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Equation [18] can be read as follows: the

probability that the observables F

follow given

evolution patte rns ’

conditioned to entropy crea-

tion rate p

is the same that they follow the time-

reversed patterns if conditioned to entropy creation

rate p

. In other words, to change the sign of

time, it is just sufficient to reverse the sign of

entropy creation rate, no ‘‘extra effort’’ is needed.

For more details, the reader is referred to Sinai

(1972, 1994), Evans et al. (1993), Gallavotti and

Cohen (1995), Gallavotti (1996, 1999), Gallavotti

and Ruelle (1997), Gallavotti et al. (2004), and

Bonetto et al. (2005).

Fractal Attractors, Pairing,

and Time Reversal

Attracting sets (i.e., sets which are the closure of

attractors) are fractal in most dissipative systems.

However, the chaotic hypothesis assumes that

fractality can be neglected. Apart from the very

interesting cases of systems close to equilibrium, in

which the closure of an attractor is the whole phase

space (under the chaotic hypothes is, i.e., if the

system is Anosov), hence not fractal, serious

problems arise in preserving validity of the fluctua-

tion theorem.

The reason is very simple : if the attractor closure

is smaller than phase space, then it is to be expected

that time reversal will change the attractor into a

repeller disjo int from it. Thus, even if the chaotic

hypothesis is assumed, so that the attr acting set

A can be considered a smooth surface, the motion

on the attractor will not be time-reversal symmetric

(as its time-reversal image will develop on the

repeller). One can say that an attracting set with

dimension lower than that of phase space in a time-

reversible system corresponds to a spontaneous

breakdown of time-reversal symmetry.

It has been noted however that there are classes

of systems, forming a large set in the space of

evolutions depending on a parameter  , in which

geometric reasons imply that if beyond a critical

value 

the attracting set becomes smaller than

phase space, then a map I

is generated mapping the

attractor A into the repeller R, and vice versa, such

that I

is the identity on A[R and I

commutes

with the evolution: therefore, the composition I  I

is a time-reversal symmetry (i.e., it anticommutes

with evolution) for the motions on the attracting set

A (as well as on the repeller R).

In other words, the time-reversal symmetry in

such systems ‘‘cannot be broken’’: if spontaneous

breakdown occurs (i.e., A is not mapped into itself

under time reversal I), a new symmetry I

spawned and I  I

is a new time-reversal symmetry

(an analogy with the spontane ous violation of time

reversal in quantum theory, where time reversal T is

violated but TCP is still a symmetry: so T plays the

role of I and CP that of I

Thus, a fluctuation relation will hold for the

phase-space contraction of the motions taking place

on the attracting set for the class of systems with the

geometric property mentioned above (technically,

the latter is called ‘‘axiom C’’ property).

This is interesting but it still is quite far from

being checkable even in numerical experiments.

There are nevertheless systems in which a ‘‘pairing

property’’ also holds: this means that, considering

the case of discrete-time maps S, the Jacobian matrix

S(x) has 2N eigenvalues that can be labeled,

in decreasing order, 

(x), ..., 

(1=2)N

(x), ..., 

(x),

with the remarkable property that (1=2)(

Nj

(x) þ



(x)) =

def

(x)isj-independent. In such systems, a

relation can be established between phase-space

contractions in the full phase space and on the

surface of the attracting set: the fluctuation theorem

for the motion on the attracting set can therefore

be related to the properties of the fluctuations of

the total phase-space contraction measured on the

attracting set (which includes the contraction trans-

versal to the attracting set) and if 2M is the

attracting set dimension and 2N is the total

dimension of phase space it is, in the analyticity

interval (p



, p



) of the function (p),

ðpÞ¼ðpÞp



½19

which is an interesting relation. It is however very

difficult to test in mechanical systems because in

such systems it seems very difficult to make the field

so high to see an attracting set thinner than the

whole phase space and still observe large

fluctuations.

For more details, the reader is referred to Dettman

and Morriss (1996) and Gallavotti (1999).

Nonequilibrium Ensembles

and Their Equivalence

Given a chaotic system, the collection of the SRB

distributions associated with the various con trol

parameters (volume, density, external forces, ...)

forms an ‘‘ensemble’’ describing the possible sta-

tionary states of the system and their statistical

properties.

As in equilibrium, one can imagine that the

system can be described equivalently in several

ways at least when the system is large (‘‘in the

Nonequilibrium Statistical Mechanics (Stationary): Overview 537

thermodynamic’’ or ‘‘macroscopic limit’’). In none-

quilibrium, equivalence can be quite different and

more structured than in equilibrium because one can

imagine to change not only the control parameters

but also the thermostatting mechanism.

It is intuitive that a system may behave in the

same way under the influence of different thermo-

stats: the important phenomenon being the extrac-

tion of heat and not the way in which it is extracted

from the system. Therefore, one should ask when

two systems are ‘‘physically equivalent,’’ that is,

when the SRB distributions associated with them

give the same statistical properties for the same

observables, at least for the very few observables

which are macroscopically relevant. The latter may

be a few more than the usual ones in equilibrium

(temperature, pressure, density, etc.) and include

currents, conducibilities, viscosities, etc., but they

will always be very few compared to the (infinite)

number of functions on phase space.

As an example, consider a system of N interacting

particles (say hard spheres) of mass m moving in a

periodic box C

of side L containing a regular array

of spherical scatterers (a basic model for electrons in

a crystal) which reflect particles elastically and are

arranged so that no straight line exists in C

which

avoids the obstacles (to eliminate obvious constants

of motion). An external field Eu acts also along the

u-direction: hence, the equations of motion are

€

¼ f

þ Eu  J

½20

where f

are the interpart icle forces and those

between scatter ers and particles, and J

are the

thermostatting forces. The following thermostat

models have been considered:

1. J

= 

(viscosity thermostat),

2. immediately after elastic collision with an obsta-

cle the velocity is rescaled to a prefixed value

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1

for some T (Drude’s thermostat),

3. J

= (E 

(Gauss’ thermostat).

