Baixe Ergotic Hypothesis in classical statistical mechanics e outras Teses (TCC) em PDF para Física, somente na Docsity! Revista Brasileira de Ensino de F́ısica, v. 29, n. 2, p. 189-201, (2007) www.sbfisica.org.br Ergodic hypothesis in classical statistical mechanics (Hipótese ergódica em mecânica estat́ıstica clássica) César R. de Oliveira1 and Thiago Werlang2 1Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, Brasil 2Departamento de F́ısica, Universidade Federal de São Carlos, São 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 discussão atual sobre aspectos f́ısicos e matemáticos da hipótese ergódica em mecânica estat́ıstica de equiĺıbrio. Então indica-se uma eventual postura para se justificar o ensemble microcanônico. Observa-se, também, que a dificuldade em se demonstrar a hipótese ergódica deveria ser esperada. Palavras-chave: hipótese ergódica, mecânica estat́ıstica, ensemble microcanônico. 1. Introduction Important physical theories are built on relations and/or equations obtained through experiments, intui- tion and analogies. Hypotheses are proposed and ex- perimentally 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 Schrödinger 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 me- chanics is a link between microscopic dynamics and macroscopic thermodynamics via an invari- ant 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 in- volves two descriptions of the same physical system, one of them time reversible (the microscopic dynamics) and the other with irreversible behavior (macroscopic ther- modynamics). 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 ar- guments 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 statisti- cal mechanics. Historical aspects and the “time arrow” will not be our main concerns (Refs. [12, 14, 18, 22, 23, 38, 40, 41]). Although most researchers accept the ideas of Boltzmann, there are some opposites, in parti- cular 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 Mañé [26] on aspects of ergodic theory via examples. 1E-mail: oliveira@dm.ufscar.br. Copyright by the Sociedade Brasileira de F́ısica. Printed in Brazil. 190 de Oliveira and Werlang An Appendix summarizes the first steps of integra- tion theory and presents selected theorems of ergodic theory; it is no more than a quick reference for the rea- ders. 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 thermodyna- mic macroscopic observables is not a simple task. By beginning with the Hamilton equations of motion of classical mechanics q̇ = ∂H ∂p , ṗ = −∂H ∂q , (1) with a general time-independent hamiltonian H = H(q, p) and vectors (the so-called microva- riables) of (cartesian) positions q = (q1, · · · , qnN ) and 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 ma- croscopic 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 dis- tribution 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 de- finitions are not clear for situations far from equilibrium (not considered here). Usually, only macrovariables are subject to expe- rimental 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) := T tξ = (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 Oξ := {ξ(t) : t ∈ IR, ξ(0) = ξ} is the orbit of ξ in phase space. It will be assumed that orbits are restricted to boun- ded (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 con- fining 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 macrosco- pic observable f , without mentioning different times. If the system is in (thermodynamic) equilibrium, in a measurement one should get the same value for each ob- servable, independently of the initial condition and the instant of time the measurement is performed; its ju- stification 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 re- called in the following. 2.2. Three approaches 2.2.1. Time averages A traditional way of introducing time averages of ob- servables 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 mo- menta 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 1 t ∫ t0+t t0 f(T sξ) ds, (2) 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 193 system; different invariant measures result in different values of space averages of functions f , and are also as- sociated to different microscopic initial conditions—see the paragraph before Theorem 5. Note that this ques- tion is closely related to the description of (strange) attractors in the theory of dynamical systems. With respect to statistical mechanics, a major re- mark