrix M�1 does not exist. Thus, the zeros of the determinant DetðMÞ define the singularities in the z-plane ofthe Fourier–Laplace transform ~F ðk;w; zÞ. In particular, isolated zeros ziðkÞ corresponding to simple poles willproduce after applying the inverse Laplace transformation an exponential time dependence of the form

F ðk;w; tÞ ¼X

i

ai eziðkÞt, (21)

with coefficients ai for each mode depending on the initial condition (F denotes the spatial Fourier transformof F ).

We performed a systematic numerical survey of the zeros of the determinant DetðMÞ for different values ofk and �. It has been found that regardless of the value of � there is a single isolated zero which behaves likez0 ¼ �Dk2

� ikV þOðk3Þ for k51. All the other zeros, both for finite k and in the limit k! 0, have negative

real parts and are located outside a band of a certain finite width around the imaginary axis. That is, theseother zeros do not have an accumulation point with vanishing real part. Therefore, for long wavelengths,there is exactly one slow diffusive mode (coefficient D) combined with convective transport with velocity V.All the others are fast kinetic modes. The presence of an unique slow mode is related to the fact thatthe mass is the only conserved quantity. No zeros with positive real part were found, reflecting the fact that theNESS is stable.


The slow mode can be obtained analytically by inserting the asymptotic formula z ¼ �Dð�Þk2� ikV ð�Þ into

the equation DetðMÞ ¼ 0 and then solving for D and V keeping only dominant terms in k. The relations (20)imply that the zeros of the determinant of the matrix Mðk; z; �Þ coincide with the zeros of the determinant ofthe matrix Mð�k; z;��Þ. It follows the symmetry relations

Dð�Þ ¼ Dð��Þ ¼ Dðj�jÞ, (22)

V ð�Þ ¼ �V ð��Þ ¼ sgnð�ÞVðj�jÞ. (23)

One finds then a unique solution

Vð�Þ ¼�½1þ e�2=�� þ ½�� 1� ð1þ �Þ e�2=��Ið�Þ

�½1� e�2=�� þ ½��þ 3þ ð1þ �Þ e�2=��Ið�Þ(24)

and

Dð�Þ ¼ ½e2=�ð�2�3 þ �4 þ 2�5 þ 8�2Ið�Þ þ 2�3Ið�Þ � 10�4Ið�Þ � 8�5Ið�Þ � 10�Ið�Þ2

� 11�2Ið�Þ2 � 2�3Ið�Þ2 þ 7�4Ið�Þ2 þ 6�5Ið�Þ2 þ 4Ið�Þ3 þ 8�Ið�Þ3 þ 12�2Ið�Þ3

þ 14�3Ið�Þ3 þ 6�4Ið�Þ3Þ þ e4=�ð�4�3 þ 2�4 � 4�5 þ 20�2Ið�Þ � 8�3Ið�Þ

� 12�4Ið�Þ þ 16�5Ið�Þ � 20�Ið�Þ2 � 2�2Ið�Þ2 þ 20�3Ið�Þ2 þ 22�4Ið�Þ2 � 12�5Ið�Þ2

þ 4Ið�Þ3 þ 16�2Ið�Þ3 þ 8�3Ið�Þ3 � 12�4Ið�Þ3Þ þ e6=�ð�2�3 � 3�4 þ 2�5

þ 12�2Ið�Þ � 26�3Ið�Þ þ 22�4Ið�Þ � 8�5Ið�Þ � 10�Ið�Þ2 � 3�2Ið�Þ2 þ 30�3Ið�Þ2

� 29�4Ið�Þ2 þ 6�5Ið�Þ2 � 8�Ið�Þ3 þ 20�2Ið�Þ3 � 22�3Ið�Þ3 þ 6�4Ið�Þ3Þ�=

½�ð�� Ið�Þ � �Ið�Þ þ e2=�ð�� 3Ið�Þ þ �Ið�ÞÞÞ3�, ð25Þ

with

Ið�Þ ¼ e1=2j�jZ 11

dw e�w2=2j�j. (26)

The drift velocity V ð�Þ (24) coincides, as expected, with the stationary average velocity VNESS found inRef. [11]. In Fig. 1 we plotted its dependence on �, showing the transition from a linear to a nonlinearresponse.

The explicit expression for the diffusion coefficient (25) is quite involved. A plot of it is presented in Fig. 1. Itshould be noticed that Dð�Þ has an interesting structure as a function of the intensity of the external field

0.0

0.5

1.0

1.5

2.0

2.5

V

0 2 4 6 8 10

ε

0.380.400.420.440.460.480.50

D

Fig. 1. Top: average velocity VNESSð�Þ in the NESS as a function of the imposed acceleration �. Bottom: field dependent diffusion

coefficient Dð�Þ.


showing a minimum for � � 1:60. The asymptotic formulae for small and large intensities are

Dð�Þ � 12� 15

8�2; j�j51, (27)

Dð�Þ �4� pffiffiffiffiffiffiffi2p3p �1=2 þ

9p� 20

ð2pÞ3=2��1=2; �b1. (28)

It should be remarked that in the limit of vanishing external field the value Dð0Þ coincides with thatfollowing at equilibrium (� ¼ 0) from the Green–Kubo formula. For large values of �, the velocity of theobstacles becomes negligible compared to that acquired by accelerated motion and their action resembles thatof stopping centers. Therefore, by purely dimensional analysis one can predict the exponent 1

2in the

dependence of Dð�Þ in (28).The appearance of a minimum in Dð�Þ shown in Fig. 1 is an interesting point which calls for interpretation.

