**The Wigner representation of classical mechanics,**

Physica A 301 (2001) 219?240

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Revista Brasileira de Ensino de Fısica, v. 29, n. 2, p. 189-201, (2007) w.sbfisica.org.br

Ergodic hypothesis in classical statistical mechanics (Hipotese ergodica em mecanica estatıstica classica)

Cesar R. de Oliveira1 and Thiago Werlang2

1Departamento de Matematica, Universidade Federal de Sao Carlos, Sao Carlos, SP, Brasil 2Departamento de Fısica, Universidade Federal de Sao Carlos, Sao Carlos, SP, Brasil Recebido em 1/6/2006; Aceito em 27/9/2006

An updated discussion on physical and mathematical aspects of the ergodic hypothesis in classical equilibrium statistical mechanics is presented. Then a practical attitude for the justification of the microcanonical ensemble is indicated. It is also remarked that the difficulty in proving the ergodic hypothesis should be expected. Keywords: ergodic hypothesis, statistical mechanics, microcanonical ensemble.

Apresenta-se uma discussao atual sobre aspectos fısicos e matematicos da hipotese ergodica em mecanica estatıstica de equilıbrio. Entao indica-se uma eventual postura para se justificar o ensemble microcanonico. Observa-se, tambem, que a dificuldade em se demonstrar a hipotese ergodica deveria ser esperada. Palavras-chave: hipotese ergodica, mecanica estatıstica, ensemble microcanonico.

1. Introduction

Important physical theories are built on relations and/or equations obtained through experiments, intuition and analogies. Hypotheses are proposed and experimentally and theoretically tested, then corrections are proposed and sometimes even “revolutions” occur. Outstanding examples are:

1. The Newton equation in classical mechanics

F = dpdt, which connects the resultant force to time variation of momentum.

2. The Schrodinger equation is accepted as the one that dictates nonrelativistic quantum dynamics.

3. General relativity presumes that gravitation is a curvature of spacetime. Its field equation relates the curvature of spacetime to the sources of the gravitational field.

4. The prescription for equilibrium statistical mechanics is a link between microscopic dynamics and macroscopic thermodynamics via an invariant probability distribution.

It is natural to wonder how to justify such kind of physical relations by means of “first principles;” at least to make them plausible. Among the examples cited above, the last one is particularly intriguing, since it involves two descriptions of the same physical system, one of them time reversible (the microscopic dynamics) and the other with irreversible behavior (macroscopic thermodynamics). The justification of such prescription is one of the most fascinating problems of physics, and here the so-called ergodic hypothesis intervenes (and it was the birth of ergodic theory).

In this paper we recall the well-known Boltzmann and Gibbs proposals for the foundation of classical (equilibrium) statistical mechanics, review the usual arguments based on the ergodic hypothesis and discuss the problem, including modern mathematical aspects. At the end, we point out an alternative attitude for the justification of the foundations of classical statistical mechanics. Historical aspects and the “time arrow” will not be our main concerns (Refs. [12, 14, 18, 2, 23, 38, 40, 41]). Although most researchers accept the ideas of Boltzmann, there are some opposites, in particular I. Prigogine and his followers (references are easily found).

Most students approaching statistical mechanics have little contact with such questions. Having eyes also for precise statements, we hope this article will be helpful as a first step to fill out this theoretical gap. We can not refrain from recommending the nice article by Prentis [31] on pedagogical experiments illustrating the foundations of statistical mechanics, as well as the article by Mane [26] on aspects of ergodic theory via examples.

1E-mail: oliveira@dm.ufscar.br.

Copyright by the Sociedade Brasileira de Fısica. Printed in Brazil.

190 de Oliveira and Werlang

An Appendix summarizes the first steps of integration theory and presents selected theorems of ergodic theory; it is no more than a quick reference for the readers.

2. Microcanonical ensemble

In this section a discussion based on intuition will be presented. Later on some points will be clarified with mathematical rigor.

2.1. Micro and macrovariables

Establishing a mechanical model for the thermodynamic macroscopic observables is not a simple task. By beginning with the Hamilton equations of motion of classical mechanics

with a general time-independent hamiltonian H = H(q,p) and vectors (the so-called microva- momenta p = (p1,· ,pnN) coordinates (N denotes the number of identical particles of the system, and n the number of degree of freedom of each particle, so that the dimension of the phase space Γ is 2nN), one introduces adequate real functions f : Γ → IR defined on Γ. A thermodynamic description is characterized by a set of parameters, the so-called thermodynamic observables which constitute the macrovariables or macroscopic observables of the system. Sometimes such identification is rather direct, as in the case of the volume, but usually each thermodynamic quantity is presumably associated with a function f (which must be empirically verifiable). Notable exceptions are the entropy and temperature, which need a probability distribution µ over phase space in order to be properly introduced; for instance, in case of a mechanical system with a well-defined kinetic energy, the temperature is identified with the phase average of the kinetic energy with respect to µ. Such probability distributions are invariant measures, as discussed ahead. Note, however, that in general small portions of phase space have a well-defined temperature, pressure, etc., since their definitions are not clear for situations far from equilibrium (not considered here).

Usually, only macrovariables are subject to experimental observations and some important observables do not depend on all microvariables; for example, the density depends only on the positions of the particles.

Given an initial condition ξ = (q,p) ∈ Γ, also called a microstate, it will be assumed that the Hamiltonian generates a unique solution ξ(t) := Ttξ = (q(t),p(t)) of Eq. (1) for all t ∈ IR (sufficient conditions can be found in texts on differential equations), and the set of points

It will be assumed that orbits are restricted to bounded (compact) sets in phase space; this is technically convenient and often a consequence of the presence of constants of motion – as energy, i.e., H(ξ(t)) is constant as function of time – and also by constraints (as confining walls). Sometimes this fact will be remembered by the expression accessible phase space.