The first two are not reversible. At least not

manifestly such, because the natural time reversal,

that is, change of velocity sign, is not a symmetry

(there might be however more hidden, hitherto

unknown, symmetries which anticommute with

time evolution) . The third is reversible and time

reversal is just the change of the velocity sign. The

third thermostat model generates a time evolution in

which the total kinetic energy K is constant.

Let 



, 

000

be the SRB distributions for the

system in a container C

with volume jC

j= L

and

density  = N=L

fixed. Imagine to tune the values

of the control parameters , T, K in such a way that

hkinetic energyi



= E, with the same E for  = 





, 

000

and consider a local observable F(

X, X) > 0

depending only on the coordi nates of the particles

located in a region   C

. Then a reasonable

conjecture is that

lim

L!1

N=L

¼

hFi





hFi



¼ lim

L!1

N=L

¼

hFi





hFi



000

¼ 1 ½21

if the limits are taken at fixed F (hence at fixed 

while L !!1). The conjecture is an open

problem: it illustrates, however, the kind of ques-

tions arising in nonequilibrium statistical mechanics.

For more details, the reader is referred to Evans

and Sarman (1993), Gallavotti (1999),andRuelle

(2000).

Outlook

The subject is (clearly) at a very early stage of

development.

1. The theory can be extended to stochastic thermo-

stats quite satisfactorily, at least as far as the

fluctuation theorem is concerned.

2. Remarkable works have appeared on the theory

of systems which are purely Hamiltonian and

(therefore) with thermostats that are infinite:

unfortunately, the infinite thermostats can be

treated, so far, only if their particles are ‘‘free’’ at

infinity (either free gases or harmonic lattices).

3. The notion of entropy turns out to be extremely

difficult to extend to stationary states and there

are even doubts that it could be actually

extended. Conceptually, this is certainly a major

open problem.

4. The statistical properties of stationary states out of

equilibrium are still quite mysterious and surpris-

ing: some exactly solvable models have appeared

recently, and attempts have been made at unveil-

ing the deep reasons for their solubility and at

deriving from them general guiding principles.

5. Numerical simulations have given a strong

impulse to the subject; in fact, one can even say

that they created it: introducing the model of

thermostat as an extra microscopic force acting on

the particles and providing the first reliable results

on the properties of systems out of equilibrium.

Simulations continue to be an essential part of the

effort of research on the field.

6. Approach to stationarity leads to many impor-

tant questions: is there a Lyapunov function

measuring the distance between an evolving

state and the stationary state towards which it

evolves? In other words, can one define an

538 Nonequilibrium Statistical Mechanics (Stationary): Overview

analogous of Boltzmann’s H-function? About this

question there have been proposals and the answer

seems affirmative, but it does not seem that it is

possible to find a universal, system-independent,

such function (search for it is related to the problem

of defining an entropy function for stationary

states: its existence is at least controversial, see the

sections ‘‘Nonequilibrium thermodynamics’’ and

‘‘Chaotic hypothesis’’).

7. Studyin g nonstationary evolution is much harder.

The problem arises when the control param eters

(force, volume, ...) change with time and the

system ‘‘undergoes a process.’’ As an example one

can ask the question of how irreversible is a given

irreversible process in which the initial state 

is a

stationary state at time t = 0, and the external

parameters F

start changing into functions F( t )

of t and tend to a limit F

as t !1. In this case,

the stationary distribution 

starts changing and

becomes a funct ion 

of t which is not stationary

but approaches another stationary distribution 

as t !1. The pro cess is, in general, irreversible

and the question is how to measure its ‘‘degree of

irreversibility’’: for simplicity we restrict attention

to very special processes in which the only

phenomenon is heat production because the

container does not change volume and the energy

also remains constants, so that the motion can be

described at all times as taking place on a fixed

energy surface. A natural quantity I associated

with the evolution from an initial stationary state

to a final stationary state throu gh a change in the

control parameters can be defined as follows.

Consider the distribution 

into which 

evolves

in time t, and consider also the SRB distribution



F(t)

corresponding to the control parameters

‘‘frozen’’ at the value at time t, that is, F(t). Let

the phase-space contraction, when the forces are

‘‘frozen’’ at the value F(t), be 

(x) = (x; F( t)). In

general 

6¼ 

F(t)

. Then,

IðfFðtÞg;

;

Þ¼

def

ð

ð



FðtÞ

ð

ÞÞ

dt ½22

can be called the degree of irreversibility of the

process: it has the property that in the limit of

infinitely slow evolution of F(t), for example, if

F(t) = F

þ(1e

t

)D (a quasistatic evolution

on timescale 

1



1

from F

to F

= F

þD),

the irreversibility degree I



!

 !0

0 if (as in the case

of Anosov evolutions, hence under the chaotic

hypothesis) the approach to a stationary stat e is

exponentially fast at fixed external forces F. The

quantity I is a time scale which could be

inte rpreted as the time needed for the process to

exhibit its irreversible nature.

The entire subject is dominated by the initial

insights of Onsager on classical nonequilibrium

thermodynamics, which concern the properties of

the infinitesimal deviations from equilibrium (i.e.,

averages of observables differentiated with respect

to the control parameters F and evaluated at F = 0).