is that the Lebesgue measure dξ is invariant under T t, that is, the natural volume measure dξ is invariant under the time evolution generated by the Hamiltonian Eqs. (1). This result is known as Liouville theorem and it is also denoted by d(T−1ξ) = dξ. Under the Hamiltonian time evolution a set in phase space can be distorted but its volume keeps the same; this is far from trivial. Definition Let T t denotes a Hamiltonian flow. The microcanonical ensemble is the invariant measure dξ (see Eq. (5)). Recall that for a Hamiltonian system the energy is conserved, that is, given an initial condition (q, p) the value of the function H(T t(q, p)) = E, ∀t, is constant under time evolution, so that the motion is restricted to the surface (manifold) H(q, p) = E; Liouville theo- rem then implies that on this restricted dynamics the measure dξ|E ‖∇H‖E , is invariant (under the assumption ∇H 6= 0 in this sur- face); dξ|E denotes the Lebesgue measure restricted to the surface of energy E. The norm of the gradient ∇H restricted to the same surface is the right correction to Lebesgue measure that takes into account different particle speeds in different portions of energy surfaces. However, it is simpler to proceed using the measure dξ, and the interested reader can consider that dξ denotes the above restricted measure, and also that Γ denotes the surface of constant energy; no difficulty will arise. If there are additional constants of motion, which are often related to symmetries of the system, then all con- served quantities must be fixed, the motion then takes place in the intersection of the corresponding surfaces and the invariant measures must be adapted to each situation [18]. Another major result is the Birkhoff theorem: if a measure µ is invariant under a dynamics τ t, then the time averages in Eq. (2) are well defined, except for in- itial conditions on a set of µ zero measure (i.e., null pro- bability). Combining with Liouville theorem one gets that for Hamiltonian systems the limit defining time averages in Eq. (3) exists a.e. (this means “almost eve- rywhere,” a short way of saying “except on a set of zero measure”) with respect to Lebesgue measure dξ. If for the initial condition ξ the time average f∗(ξ) does exist, consider another initial time t1; then ∫ t1+t t1 f(T sξ) ds = ∫ t0 t1 f(T sξ) ds + ∫ t0+t t0 f(T sξ) ds + ∫ t1+t t0+t f(T sξ) ds. (9) Now ∫ t1+t t0+t f(T sξ) ds = ∫ t1 t0 f(T s+tξ) ds and so for reasonable (e.g., bounded) functions f this integral, as well as ∫ t0 t1 f(T sξ) ds, are bounded; therefore after divi- ding by t and taking t →∞ both vanish. Hence f∗(ξ) = lim t→∞ 1 t ∫ t1+t t1 f(T sξ) ds = lim t→∞ 1 t ∫ t0+t t0 f(T sξ) ds, (10) and a.e. the time average does not depend on the initial time. The exclusion of sets of measure zero is not just a mathematical preciosity. For example, for a gas in a box, consider an initial condition so that the motion of all particles are perpendicular to two opposite faces resulting in null pressure on the other faces of the box; another situation is such that all particles are confined in a small portion of the box; such initial conditions are not found in practice and the mathematical formalism is wise enough to include them in a set of Lebesgue measure zero for which the results do not apply. It is outstanding that the volume measure dξ is a macroscopic equilibrium for Hamiltonian mechanics (Liouville theorem), and that merely the existence of this equilibrium has resulted in well-defined time ave- rages and their independence on the initial time in a set of full volume. Although apparently we have ans- wered some of the questions proposed at the beginning of this section, an important point is, however, miss- ing: we are not sure that the microcanonical ensemble dξ is the equilibrium to be implemented in statistical mechanics, that is, why is this the invariant measure to be considered? This is the wanted justification of the ergodic hypothesis, a supposition not fully justified yet—as discussed ahead. An invariant measure µ with respect to a general dynamics τ t is called ergodic (or the pair (τ t, µ) is er- godic) if for every set A ⊂ Ω with τ t(A) = A, one has either µ(A) = 0 or µ(Ω \ A) = 0, i.e., every invariant set under the dynamics has zero or full measure. Then ergodicity means that the only nontrivial (that is, with nonzero measure) invariant set is just the whole set; in other words, ergodicity is equivalent to the set Ω be in- decomposable under the dynamics (τ t, µ). If it is clear what is the invariant measure under consideration, one also says that the system or the dynamics τ t is ergodic. This is just one possible way of defining ergodic measu- res, and a more detailed discussion is presented in the Appendix—compare with Definition 3 and Theorem 4. 