The number f(ξ(t)) should describe the value of the macroscopic observable represented by f, at the instant of time t, if it is known that at time t = 0 the system was in the microstate ξ. In principle, different initial conditions will give different values of the macroscopic observable f, without mentioning different times. If the system is in (thermodynamic) equilibrium, in a measurement one should get the same value for each observable, independently of the initial condition and the instant of time the measurement is performed; its justification is at the root of the foundation of statistical mechanics. Note also that the notion of macroscopic equilibrium, from the mechanical (microscopic) point of view, must be defined and properly related to the thermodynamic one.

In the physics literature there are three traditional approaches to deal with the questions discussed in the last paragraph: 1) time averages, 2) density function and 3) equal a priori probability. They are not at all independent, are subject of objections, and will be recalled in the following.

2.2. Three approaches 2.2.1. Time averages

A traditional way of introducing time averages of observables follows. Given a phase space function f that should correspond to a macroscopic physical quantity, the measurements of the precise values f(ξ(t)) are not possible since the knowing of detailed positions and momenta of the particles of the system would be necessary; it is then supposed that the result of a measurement is the time average of f.

It is also argued that each measurement of a ma- croscopic observable at time t0 takes, actually, certain interval of time to be realized; in such interval the mi- crostate ξ(t) changes and so different values of f(ξ(t)) are generated, and the time average may emerge as “constant” (i.e., independent of t0 and t). Next one asserts that the macroscopic interval of time for the measurement is extremely large from the microscopic point of view, so that one may take the

Ergodic hypothesis in classical statistical mechanics 191

This limit does not depend on the initial time t0 (see ahead) and arguing that for such large microscopic in- tervals of time the system visits all open sets of the phase space Γ during the measurement process, it sounds reasonable that this limit should coincide with the average value of f over Γ, defined by

For such integral be meaningful one has to assume that f is an integrable function (see the Appendix). Summing up, it is expected that which turns out to be the main version of the ergodic hypothesis (mathematically there are other equivalent formulations; see the end of Section 3 and Theorem 4 in the Appendix). Note that if this relation holds, then f∗ does not depend neither on the initial microstate ξ (excluding a set of measure zero) nor on the initial time, and it describes the equilibrium. Since a measu- rement for the constant function f1(ξ) = 1 must result the obvious value 1, one assumes that ∫ dξ = Ndqdp, where N denotes a normalization factor, dq = dq1 ···dqnN, dp = dp1 ···dpnN and so dξ is Lebesgue measure. Often the accessible phase space is supposed to be compact so that the Lebesgue measure is normalized and constant functions are integrable.

The assumption that 〈f〉 corresponds to the equilibrium value for each observable f is the so-called microcanonical ensemble or microcanonical measure, as discussed in Section 3. Note the prominent role played by Lebesgue measure in the discussion.

2.2.2. Density function

The initial idea is due to Maxwell and was then developed by Gibbs. Consider a huge collection of identical systems, each with its own initial condition at time t0 (some authors call this collection an “ensemble;” here an ensemble will mean a probability distribution, i.e., a probability measure as defined in the Appendix); the “typical behavior” of such collection would correspond to equilibrium. This behavior is characterized by an initial positive density function ρt : Γ → IR+. If ρt(ξ) is the corresponding density function at time t, then∫ A ρt(ξ)dξ indicates the average number of microstates that will occupy the set A ⊂ Γ at t (i.e., a probability);so ∫

Γ ρt(ξ)dξ = 1 due to the normalization of the total probability. The value of an observable f at time t would be ∫

Frequently the condition for equilibrium is written in the form dρt(ξ) note that if one makes explicit the time derivative and uses Hamiltonian given by Eq. (1), then it follows that (7) is exactly the famous Liouville equation. This equa- tion has the immediate solution ρt = constant, which is equivalent to the invariance of Lebesgue measure dξ

(see Proposition 1 and Eq. (8)). By taking this solution for granted, that is, by ignoring all other possible solutions, it results in the observable value 〈f〉 above, recovering the microcanonical ensemble as well as the conclusions on equilibrium of the previous subsection. Note again the introduction of averages, which is related to probability, and the latter becomes the main ingredient carrying out the micro-macro connection.

2.2.3. Equal a priori probability

This is the argument for introducing the microcanonical ensemble frequently invoked in textbooks on statistical mechanics. It is a variant of (sometimes complementary to) the previous discussion and, as stated here, the time evolution does not appear explicitly in the argument, and it is as follows.

Since there are no clear reason for certain accessible microstates be more probable than others, one postulates that for an isolated system all microstates are equally probable (maybe under the influence of Laplace’s Principle of Insufficient Reason) [36, 39]. In symbols, this is equivalent to taking the density func- tion ρt = constant in Eq. (6), that is, exactly the microcanonical ensemble.

In general, textbooks on statistical mechanics do not mention if the models considered are ergodic or not (see Section 3), since the microcanonical ensemble is introduced by means of a postulate instead of a discussion taking into account the dynamics. Due to the fruitfulness of statistical mechanics with respect to applications, some researchers consider that such discussion about ergodicity is not necessary. Another reason this discussion is avoided in textbooks is the high degree of abstraction that ergodic theory has currently reached, with a mathematical apparatus beyond the scope of such textbooks; the consequence is that it is not usually presented to physicists.

The next step here is to discuss some mathematical issues related to the ergodic hypothesis (including its precise definition) and the microcanonical ensemble.

3. A mathematical digression

There is a series of (interesting) questions, discussed in the last section, that an attentive reader could ask for more convincing explanations. For example:

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