The present efforts are devoted to studying proper-

ties at F 6¼0. In this direction, the classical theory

provides certainly firm constraints (like Onsager

reciprocity or Green–Kubo relations or fluctuation–

dissipation theorem) but at a technical level, it gives

little help to enter the terra incognita of none-

quilibrium thermodynamics of stationary states.

For more details, the reader is referred to

Kurchan (1998), Lebowitz and Spohn (1999),

Maes (1999), Eck mann et al. (1999), Bonetto

et al. (2000, 2005), Eckmann and Young (2005),

Derrida et al. (2001), Bertini et al. (2001), Evans

and Morriss (1990), Evans et al. (1993), Goldstein

and Lebowitz (2004), and Gallavotti (2004).

See also: Adiabatic Piston; Chaos and Attractors;

Ergodic Theory; Lie, Symplectic, and Poisson Groupoids

and Their Lie Algebroids; Macroscopic Fluctuations and

Thermodynamic Functionals; Nonequilibrium Statistical

Mechanics: Dynamical Systems Approach; Quantum

Dynamical Semigroups; Random Dynamical Systems.

Further Reading

Bertini L, De Sole A, Gabrielli D, Jona G, and Landim C (2001)

Fluctuations in stationary nonequilibrium states of irreversible

processes. Physical Review Letters 87: 040601.

Bonetto F, Gallavotti G, Giuliani A, and Zamponi F (2005)

Chaotic hypothesis, fluctuation theorem and singularities,

mp_Ar Xiv 05-257, cond-mat/0507672.

Bonetto F, Lebowitz JL, and Rey-Bellet L (2000) Fourier’s law: a

challenge to theorists. In: Streater R (ed.) Mathematical

Physics 2000, pp. 128–150. London: Imperial College Press.

de Groot S and Mazur P (1984) Non-Equilibrium Thermody-

namics, (reprint). New York: Dover.

Derrida B, Lebowitz JL, and Speer E (2001) Free energy

functional for nonequilibrium systems: an exactly solvable

case. Physical Review Letters.

Dettman C and Morriss GP (1996) Proof of conjugate pairing for

an isokinetic thermostat. Physical Review E 53: 5545–5549.

Eckmann JP, Pillet CA, and Rey Bellet L (1999) Non-equilibrium

statistical mechanics of anharmonic chains coupled to two

heat baths at different temperatures. Communications in

Mathematical Physics 201: 657–697.

Eckmann JP and Young LS (2005) Nonequilibrium energy profiles for

a class of 1D models. Communications in Mathematical Physics.

Evans DJ, Cohen EGD, and Morriss G (1993) Probability of

second law violations in shearing steady flows. Physical

Review Letters 70: 2401–2404.

Evans DJ and Morriss GP (1990) Statistical Mechanics of

Nonequilibrium Fluids. New York: Academic Press.

Nonequilibrium Statistical Mechanics (Stationary): Overview 539

Evans DJ and Sarman S (1993) Equivalence of thermostatted

nonlinear responses. Physical Review E 48: 65–70.

Gallavotti G (1996) Chaotic hypothesis: Onsager reciprocity and

fluctuation–dissipation theorem. Journal of Statistical Physics

84: 899–926.

Gallavotti G (1998) Chaotic dynamics, fluctuations, non-equili-

brium ensembles. Chaos 8: 384–392.

Gallavotti G (1999) Statistical Mechanics. Berlin: Springer.

Gallavotti G (2004) Entropy production in nonequilibrium

stationary states: a point of view. Chaos 14: 680–690.

Gallavotti G, Bonetto F, and Gentile G (2004) Aspects of the

ergodic, qualitative and statistical theory of motion.

pp. 1–434. Berlin: Springer.

Gallavotti G and Cohen EGD (1995) Dynamical ensembles in

non-equilibrium statistical mechanics. Physical Review Letters

74: 2694–2697.

Gallavotti G and Ruelle D (1997) SRB states and nonequilibrium

statistical mechanics close to equilibrium. Communications in

Mathematical Physics 190: 279–285.

Goldstein S and Lebowitz J (2004) On the (Boltzmann) entropy of

nonequilibrium systems. Physica D 193: 53–66.

Kurchan J (1998) Fluctuation theorem for stochastic dynamics.

Journal of Physics A 31: 3719–3729.

Lebowitz JL (1993) Boltzmann’s entropy and time’s arrow.

Physics Today (September): 32–38.

Lebowitz JL and Spohn H (1999) The Gallavotti–Cohen fluctua-

tion theorem for stochastic dynamics. Journal of Statistical

Physics 95: 333–365.

Maes C (1999) The fluctuation theorem as a Gibbs property.

Journal of Statistical Physics 95: 367–392.

Ruelle D (1976) A measure associated with axiom A attractors.

American Journal of Mathematics 98: 619–654.

Ruelle D (1996) Positivity of entropy production in nonequilibrium

statistical mechanics. Journal of Statistical Physics 85: 1–25.

Ruelle D (1997) Entropy production in nonequilibrium statistical

mechanics. Communications in Mathematical Physics 189:

365–371.

Ruelle D (1999) Smooth dynamics and new theoretical ideas in

non-equilibrium statistical mechanics. Journal of Statistical

Physics 95: 393–468.

Ruelle D (2000) A remark on the equivalence of isokinetic and

isoenergetic thermostats in the thermodynamic limit. Journal

of Statistical Physics 100: 757–763.

Sinai YaG (1972) Gibbs measures in ergodic theory. Russian

Mathematical Surveys 166: 21–69.

Sinai YaG (1994) Topics in Ergodic Theory. Princeton Mathe-

matical Series, vol. 44. Princeton: Princeton University Press.