The minimum can be understood by analyzing the partial collision frequencies with obstacles moving withvelocities þ1 or �1. In the NESS, these collision frequencies are given by

m0�ð�Þ ¼Z

dwjw� 1jF0ðw; �Þ, (29)

where the average is computed with the use of the NESS distribution F0ðw; �Þ given in Eq. (40) of Ref. [11]. Asshown in Fig. 2, the collision frequency m0� with obstacles moving to the right (in the direction of theaccelerating field) has a minimum at � ¼ 1:37. In order to understand this fact consider a particle that has justcollided with an obstacle with velocity �1 getting instantaneously its velocity. If the field is weak, it willcontinue moving to the left encountering obstacles with velocity þ1 and will be sent again towards those withvelocity �1 the collision frequency m0� remaining high. Also, if the field is large enough, the particle will turnrapidly to the right and gain a large velocity leading again to a large value of m0�. There are, however,intermediate values of acceleration where the particle remains a long time turning to the right under the actionof the field before the next collision occurs without getting large velocities and thus reducing the collision rate.This fact explains the origin of the minimum in m0�ð�Þ. Finally, the presence of the minimum in m0� can berelated to the observed minimum of Dð�Þ because the lowering of the collision rate for a range of values of �produces an evolution that is closer to a deterministic one, and thus less diffusive.

Knowing the analytic solution of the initial value problem for the kinetic equation (3) we could also verify ifthe diffusion coefficient (25) follows from the Green–Kubo formula when the NESS distribution is used in theevaluation of the autocorrelation function. We thus considered the Green–Kubo expression

DGK ¼ limz!0

Z 10

dt e�zthðw� V ÞðtÞðw� V Þð0ÞiNESS ð30Þ

¼ limz!0

Z 10

dt e�ztZ

dwðw� V ÞFGKðw; tÞ, ð31Þ

0 21 3 4 5ε

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

µ 0+, µ

0−

Fig. 2. Partial collision frequencies in the NESS, solid line m0þ, dashed line m0�.


where FGKðw; tÞ is the solution of Eq. (3) with initial condition FGKðw; 0Þ ¼ ðw� V ÞF0ðwÞ and V is the NESSdrift velocity. The computation process is as follows. Given the initial condition FGKðw; 0Þ, Eq. (11) is solvedfor ~m� (note that z must be nonzero because otherwise the inverse matrix Mðk ¼ 0; zÞ�1 does not exist). Then,~FGKðw; zÞ is obtained and the coefficient DGK can be found from the relation

DGK ¼ limz!0

Zdwðw� V Þ ~FGKðw; zÞ. (32)

We performed the calculation obtaining a finite value which coincides with (25). We thus conclude that whenthe system is in the NESS, both in the linear and in the nonlinear response regimes, the diffusion coefficient isgiven by the Green–Kubo formula provided the NESS distribution is used in the averaging process and thefluctuations of the velocity around the non-zero mean value VNESS are considered. This conclusion is alsoconfirmed by an analogous result obtained for a simpler one-dimensional case where only nonlinear transportis present [12].

Finally, we have checked that beyond the linear response regime the system does not obey thefluctuation-dissipation theorem, relating the diffusion coefficient D and the mobility m. The mobilityis defined as m ¼ dVNESS=d�. Evaluating the non-equilibrium stationary temperature T ¼ hðw� VNESSÞ

2i

using the stationary distribution F 0, it is directly checked that Dð�ÞaTð�Þmð�Þ. The equality is onlysatisfied at the equilibrium case � ¼ 0. A similar phenomenon has been predicted in granulargases [13].

4. Concluding comments

We have considered a one-dimensional system of particles absorbing energy from a constant and uniformexternal field, and dissipating it via collisions with a bath of scattering obstacles with a fixed velocitydistribution (2). In this situation the appropriate linear Boltzmann equation (3) predicts the approach to anon-equilibrium steady state (NESS) characterized by a mass flux that can present both linear and nonlinearresponse depending on the strength of the field. It has been found that the evolution of arbitrary initialconditions towards the NESS involves short time-scale kinetic modes producing the stationary velocitydistribution followed by the slow long wavelength hydrodynamic diffusion superposed on a convective currentflowing with the average velocity of the NESS.

Analytic calculations show an interesting structure in the dependence of the diffusion coefficient on theintensity of the external field, with a minimum at intermediate values of the field. The minimum is related tothe presence of another minimum in the collision frequency between the particles and the obstacles, thusreducing the number of randomizing scattering processes. Furthermore, we have also shown that the value ofthe diffusion coefficient could be obtained from a Green–Kubo formula by considering the time displacedpeculiar velocity correlation function (subtracting the average NESS velocity) averaged over the NESSdistribution.

It thus appears that the modes of approach to the NESS when only the mass remains as a conservedquantity involve the corresponding single hydrodynamic mode which is the classical process of spatialdiffusion. The diffusion coefficient does depend on the external field intensity and can be obtained at any valueof the field from an appropriately generalized Green–Kubo formula. We conclude that the approach to NESSshows qualitative analogy to that of reaching equilibrium. Hopefully, this conclusion remains valid for a largeclass of systems in which the stationary state results from the balance between the energy flow absorbed froman external field and collisional coupling to a kind of thermostat.

Acknowledgements

The authors thank P. Cordero for helpful discussions. This work has been partly financed by Fondecytresearch Grants 1030993 and 7040123 and Fondap Grant 11980002.


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[5] J. Karkheck (Ed.), Dynamics: Models and Methods for Non-Equilibrium Many Body Systems, Kluwer, London, 2000 (Chapter 3).

[6] Granular Gas Dynamics, in: T. Poschel, N. Brilliantov (Eds.), Lecture Notes in Physics, vol. 624, Springer, Berlin, 2003.

[7] R. Soto, J. Piasecki, M. Mareschal, Phys. Rev. E 64 (2001) 031306.

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Approach to a non-equilibrium steady state