194 de Oliveira and Werlang A consequence of the Birkhoff theorem is: if (τ t, µ) is ergodic and f is an integrable function, then for µ-a.e. the time averages f∗(ω) exist, are constant and equal the space average, that is f∗(ω) = ∫ Ω f(ω) dµ(ω). (11) In the context of Hamiltonian dynamics, if (T t, dξ) is ergodic, then the time averages f∗(ξ) are constant Le- besgue a.e. and Eq. (5) holds, so that for justifying the equality between time and space averages one has to prove that the volume measure dξ in phase space is ergodic under the dynamics generated by the Hamilto- nian equations: a chief mathematical problem. Consider now density functions ρt (see Eq. (6) and the discussion in Sections 2.2.2 and 2.2.3), that is, the particular class of measures absolutely continuous with respect to Lebesgue of the form dν(ξ) = ρt(ξ) dξ. (12) Note first that without Liouville theorem the dis- cussion about density functions in Subsection 2.2.2, in- cluding condition given by Eq. (7) for “equilibrium,” would be incorrect. Indeed, for the average value 〈f〉 of f over Γ at time t we have ∫ Γ f(T tξ) ρ0(ξ) dξ = ∫ Γ f(ξ) ρ0(T−tξ) d(T−tξ) = ∫ Γ f(ξ) ρ0(T−tξ) dξ. (13) Liouville theorem was applied in the last equality and the expression ρt(ξ) = ρ0(T−tξ) has been revealed. As a consequence of another important result of the abstract ergodic theory, which is stated in Theorem 5, one has that if (T t, dξ) is ergodic, then the condition for ν in Eq. (12) (i.e., measures defined via density functi- ons) be invariant under T t implies that the (integrable) function ρt(ξ) = 1, that is, it is necessarily constant. Again we have an important consequence of the (pre- supposed) ergodicity of dξ with respect to the Hamil- tonian evolution: the mathematical justification of the equal a priori probability and the use of the mentioned trivial solution of Eq. (7). See also the discussions in Refs. [10, 23, 25]. As a last consequence of the ergodicity of dξ we re- mark that it implies that a.e. the orbits Oξ are dense in the accessible phase space Γ, that is, each orbit inter- sects every open set of Γ. This property may not hold for general ergodic measures. The time is ripe for a precise statement of the Ergodic hypothesis in statistical mechanics: The microcanonical ensemble dξ is ergodic with respect to the Hamiltonian dynamics. In the next section we discuss the situation related to this supposition. 4. Discussion on the ergodic hypothesis From the discussion in the preceding section, if the ergo- dic hypothesis holds, then we have a precise mechanical definition of macroscopic equilibrium dξ, the existence a.e. of time averages of integrable functions f repre- senting macroscopic observables, the equality of such averages with space averages (i.e., relation (5)) and a justification of the adoption of the microcanonical en- semble. In summary, the statistical mechanics prescrip- tion would be justified. In this section many aspects related to the ergodic hypothesis in classical statistical mechanics are discus- sed. We have tried to cover the main current points, including some numerical indications. 4.1. Success and nonergodicity Before going on, we present an example of a Hamil- tonian system that is not ergodic. Consider two in- dependent harmonic oscillators; since the phase space can be decomposed in two independent parts, one for each oscillator and both of nonzero dξ measure, the pair (T t, dξ) is not ergodic. The argument easily ge- neralizes to noninteracting systems with finitely many particles, so that for the ergodicity of (T t, dξ) the in- teraction among its particles is fundamental. For the so-called gas of hard spheres whose partic- les interact only via elastic collisions, it was recently demonstrated by Simányi [35] that elastically colliding N ≥ 2 hard balls, of the same radius and arbitrary masses, on the flat torus of any finite dimension is er- godic (also mixing, see ahead); the energy is fixed and the total momentum is zero. This is a generalization of results by Sinai around 1970 for N = 2. There are some classes of two-particle mechanical systems, i.e., the Ha- miltonian is the sum of