Nonequilibrium Statistical Mechanics:

Dynamical Systems Approach

P Butta` and C Marchioro, Universita

di Roma

‘‘La Sapienza,’’ Rome, Italy

Time Evolution of Infinite-Particle

Systems

A preliminary problem in the rigorous study of

nonequilibrium statistical mechanics is to give a

precise sense to the time evolution of infinitely

extended systems. In fact, statistical mechanics deals

with systems composed by a very large number of

bodies (of the order of 10

) and studies the

properties of such systems which are related to

their large number of degrees of freedom. Mathe-

matically, this aspect is stressed by introducing the

so-called ‘‘thermodynamical limit,’’ that is, by

defining and analyzing systems with infinite degrees

of freedom. For particle systems, the problem can be

formulated in the following way. A phase point of

the system is an infinite sequence {(x

, v

)}

i2N

of the

positions and velocities of the particles, and its time

evolution is characterized by the solutions of the

Newton equations:

€

ðtÞ¼

j2N :j6¼i

ðtÞx

ðtÞ



; i 2 N ½ 1

where m is the mass of each particle, F(x) = r(x),

and  is a two-body potential. Equation [1] must be

completed by the initial data {(x

(0), v

(0))}

i2N

.The

time evolution of a phase point implies in a natural

way the time evolution of functions on the phase

space, which are the observables to be compared with

experiments.

The existence of a solution to eqn [1] is not

obvious, because the classical theorem of existenc e

and uniqueness for the Cauc hy problem of the

Newton equations depends on the number of

degrees of freedom of the system. The mai n

difficulty is that a priori the time evolution can

bring infinitely many particles in a bounded region

within a finite time, so that the right-hand side of

eqn [1] becomes meaningless. Without any hypoth-

esis on the initial conditions, this can happen, as

shown by the following simple example. Consider a

system of free (noninteracting) particles moving

on the real line with initial conditions x

= i, v

= i,

i 2 N. It is clear that at time t = 1 all the particles

are at the origin. To forbid this ‘‘coll apse,’’ we must

restrict the allowed initial conditions, but we cannot

be too drastic. For instance, we could surely avoid

these pathologies by choosing the initial velocities

uniformly bounded and the initial distribution of

particles locally finite. But the set of such data is

exceptional with respect to the Gibbs state (as it can be

easily shown using that, at equilibrium, the velocities are

independent identically distributed Gaussian variables).

In conclusion, we must construct the dynamics for initial

conditions which are chosen in a set sufficiently large to

540 Nonequilibrium Statistical Mechanics: Dynamical Systems Approach

be the support of states of interest from a thermo-

dynamical point of view.

The difficulty of the problem increases with the

spatial dimension d, as it is shown by the following

example. Let the potential  be smooth enough and

short range and assume that, initially, the velocities

and the density are bounded, that is,

sup

j < 1; sup

2R

;R>1

NðX; ; RÞ

< 1½2

where X = {(x

, v

)}

i2N

is the particle configuration and

N(X; , R) is the number of particles in the ball of radius

R, centered at .IfV(t) denotes the modulus of the

maximal velocity carried by the particle during the time

[0, t]andX(t) the evolved configuration, the conserva-

tion of the particles number yields

NðXðtÞ; ; R

ÞNðXð0Þ; ; RðtÞÞ  const: RðtÞ

½3

where

RðtÞ¼R

dsVðsÞ½4

On the other hand, V(s) is controlled by the force,

which turns out to be bounded by sup



N(X(s); , r),

where r > 0 is the range of the potential. By virtue of

eqns [3] and [4], we arrive at the integral inequality:

RðtÞR

þ const : t þ const:

dsRðsÞ

½5

which is solvable globally in time only if d = 1.

In the case of interest, from a thermodynamical

point of view, we also need to allow fluctuations of

the density and velocities, which add further

difficulties. The existence, uniqueness, and locality

of the motion has been solved in dimension d = 1 for

almost all relevant interactions (Lanford 1968,

Dobrushin and Fritz 1977), and in dimension d = 2

for interactions not too singular at the origin (Fritz

and Dobrushin 1977). (This does not cover, for

instance, the hard-core interactions, wher e it is still

an open problem to investigate whether the

dynamics evolves toward a close-packing situation.)

Finally, in dimension d = 3, the result has recently

been proved only for bounded, non-negative, finite-

range interactions (Caglioti et al. 2000 ).

We state the result for the three-dimensional case.

Let the interaction  depend only on the mutual

distance, be twice differentiable, positive in the

origin and, for the moment, also non-negative and

compactly supported. We assume that the initial

data have bounded local energies and densities, with

at most logarithmic divergences in velocities and

densities. More precisely, we define

QðX; ; RÞ¼

i2N

ðjx

 jRÞ



j:j6¼i

 x



þ 1

½6

where (A) denotes the characteristic function of the

set A so that eqn [6] gives the energy and density

contained in a ball centered at  with radius R.

Define



ðXÞ¼sup



sup

R:R>



ðÞ

QðX; ; RÞ

½7

where >0 and





ðxÞ¼

log



ðe þjxjÞ; x 2 R

½8

We denote by X



the set of the phase points X such

that Q



(X) < 1. It is possible to prove that for any

  1/3, X



has full measure with respect to any

Gibbs measure.

We define the partial dynamics t 7!X

(n)

(t) as the

solutions to eqn [1] obtained by neglecting all the

particles which are initially outside the ball of radius

n and centered at the origin.

Theorem If X 2X



there exists a unique flow

X ! X(t) 2X

ð3/2Þ

satisfying eqn [1] with X(0) = X.

Moreover, the partial dynamics locally converges to

X(t) as n !1.

The result has been extended to bounded super-

stable long-range interact ions. The (nontrivial) proof

is based on several steps: we introduce a mollified

version on the local energy and study its evolution in

time under the partial dynamics. The energy

conservation allows us to prove that the local energy

grows at most as the cube of the maximal velocity.

On the other hand, a suitable time average allows us

to control the maximal velocity via the local energy

in an appropriate way. The result is achieved by

letting n !1.

Long-Time Behavior

Existence and locality of the dynamics is only a first,

preliminary, step. The next and much more subtle

question concerns the asymptotic (in time) and the

statistical properties of the motion. Here, the main

problem is the absence of simple but nontrivial

models. Let us explain this point by a comparison

with the situation in equilibrium statistical

mechanics. In this case, even the simpler model,

the free-partic le system, exhibits all the relevant

Nonequilibrium Statistical Mechanics: Dynamical Systems Approach 541

thermodynamical properties of real systems away

from the critical regime. In fact, the effort is

often reduced to rigorously proving that the real

systems away from the critical region behave as a

free-particle system. The presence of the interaction

is instead essential to describe phase transitions.

In the case of nonequilibrium statistical mechanics

there are very few solvable models (free particles,

chain of oscillators, hard-core system in one dimen-

sion), and typically they do not catch the essential

properties of the real systems. For example, let us

consider a system which is close to equilibrium and

ask whether it converges to the corresponding Gibbs

state. Two possible mechanisms usually come together:

the dispersive properties of the matter (by which

perturbations ‘ ‘escap e’’ to infinity) and the mixing

properties (by which perturbations are ‘ ‘spread’’ and

disappear). The former is present also in the free-particle

system, being responsible of its ergodic properties. The

latter requires a deep analysis of the dynamics of

interacting-particle systems and it is too difficult to be

analyzed except in rare cases.

We just mention the case of systems with

instantaneous interaction, which are simple enough

to be studied but nevertheless exhibit a nontrivial

long-time behavior. We recall in particular the

famous Sinai’s billiard: a particle moving freely in

a two-dimensional torus except for elastic collisions

with the boundary of a convex obstacle. As proved

by Sinai (1970), this system has strong ergodic

properties. Sinai’s billiard can be proved to be

equivalent to the ‘‘Lorentz gas’’ in which the

obstacles are dislocated in a periodic way.

Bunimovich and Sinai (1981) proved that when

the obstacles are close enough to each other, the

diffusive (weak) limit of the particle motion is the

Wiener process. This remarkable result gives a

rigorous deriv ation of Brownia n motion from a

Hamiltonian system.

More recently, similar questions have been inves-

tigated in the case of a charged particle subject to a

constant electric field and interacting with a medium

described by a particle system. Several rigorous

results have been obtained on this subject. We only

recall those by Boldrighini and Soloveitchik (1995,

1997). In the context of a simplified model, the

asymptotic motion of the charged particle is

described as a drift plus a Brownian motion, and

the Einstein relation between the drift and the

diffusion constant is establish ed.

Mean-Field Limit

The validity of any model is related to some

approximation limit. In statistical mechanics, we

encounter one of the most important ones, the

‘‘thermodynamical limit,’’ used to stress the effect of

large number of particles. Here we briefly discuss the

‘‘mean-field limit.’’ For the kinetic, Boltzmann–Grad

limit, see Boltzmann Equation (Classical and Quan-

tum) and Kinetic Equations.

We consider N particles of mass m mutually

interacting via the force F. The equations of motion are

€

ðtÞ¼

j¼1;...;N:j6¼i

Fðx

ðtÞx

ðtÞÞ

ðx

ð0Þ;

ð0ÞÞ ¼ ðx

; v

i ¼ 1; ...; N

½9

We consider a system with N very large, the mass m

of each particle very small, and the interaction very

weak. An interesting situation arises when the

quantities N, m, and F are linked by the relations

m ¼

; F ¼

½10

for some function G. Of course, M is the total mass

of the system.

We are interested in investigating the limit N !1.

We assume that the initial data are chosen in a way

that the empirical measure N

1



weakly

converges (as N !1) to the absolutely continuous

measure f

(x, v)dx dv with some smooth density

(x, v). We ask whether at some positive time t > 0

the empirical measure N

1



(t)



(t)

weakly con-

verges to f (x, v, t)dx dv with a density f ( x, v, t)

satisfying some limiting evolution equation.

Formally, it is easy to find this equation: by the

Liouville theorem, a continuous medium in which

each point moves under the action of an acceleration

field behaves as an incompressible fluid. The

continuity equation becomes

f ðx; v; tÞþv r

f ðx; v; tÞþE r

f ðx; v; tÞ¼0

f ðx; v; 0Þ¼f

ðx;vÞ

½11

where

Eðx; tÞ¼

dyGðx  yÞðy; tÞ½12

ðx; tÞ¼

dv f ðx; v; tÞ½13

This equation can be studied by following the

characteristics, for which it suffices to look at the

pair of functions

ðx; vÞ7!ðXðx; v; tÞ; Vðx; v; tÞÞ ; f

ðx; vÞ7!f ðx; v; tÞ

542 Nonequilibrium Statistical Mechanics: Dynamical Systems Approach

where (x, v) 2 R

 R

and t 2 R, solutions of

Xðx; v; tÞ¼Vðx; v; tÞ;

Vðx; v; tÞ¼Eðx; t Þ

Xðx; v; 0Þ¼x; Vðx; v; 0Þ¼v

f ðXðx; v; tÞ; Vð x; v; t Þ; tÞ¼f

ðx; vÞ

½14

This is a weak formulation of eqn [11], in the sense

that any smooth solution to eqn [11] satisfies eqn

[14], but this last equation in meaningful also for

nonsmooth functions. This is a weak version of the

Vlasov equ ation and its measure solutions will play

an important role in the sequel.