the kinetic energy K and the potential energy U , on the two-dimensional torus stu- died by Donnay and Liverani [9], whose potential U is radially symmetric and vanishes outside a disk, for which ergodicity is also proved. These can be conside- red the most realistic models in statistical mechanics where the ergodic hypothesis has been rigorously esta- blished. What is then the attitude when other interac- tions are present? Unfortunately a satisfactory answer is still missing. Some authors (as in most textbooks) suggest that the postulate of equal a priori probability should be invoked and taking the experimental results as a final confirmation. Some workers in the area have argued that ergo- dic theory did not success in explaining the positive results of statistical mechanics. For instance, Earman and Rédei [10] claim that many models for which sta- tistical mechanics works are likely not ergodic and so nonergodic properties must be invoked. As an alterna- tive they have proposed that an ergodic-like behavior, Ergodic hypothesis in classical statistical mechanics 195 i.e., the validity of Eq. (5) should hold only for a fi- nite set of observables f (see also Eq. [24], where it is proposed to restrict ergodicity and mixing to suitable variables). They assert that for each model the equi- librium statistical mechanics predicts values for only a finite set of observables, and one should investigate whether the ergodic-like behavior holds only for such set, which may differ from system to system. There are some criticisms on the assumption that the outcome of a measurement can be described by in- finite time averages; for example, only results concer- ning quantities in equilibrium could be obtained in this way (see Refs. [16] p. 84 and [37] p. 176). Note that in this work only equilibrium statistical mechanics is con- sidered. Also that sets of Lebesgue measure zero can be neglected has the opposition of some authors; the interested reader is referred to Ref. [25]. Another suggestive argument employed for the ju- stification of the microcanonical ensemble comes from analogies with the second law of thermodynamics. Se- parate the phase space Γ into a finite number of M disjoint cells and associate a probability pj of a repre- sentative of the microstate be in the j-th cell; then we have the constrain ∑M j=1 pj = 1. Such separation of phase space into cells of nonzero volume is usually cal- led a coarse grain partition and associated to each of them one defines the Gibbs coarse grain entropy S(pj) = − M∑ j=1 pj ln pj . Now impose that the equilibrium is attained for the dis- tribution of pj with maximum entropy under the above constrain; it is left as an exercise to conclude that such maximum is obtained for pj = 1/M , for all j, that is, equal probability. In the formal limit of infinite many cells (with vanishing sizes) one gets an indication for the validity of the equal a priori probability. Note, however, that the microscopic evolution does not enter explicitly in the argument, as should be expected, and that this argument only shift the actual problem to another one. Since the idea of separating the phase space into finitely many cells was mentioned, it is worth recal- ling Boltzmann reasoning that led him to the “origi- nal formulation of the ergodic hypothesis” [14] in the 1870’s. Under time evolution Boltzmann supposed that the cells are cyclically permuted; then it became natu- ral to assume that time averages could be performed by averaging over cells. In the limit of infinite cells one would get Eq. (5). There is a clear general uncomfortable reaction in the literature with respect to the missing proofs of the ergodic hypothesis for a large class of particle inter- actions. In our opinion such great difficulty for ergo- dic proofs should be expected due to the possibility of coexistence of different phases. More precisely, sup- pose that (T t, dξ) is not ergodic; another mathematical result says that every invariant measure is a convex combination of ergodic ones (see Theorem 5). In case dξ = λdµ1 + (1 − λ)dµ2, with 0 < λ < 1 and (T t, µ1) and (T t, µ2) ergodic with µ1 ⊥ µ2, then, given an ob- servable f , there are two disjoint sets A1, A2 ⊂ Γ with µ1(A1) = 1 = µ2(A2) and µ1(A2) = 0 = µ2(A1), so that by Birkhoff theorem and ergodicity, time averages of f exist for initial conditions ξ ∈ Aj and resulting in∫ Γ f(ξ) dµj(ξ), j = 1, 2 (see Eq.(11)). That is, there are two different important equilibrium for the system, one for each ergodic component µ1, µ2, which one can interpret as the coexistence of different phases. The argument generalizes for more than two ergodic mea- sures in the decomposition. From this point of view, the results of Sinai and Simányi [35] can be interpre- ted as a proof of just one phase for the gas of finitely many hard spheres. This is closely related to a rigo- rous approach to the description of phase transitions when one works directly with infinite volume systems, in which the corresponding thermodynamic limit is ta- ken on probability measures; the reader is referred to Refs. [19, 15] and references there in, and for an intro- duction to Gibbs states in dynamical systems see Ref. [17]. Without going into a detailed discussion of none- quilibrium, we observe that the usual justification why systems approach to equilibrium under time evolution [20] is via the property of mixing, which was intuitively explored by Gibbs (through the well-known example of stirring a drop of ink in a liquid [23]), but its mathema- tical definition is due to von Neumann in 1932: (τ t, µ) is mixing (µ invariant) if for any A,B ⊂ Ω one has lim t→∞ µ(τ t(A) ∩B) = µ(A) µ(B). (14) This expression means that as time increases the por- tion of a given measurable set A that then resides any set B is proportional to the measure of B; thus, ac- cording to the probability measure µ, τ t(A) becomes uniformly distributed over Ω. If this condition is sa- tisfied then, necessarily, the system is ergodic; in fact, if τ t(A) = A and by taking B = A in (14), it follows that µ(A) = µ(A)2, so that µ(A) = 0 or µ(A) = 1, i.e., (τ t, µ) is ergodic. Therefore, the natural condition claimed to assure convergence to equilibrium implies ergodicity. Although this proof sounds simple today, it hides a complex set of developments and such implica- tion was not known by Boltzmann and Gibbs, and was first proved many years after their works on the foun- dations of statistical mechanics. Certainly, it is usually harder to show that a system is mixing than its ergodi- city (in case such properties hold). Lombardi [24] argued that although some authors take a pragmatic position to the microcanonical en- semble [36, 39], in accepting that mixing of dξ plays a significant role in the description of the approach to macroscopic equilibrium dξ then, according to the pre- ceding paragraph, ergodicity of dξ comes up again. Ne- 198 de Oliveira and Werlang have not dealt with: the problem of irreversibility; the evolution toward equilibrium and the arrow of time; Kinchin approach [18, 2] to the ergodic question; er- godicity and phase transitions of systems with an infi- nite number of degrees of freedom [19, 15]. Note that in order to discuss situations far from equilibrium one should go a step further the ideas presented above, in particular for estimating the increase of entropy, and so on. Finally, we mention a personal view expressed by G. Gallavotti; in his opinion, despite the lack of pro- per mathematical tools Boltzmann understood many points related to the ergodic hypothesis better than we do now. Acknowledgments We thank the anonymous referee for valuable suggesti- ons. C.R. de O. thanks the partial support by CNPq. T.W. thanks a scholarship by FAPESP. Appendix A short discussion on ergodic theory The (mathematical) ergodic theory can be considered a branch of dynamical systems, particularly of those that preserve a measure. After a discussion of (Lebes- gue) integration theory, some abstract results related to ergodicity are presented. It is expected that this col- lection of results and ideas could clarify many of the statements in the main body of this article. Details of the traditional subject of measure and integration can be found, for instance, in Refs. [6, 32], and of ergodic theory in Refs. [5, 14, 26, 27, 29, 42]. Measure and integration Measure theory is a generalization of the concept of length, area, etc., as well as of probability and density. For example, in the case of the real line IR the natural length of an interval (a, b) is (b − a), its so-called Le- besgue measure, and one tries to extend this notion to all subsets of IR (generalized lengths can also be con- sidered, e.g., (b2 − a2), resulting in other measures). However, by using the Axiom of Choice it is possible to construct subsets of IR for which their Lebesgue mea- sure depends on the way the set is decomposed. So, clearly such sets can not be considered with a well- defined length, and one says they are not measurable. A consequence of this remark is that each measure must be defined on a specific domain of measurable sets, cal- led a σ-algebra; and the theory gets rather involved. Definition 1 A σ-algebra in a set Ω is a collection A of subsets of Ω (and each element of A is called a mea- surable set) so that 1. Ω and ∅ belong to A. 2. If A ∈ A then Ω \A ∈ A. 3. If A1, A2, A3, · · · are elements of A, then ∪∞j=1Aj ∈ A. The Borel σ-algebra B in a topological space is the smallest σ-algebra that contains all open sets; in this case each measurable set is also called a Borel set. Definition 2 A (positive) measure µ on a σ-algebra A is a function µ : A → [0,∞] (in some cases infinity is allowed) so that if A1, A2, A3, · · · are pairwise disjoint elements of A, then µ (∪∞j=1Aj ) = ∑∞ j=1 µ(Aj). From the definitions a series of properties of mea- sures follows. For example: µ(∅) = 0; if A1, A2 are measurable sets with µ(A1) < ∞ and A2 ⊂ A1, then µ(A1 \A2) = µ(A1)− µ(A2). A measure µ is finite if µ(Ω) < ∞ and σ−finite if Ω = ∪∞j=1Aj with µ(Aj) < ∞, ∀j. A measure µ is a probability measure if µ(Ω) = 1. A Borel measure is one defined on the Borel sets. Here all considered measures are Borel and σ-finite. In open subsets of IRn and differentiable manifolds it is possible to introduce Lebesgue measure on the Borel sets by extending the notion of length, area, and so on, from the respective definitions on intervals, rectangles, etc. (Lebesgue measure is, in fact, a slight generali- zation of this construction, but this is not important here). Sets of measure zero play a very important role in the theory, particularly in applications. If µ(A) is inter- preted as a probability of the occurrence of the events (points) of A, then if µ(A) = 0 with A 6= ∅ there are events that are not expected to occur; it is related to the famous expression “possible but improbable events.” A shorthand notation for a property that holds except on a set of µ measure zero is µ-a.e. (almost everywhere). As an illustration, consider a gas with 1023 molecu- les in a box at certain temperature; a possibility is all molecules moving parallel to each other giving nonzero pressure to just two sides of the box; however, such par- ticular situation is improbable and is not expected to be found in practice, so the whole set of such configu- rations must have zero measure. With a measure µ at hand it is possible to integrate certain positive functions f : Ω → [0,∞) with respect to µ; for instance if f = χA is the characteristic function of the measurable set A, that is, χA(ω) = 1 if ω ∈ A and zero otherwise, then one defines ∫ Ω f dµ = ∫ Ω f(ω) dµ(ω) := µ(A), and extends it linearly, that is, for f = ∑n j=1 ajχAj , aj ∈ IR, Aj measurable, 1 ≤ j ≤ n (the so-called sim- ple functions), then ∫ Ω f(ω) dµ(ω) := n∑ j=1 ajµ(Aj). Ergodic hypothesis in classical statistical mechanics 199 Functions f that can be approximated by simple func- tions fn in a pointwise way are called measurable func- tions, and their integrals are defined by the correspon- ding limit ∫ Ω f dµ := lim n→∞ ∫ Ω fn dµ. For not necessarily positive functions, consider its positive f+(ω) = max{0, f(ω)} and negative f−(ω) = max{0,−f(ω)} parts, so that f = f+ − f− and define ∫ Ω f dµ := ∫ Ω f+ dµ− ∫ Ω f− dµ. If both integrals in this difference is finite one says that f is integrable (or µ-integrable if one wishes to specify the measure) and the space of such functions is deno- ted by L1µ(Ω) (sometimes complex-valued functions are allowed). Fix a set Ω and a σ-algebra A. Given two measures µ, ν on A, then µ is absolutely continuous with respect to ν, denoted by µ ¿ ν, if µ(A) = 0 in case ν(A) = 0. µ and ν are equivalent if µ ¿ ν and ν ¿ µ. An interes- ting way of generating a measure µ from another given measure ν is by means of densities dµ(ω) = ρ(ω) dν(ω), that is ∫ Ω f dµ = ∫ Ω fρ dν, for some positive ν-integrable function (the density) ρ : Ω → [0,∞). Theorem 1 (Radon-Nikodym) µ ¿ ν if, and only if, there exists a density ρ as above so that dµ(ω) = ρ(ω) dν(ω). Two measures µ and ν are mutually singular if there exist a measurable set A so that µ(A) = 0 = ν(Ω \ A), which is denoted by µ ⊥ ν. Note that if µ ⊥ ν, then the concepts of µ−a.e. and ν−a.e. apply to really different situations! Basic concepts of ergodic theory Let Ω be a compact metric space and τ t : Ω → Ω be a flow, that is τ t is a continuous and invertible map for each t ∈ IR (identified with time), τ0ω = ω, ∀ω ∈ Ω, τ t+s = τ tτs, ∀t, s ∈ IR. A Borel measure µ is inva- riant for the flow τ t if µ(τ tA) = µ(A), ∀A ∈ B and t ∈ IR. The existence of invariant measures in this case is assured by Krylov-Bogolioubov theorem. Proposition 1 The measure µ is invariant under the flow τ t if, and only if, for all continuous function f : Ω → IR one has ∫ X f(τ t(ω)) dµ(ω) = ∫ X f(ω) dµ(ω), ∀t. The presence of an invariant measure implies the existence of time averages 1 t ∫ t 0 f(τ sω) ds, as t →∞, of certain functions f : Ω → IR. This is the main con- tent of theorems usually denominated “ergodic theo- rems.” Here two of such results will be stated, von Neumann Mean Ergodic Theorem and Birkhoff Point- wise Ergodic Theorem, both proved around 1931. The interested reader in a rather complete treatment of er- godic theorems and their variants are referred to Ref. [21]. Theorem 2 (von Neumann) Let f : Ω → IR be a function so that its square |f |2 is integrable and µ an invariant measure for the flow τ t. Then there exists a function f̃ : Ω → IR so that |f̃ |2 is integrable and lim t→∞ ∫ Ω ∣∣∣∣ 1 t ∫ t 0 f(τsω) ds− f̃(ω) ∣∣∣∣ 2 dµ(ω) = 0. (15) Theorem 3 (Birkhoff) Let µ be an invariant mea- sure for the flow τ t. If f : Ω → IR is integrable, then i f∗(ω) := limt→∞ 1t ∫ t 0 f(τsω) ds exists µ-a.e. and the function f∗ is also integrable. ii f∗(τ tω) = f∗(ω) µ-a.e., that is, f∗ is constant over orbits. iii ∫ Ω f∗(ω) dµ(ω) = ∫ Ω f(ω) dµ(ω). The proof of Theorem 2 is much simpler than the proof of Birkhoff theorem 3, but the former gives no information on the existence of time averages of indi- vidual initial conditions ω, since an integral is present before the limit t → ∞. Item ii in Birkhoff theorem should be expected. Item iii says that “space average” of f coincides with time average of f∗, and an import- ant particular case is when the latter is constant. Definition 3 If for each integrable f the time average f∗ is constant µ−a.e., then the pair (τ t, µ) is called ergodic. Note that this implies f∗(ω) = ∫ Ω f(ω′) dµ(ω′), µ− a.e., that is, the equality of space and time averages. If the flow τ t is fixed, it is also common to say that the invariant measure µ is ergodic. Under the conditi- ons of Krylov-Bogolioubov result mentioned above, it is possible to show that ergodic measures exist. It is worth considering f = χB in case µ is ergodic χ∗B(ω) = lim t→∞ 1 t ∫ t 0 χB(τsω) ds = µ(B), µ− a.e., that is, τ tω visits each set B with frequency equal to the measure of B. 200 de Oliveira and Werlang There is a number of characterizations of ergodi- city, and we mention some of them ahead. A measura- ble set A is invariant if τ t(A) = A and µ-invariant if µ(τ tA∆A) = 0, ∀t ∈ IR (recall that A∆B = (A \ B) ∪ (B \ A) is the symmetric difference between the sets A and B), and a measurable function f is invariant if f ◦ τ t = f µ−a.e.; for simplicity, in what follows invari- ant measures are supposed to be probability measures. Theorem 4 Let µ be an invariant probability measure for τ t. The following assertions are equivalent: i) (τ t, µ) is ergodic. ii) Every invariant measurable function is constant µ−a.e. iii) Every µ-invariant set A ∈ A has µ measure 0 or 1. iv) Every invariant set A ∈ A has µ measure 0 or 1. The condition in item iv) of Theorem 4 is called indecomposability; sometimes this nomenclature is also used to the condition in item iii). Unfortunately none of the characterizations in this theorem is easy to verify for models in statistical mechanics; however, they have been checked for several dynamical systems in mathe- matics (see the cited books above). It is important to realize that consequences of ergo- dicity depends on the invariant measure under conside- ration; for example, if µ and ν are ergodic (with respect to the same flow), for each function f Birkhoff theorem implies there are sets A,B, with µ(A) = 1 = ν(B), so that time averages f∗(ω) exist for any initial con- dition ω ∈ A, resulting in ∫ Ω fdµ, as well as for any ω ∈ B, but now resulting in ∫ Ω fdν. Since in general∫ Ω fdµ 6= ∫ Ω fdν, different ergodic measures are related to different values of time averages and over different sets of initial conditions. Theorem 5 Let ν ¿ µ. If ν is invariant and µ ergo- dic, then ν = µ. Theorem 6 Let Ω be a compact metric space, τ t a con- tinuous flow on Ω and µ an invariant probability mea- sure. 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