Equations [11]–[14] are called Vlasov equations,

after Vlasov, who first introduced them in plasma

physics. They have a Hamiltonian structure and

conserve several quantities: the total mass, the total

energy, the Liouville measure dx dv, and in general

each moment of this measure.

The existence and uniqueness of the solutions

has been studied in many papers. Two cases have

to be considered, depending on whether the total

mass

M ¼

dx dv f

ðx; vÞ½15

is finite or not. We start with the first case. If the

interaction G is bounded, the analysis is easy. On

the other hand, in plasma physics one d eals with

the Coulomb interaction, which is singular at the

origin. In this case (where eqn [11] is usually

called the Vlasov–Poisson equation), e xistence and

uniqueness can still be proved, but it is not

straightforward, especially in dimension d = 3.

The case with the complete Lorentz force, also

taking into account the relativistic effect, is much

more difficult.

For infinite total mass, the problem has been

solved recently in three (or lower) dimensions for

bounded, non-negative, finite-range interactions,

and in two dimensions for singular Helmholtz

interactions.

Another way to relate the Vlasov equation with

the particle systems is to consider the usual

transition from microscopic to m acroscopic evolu-

tions based on a separation between microscopic

and macroscopic scales. Moreover, the force

between the particles is due to a long-range pair

interaction of the Kac type, in which the range

parameter tends to infinity as the ratio "

1

between the macro and the micro spatial scale:

F(x

 x

) = "

2dþ1

G("x

 "x

). Finally, the mass of

the particles is proportional to "

: m = "

.After

rescaling space and time by a factor ",inthe

macroscopic variables (, r) = ("t, "x), the equa-

tions of motion (eqn [9])become

d

j:j6¼i

Gðr

 r

Þ½16

Then eqn [14] is the limiting equation as " ! 0.

Other Models

We mention another model of larger interest. We

introduce it in the simplest formulation, leaving

possible generalizations to the reader.

We consider an infinite chain of anharmonic

oscillators, with Hamiltonian H given by

Hðq; pÞ

i2Z

þ aq

þ b

j:jijj¼1

ðq

 q

þ cq

þ d

½17

where q

, p

2 R, a  0, b, c, d > 0.

When a = 0, it reduces to the well-known chain of

harmonic oscillators, which is integrable and widely

studied in the literature.

The time evolution defined by the Hamiltonian in

eqn [17] exists and it is unique for initial data

chosen in a set large enough to be the support of

any reasonable thermodynamic (equilibrium or

nonequilibrium) state. This can be achieved by

proving integral inequalities for the ‘‘Lyapunov

function’’

Lðq; pÞ¼sup

i2Z

þ aq

þ d



jijþ1

It is interesting to note that uniqueness holds only in

a class of data such that the position of the ith

oscillator does not increase too much as jij!1.

For example, besides the stationary solution

(t) = 0, i 2 Z, we can construct a different solution

corresponding to the same initial conditions

(0) = 0, p

(0) = 0, i 2 Z. In fact, by imposing

(t) = t

and q

(t) = q

i

(t), we can solve recursively

the equations of motion and obtain a nonzero

solution q

(t), which however increases superexpo-

nentially as jij!1.

The Hamiltonian dynamical systems (classical or

quantum) are surely quite faithful descriptions of

real systems, but they are too difficult to study.

Mainly it is not known how to prove good

dynamical mixing for deterministic evolutions with

many degree s of freedom. Therefore, stochastic

evolutions have been introduced to model the real

systems. More precisely, one renounces a full

description of the microscopic dynamics, introdu-

cing simplified models where the effects of the

Nonequilibrium Statistical Mechanics: Dynamical Systems Approach 543

‘‘hidden degrees of freedom’’ are taken into account

by adding suitable stochastic forces. Many useful

results have been obtained, which show that these

stochastic model systems exhibit a macroscopic

behavior much closer to that observed in nature.

The main criticism concerns the role of stochasticity,

which in these models is introduced ab initio.In

other words, if one believes that the statistical

properties of the deterministic motion on the small

scale determine the collective behavior of systems

with many degrees of freedom, then these properties

do have to be proved for a true understanding of

nonequilibrium phenomena.

See also: Adiabatic Piston; Boltzmann Equation

(Classical and Quantum); Fourier Law; Kinetic Equations;

Nonequilibrium Statistical Mechanics (Stationary):

Overview; Nonequilibrium Statistical Mechanics:

Interaction between Theory and Numerical Simulations.

Further Reading

Boldrighini C and Soloveitchik M (1995) Drift and diffusion for a

mechanical system. Probability Theory and Related Fields

103: 349–379.

Boldrighini C and Soloveitchik M (1997) On the Einstein relation

for a mechanical system. Probability Theory and Related

Fields 107: 493–515.

Bunimovich LA and Sinai YG (1981) Statistical properties of

Lorentz gas with periodic configuration of scatterers. Com-

munications in Mathematical Physics 78: 479–497.

Caglioti E, Marchioro C, and Pulvirenti M (2000) Non-equilibrium

dynamics of three-dimensional infinite particle systems. Com-

munications in Mathematical Physics 215: 25–43.

Cornfeld IP, Fomin SV, and Sinai YG (1982) Ergodic theory. In:

Artin M et al. (eds.) Grundlehren der Mathematischen

Wissenschaften (Fundamental Principles of Mathematical

Sciences), vol. 245. New York: Springer.

Dobrushin RL and Fritz J (1977) Non-equilibrium dynamics of one-

dimensional infinite particle systems with a hard-core interac-

tion. Communications in Mathematical Physics 55: 275–292.

Fritz J and Dobrushin RL (1977) Non-equilibrium dynamics of two-

dimensional infinite particle systems with a singular interaction.

Communications in Mathematical Physics 57: 67–81.

Lanford OE III (1968) Classical mechanics of one-dimensional

systems with infinitely many particles. I An existence theorem.

Communications in Mathematical Physics 9: 176–191.

Lanford Oscar E, III (1975) Time evolution of large classical

systems. In: Ehlers J, Hepp K, and Weidendmu¨ ller HA (eds.)

Dynamical Systems, Theory and Applications (Recontres,

Battelle Res. Inst., Seattle, WA, 1974), Lecture Notes in

Physics, vol. 38, pp. 1–111. Berlin: Springer.

Neunzert H (1984) An introduction to the nonlinear Boltzmann–

Vlasov equation. In: Dold A and Eckmann B (eds.) Kinetic

Theories and the Boltzmann Equation (Montecatini, 1981),

Lecture Notes in Mathematics, vol. 1048, pp. 60–110. Berlin:

Springer.

Sinai TG (1970) Dynamical systems with elastic reflections.

Ergodic properties of dispersing billiards. (Russian) Uspehi

Mat. Nauk 25: 141–192.

Spohn H (1991) Large Scale Dynamics of Interacting Particles,

Texts and Monographs in Physics. Berlin: Springer.

Sza´sz D (ed.) (2000) Hard Ball Systems and the Lorentz Gas.

Encyclopædia of Mathematical Sciences, vol. 101. Berlin:

Springer.

Nonequilibrium Statistical Mechanics: Interaction between Theory

and Numerical Simulations

R Livi, Universita

di Firenze, Sesto Fiorentino, Italy

Introduction

Nonequilibrium statistical mechanics concerns a

wide range of fundamental problems and applica-

tions. Perturbative m ethods are quite effective for

approaching weakly nonlinear problems, usually

relying upon effective coarse-grained equations.

The attempt of obtaining a microscopic description

of genuine nonlinear problems demands the com-

bined use of theoretical methods and numerical

simulations. The proprotypic case is the numerical

experiment performed by Fermi, Pasta, and Ulam

in 1955. As we discuss in the following s ection, the

main questions, w hich had inspired this experi-

ment, remained without an answer for a long time,

while new puzzling problems emerged. Despite its

apparent failure, the Fermi–Pasta–Ulam (FPU)

experiment represents a remarkable example in

the history of science of how a good guess may be

the source of many fruitful achievements. Part of

them are discussed in the section on e nergy

relaxation in nonlinear chains, where we summar-

ize the present understanding of the very slow

relaxation mechanism, characterizing the dynamics

of nonlinear chains of oscillators, like the FPU

model, at low energies. Next, we report one further

success of the interplay between theory and

numerics, that is, the formulation of a generalized

fluctuation–dissipation relation for stationary pro-

cesses. Finally, we survey the main achievements

concerning the study of anomalous transport

properties in low-dimensional systems. In particu-

lar, we focus our attention on the heat conduction

in nonlinear lattices. Lacking a general hydrody-

namic theory, also in this case computer simula-

tions and theoretical arguments have greatly

544 Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

contributed to clarify the general scenario, unveil-

ing surprising aspects, which, up to a few years

ago, were completely unexpected.

The Numerical Experiment by Fermi,

Pasta, and Ulam

The impressive progress of electronic technology

during World War II made possible the design of the

first digital computers. The equally impressive

budgets for their production and maintenance

could only be justified by their employment in

classified military research. Nonetheless, some of

the outstanding scientists involved in these

researches, like E Fermi, immediately realized the

great potential of these new machines for tackling

also some fundamental problems in basic science.

Fermi had in his mind a crucial and still open

physical problem. In 1914 the Dutch physicist

P Debye had suggested that the finiteness of thermal

conductivity in crystals should be due to the

nonlinear forces acting among the constituent

atoms. Forty years later a microscopic theory of

transport processes, including nonlinear effects, was

still lacking. Actually, technical difficulties pre-

vented a theoretical approach based on analytic

methods. Numerical integration of the equations of

motion by a digital machine appeared to Fermi as

an effective way for tackling this problem. In

collaboration with the mathematician S Ulam and the

physicist J Pasta, Fermi used MANIAC 1 (a proto-

type digital computer installed at Los Alamos National

Laboratories, USA) for integrating the dynamical

equations of the simplest mathematical model of

an anharmonic crystal: a chain of N harmonic oscilla-

tors, coupled by nonlinear forces. Its Hamiltonian

reads

H ¼

i¼1

ðq

iþ1

 q



ðq

iþ1

 q



ðq

iþ1

 q

½1

where ! is the harmonic frequency, while  and 

are the positive coupling constants of the nonlinear

terms. The integer space index i labels the oscillators

along the chain, while q

and p

are the displacement

from the equilibrium position and the momentum of

the ith oscillator, respectively. The potential energy

is the general form taken by any nonlinear interac-

tion potential, when expanded, up to fourth order,

around its equilibrium position. This choice guaran-

tees the boundedness of trajectories for any finite

energy.

Accordingly, the model contains the minimal

basic ingredients, needed for testing the conjecture

about the finiteness of thermal conductivity.

The equations of motion

;

¼

½2

were integrated numerically by an algorithm, where

space and time derivatives were approximated by

proper finite-difference expressions.

The choice of the initial conditions was motivated

by a further basic questi on concerning Fermi and his

collaborators. In fact, they aimed at verifying also a

common belief that had never been proved rigor-

ously: in an isolated mechanical system with many

degrees of free dom (i.e., made of a large number of

oscillators), a generic nonlinear interaction among

them should eventually yield equilibrium through

‘‘thermalization’’ of the energy. On the basis of

physical intuition, nobody would object to this

expectation if the mechanical system would start

its evolut ion from an initial state very close to

thermodynamic equilibrium. Nonetheless, the same

should be observed by considering an initial state,

where energy is supplied to a small subset of

oscillatory modes of the crystal. At variance with a

finite system of linear osci llators, where each

initially excited mode keeps its energy constant,

nonlinear terms should make the energy flow

towards all oscillatory modes, until thermal equili-

brium is eventually reached. Thermalization corre-

sponds to energy equipartition among all the modes.

This statement has to be interpreted in a statistical

sense: the time averages of the energies contained in

the modes converge to the same constant value. But

if this was the case, one further fundamental aspect

concerning the evolution towards thermodynamic

equilibrium could be checked. In the formulation of

his transport equation, L Boltzmann had conjec-

tured that thermodynamic irreversibility can emerge

from microscopic reversible dynamics (which is

the case of eqns [2]). The paradoxical implication

of Boltzmann’ s conjecture was pointed out by

H Poincare´, who had proved that any isolated

Hamiltonian system necessarily evolves towards an

almost-recurrent dynamics. This is manifestly

incompatible with the second law of thermody-

namics, which implies that thermodynamic systems,

in the absence of a supplied energy flux, have to

evolve irreversibly towards their equilibrium state.

In this perspective, the FPU numerical experiment

was intended to test also if and how equilibrium is

approached by a relatively large number of non-

linearly coupled oscillators, obeying the classical

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations 545

laws of Newtonian mechanics. Furthermore, the

measurement of the time interval needed for

approaching the equilibrium state, that is, the

‘‘relaxation time’’ of the chain of oscillators, would

have provided an indirect determination of thermal

conductivity. In fact, according to elementary kinetic

theory, the relaxation time, 

, represents an

estimate of the timescale of energy exchanges inside

the crystal : Debye’s argument predicts that thermal

conductivity  is proportional to the specific heat at

constant volume of the crystal, C

, and inversely

proportional to 

, in formulas  / C

=

Fermi, Pasta, and Ulam considered relatively short

chains, up to 64 oscillators – a size that alre ady

challenged the limits of the computational power of

MANIAC 1. They imposed fixed bounda ry condi-

tions (i.e., the particles at the chain boundaries

interact with infinite mass walls) and the energy was

initially stored just in one of the long-wavelength

oscillatory modes.

A very s urprising and unexpected scenario

showed up. Contrary to any intuition, the energy

did not flow to the higher modes, but was

exchanged only among a small number of long-

wavelength modes, before flowing back almost

exactly to the initial state, thus yielding a recurrent

behavior.

Although nonlinearities were at work, neither a

tendency towards thermalization, nor a mixing rate

of the energy could be identified. The dynamics

exhibited regular features very close to those of an

integrable system.

Fermi guessed that they were facing a very

important result, but he was also quite disappointed

by the difficulties in finding a convincing explana-

tion. This lacking, he had decided not to publish the

results in a scientific review, which remained

confined into a Los Alamos report for almost one

decade. In fact, he died in 1955, the same year of

publication of the report.

The results were finally published i n 1965, in a

volume containing his collected papers (Fermi et al.

1965), and they immediately raised a renewed

interest in the scientific communit y. Despite the

failure in answering all the questions that had been

raised, the FPU numerical experiment represents a

crucial scientific achievement, which determined

many subsequent scientific progresses. The implica-

tions about nonequilibrium will be widely dis-

cussed in the following sections. H ere, we want to

conclude by mentioning the important develop-

ments, inspired by the FPU experiment, that led to

the discovery of solitons by Zabusky and Kruskal

in 1965.

Slow and Fast Energy Relaxation

in Nonlinear Chains

The results of the FPU numerical experiment

indicate that the energy initially supplied to long-

wavelength oscillatory (Fouri er) modes remains

localized for a very long time in a small subset of

long-wavelength modes. This time can be exceed-

ingly larger than any typical timescale of the model

(e.g., !

1

, i.e., the inverse of the harmonic frequency

in [1]). An explanation of this apparently bizarre

scenario has been tackled by combining theoretical

approaches with numerical studies. A complet e

account of the many contributions in this direction

being beyond the scope of this text, we shall

summarize the two main lines along which this

problem has been considered.

The Resonance-Overlap Criterion

The almost-recurrent behavior of single-mode exci-

tations studied in the FPU experiment can be

explained by the resonance-overl ap criterion, intro-

duced in 1959 by the Russian scientist B Chirikov.

Moreover, this criterion provides a quantitative

estimate of the value of the energy density, above

which the regular motion observed in the FPU

experiment should be definitely lost.

In order to provide the reader with an illustration

of this criterion, we have to introduce a few simple

mathematical ingredients.

The Hamiltonian [1] can be rewritten in terms of

linear normal Fourier coordinates, (Q

(t), P

(t)), as

follows:

H ¼

þ !



þ V

ðfQ

gÞ

þ V

ðfQ

gÞ ½3

Here, we have used the shorthand notation V

({Q

})

for the lengthy explicit expressions, in the new set of

coordinates, of the nonlinear potentials of [1].

Without prejudice of generality, we can impose

periodic boundary conditions to the FPU chain: the

frequency of the kth normal mode is given by the

expression !

= 2 sin(k=N). The coupling constants

 and  control the energy exchange among the

normal modes, due to nonlinear interactions.

For the sake of space, we give here a brief sketch

of Chirikov’s criterion for the FPU -model (this

model amounts to take  = 0in[3], i.e., to exclude

the cubic part of the nonlinear potential).

By making reference to the initial conditions of

the FPU experiment, we can consider a single

excited mode, so that the Hamiltonian [3] can be

546 Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations