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Ergodic Theorem, Ergodic Theory, and Statistical Mechanics Calvin C

Ergodic Theorem, Ergodic Theory, and Statistical Mechanics Calvin C

PERSPECTIVE PERSPECTIVE Ergodic theorem, ergodic , and Calvin C. Moore1 Department of , University of California, Berkeley, CA 94720

Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved January 9, 2015 (received for review November 13, 2014)

This perspective highlights the mean ergodic theorem established by and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject—namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman’s to the Ehrenfests’ quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics.

George D. Birkhoff (1) and John von (a concept to be defined below). First of all, container. The molecules are in motion, Neumann (2) published separate and vir- these two papers provided a key insight into colliding with each other and with the hard tually simultaneous path-breaking papers a 60-y-old fundamental problem of statistical walls of the container. The molecules can be in which the two authors proved slightly mechanics, namely the rationale for the hy- assumed for instance to be hard spheres different versions of what came to be known pothesis that time averages can be set equal to (billiard balls) bouncing off each other or (as a result of these papers) as the ergodic phase averages, but also initiated a new field of alternately may be assumed to be polyatomic theorem. The techniques that they used were mathematical research called ergodic theory, molecules with internal structure and where strikingly different, but they arrived at very which has thrived for more than 80 y. Sub- collisions are governed by short-range re- similar results. The ergodic theorem, when sequent research in ergodic theory since 1932 pelling potentials. One may also choose to applied say to a mechanical system such as has further expanded the connection between include the effects of external forces, such as one might meet in statistical mechanics or in the ergodic theorem and this core hypothesis gravity on the molecules. We assume that the celestial mechanics, allows one to conclude of statistical mechanics. phase M consists of a surface of constant remarkable results about the average behavior The justification for this hypothesis is . This assumption, together with the of the system over long periods of time, pro- a problem that the originators of statistical finite extent of the container, ensures that M mechanics, J. C. Maxwell (3) and L. Boltzmann vided that the system is metrically transitive is compact and that the invariant de- (4), wrestled with beginning in the 1870s as rived from Liouville’stheoremisfinite.The did other early workers, but without math- equations of motion, say in Hamiltonian form, ematical success. J. W. Gibbs in his 1902 can be written in local coordinates as a first- work (5) argued for his version of the hy- order system of ordinary differential equations pothesis based on the fact that using it gives results consistent with experiments. The dxi=dt = Xiðx1; ...xnÞ: 1931–1932 ergodic theorem applied to the of a mechanical system that First, the number of variables in the equa- arises in statistical mechanics and to the tions is enormous, perhaps on the order of one-parameter group of homeomorphisms Avogadro’s number, and the equations are representing the time evolution of the system quite complex. The system is perfectly deter- asserts that for orbits, the time ministic in principle; hence, given the initial average of an integrable function on phase positions and momenta of all of the mole- space is equal to its phase average, provided cules at an initial time, the system evolves that the one-parameter group is metrically transitive. Hence, the ergodic theorem trans- forms the question of equality of time and Author contributions: C.C.M. wrote the paper. phase averages into the question of whether The author declares no conflict of interest. the one-parameter group representing the This article is a PNAS Direct Submission. time evolution of the system is metrically This article is part of the special series of PNAS 100th Anniversary transitive. articles to commemorate exceptional research published in PNAS over the last century. See the companion articles, “Proof of the To be more specific about statistical me- ergodic theorem” on page 656 in issue 12 of 17 and “Proof George D. Birkhoff. Image courtesy of the chanics systems, consider a typical situation of the quasi-ergodic hypothesis” on page 70 in issue 1 of volume American Mathematical Society (www. in dynamics where one has a macroscopic 18, and see Core Concepts on page 1914. ams.org). quantity of a dilute gas enclosed in a finite 1Email: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1421798112 PNAS | February 17, 2015 | vol. 112 | no. 7 | 1907–1911 Downloaded by guest on September 26, 2021 that a physical measurement takes a short can be identified as the conditional expecta- period to perform, but this short time period tion of f onto the sigma field of invariant sets, is a long period for the physical system be- to use the language of ). In cause, for instance, each molecule will on thecaseofmetrictransitivity,thisfunction average experience billions of collisions per is just the constant function equal to the second in a typical gas. In fact, this line of integral of f over M.vonNeumannproved reasoning leads to the idea that the result of that these functions of x converge in mean the measurement is best represented by the square [that is in L2ðM; μÞ] to the orthogonal limit as T tends to infinity of the time average projection of the function f onto the closed above. The first problem is, of course, to subspace of invariant functions, which in the know that this limit exists, except of course metrically transitive case is one dimensional for a negligible set of x.Thentheassumption consisting of the constant functions. Birkhoff of equality of time averages and phase aver- assumes the function f is bounded and mea- ageswouldassertthatthelimitofthetime surable, whereas von Neumann assumes the average above is independent of x and equal more general condition that the function f is to the phase average square integrable. Although both theorems Z were originally formulated and proved for f ðvÞdμðvÞ; measure preserving one parameter groups generated by first-order differential equations M on compact manifolds, subsequent work has John von Neumann. Image courtesy of the shown using the same arguments that these x ∈ M μ Shelby White and Leon Levy Archives Center, for all but a neglible set of where is results are valid for a much broader class of Institute for Advanced Study, Princeton. the Liouville . The signifi- dynamical systems including one-parameter cant and useful point is this phase average families of measure preserving transforma- can be calculated in many cases, whereas tions of a finite measure space, which may deterministically. However, there is no chance the time average cannot be calculated. not necessarily be defined by systems of dif- of knowing these initial conditions exactly Another way of phrasing this equality is ferential equations. Later work also showed and little chance of integrating these equations f to use for the indicator function of a that Birkhoff’s theorem holds for an integra- to find the solutions. Given therefore that we A M measurable of (a function that ble function f. Thus, these theorems are the- only have partial information about the sys- A A is equal to 1 on and 0 outside of ). Then orems about one-parameter groups of au- tem, a statistical approach to the analysis of the time averages above record the fraction of tomorphisms of measure with no such systems is appropriate and necessary. A time that the spends in ,andthebasic mention of topology. The theorems also Maxwell (6) and Boltzmann (7) began such hypothesis of statistical mechanics asserts clarify what is meant by the informal term a project, which was further developed and that this is equal for almost all orbits to the in statistical mechanics, negligble set, namely, elaborated by Gibbs (5). A Liouville measure of (assuming that the it is a set of μ that measures zero. It should The system of differential equations above M measure of the total space has been nor- be added that the time average does not have generates a , which we denote PtðxÞ malized to 1). However, another way of ex- to be taken from −T to T, but can be taken where x = ðx ; ...xnÞ is a point in the phase 1 pressing this equality is to assert that for all over any intervals from T to T ,aslongas space M representing the system at time 1 2 but a negligible set of states of the gas, the the difference T2 − T1 tends to infinity. t = 0. PtðxÞ is the position to which the sys- observed value of a function f will be equal Before moving on to subsequent develop- tem moves after time t so that PtðxÞ is the to the average value of f taken over M that ments in ergodic theory, it is worth pausing solution of the differential equation with initial is an average value of f taken over an en- x t = P to examine the sequence of events leading to value at time 0. t is a homeomorphism semble (to use Gibbs’s language) of all pos- M t the proofs and publication of the two ergodic of onto itself defined for all (as the system sible states with the same energy. The theorems: the pointwise ergodic theorem of is time reversible) and satisfies the group prop- constant energy surface M with its invari- P P = P Birkhoff and the mean ergodic theorem of erty t s t+s. It also preserves the Liouville ant volume element here is what Gibbs μ von Neumann. Much of this was laid out measure , which is finite in this case. called the . f in a subsequent paper of Birkhoff and Now if is an integrable function on the The ergodic theorems of Birkhoff and von M Koopman (8) in the March 1932 issue of the phase space ,onemayarguethatifone Neumann assert first of all of the existence of f PNAS. von Neumann was very much aware makes a physical measurement of on a sys- the time limit for T → ∞ for any one param- x ∈ M of the results of M. H. Stone (9) on spectral tem that is in state , what one gets, at eter measure preserving group, and then, as- theory of one-parameter groups of unitary least for all but a negligible number of states suming that Pt is metrically transitive, they x operators and the results of Koopman (10) , is a time average assert the equality that used Stone’s results to analyze one- parameter groups of measure preserving ZT ZT Z transformations. Koopman had indeed sug- −1 1=2T f ðPtðxÞÞdt: lim ð2TÞ f ðPtðxÞÞdt = f ðvÞdμðvÞ: T→∞ gested to von Neumann that he might use −T −T M these results to resolve the problem of equal- ity of time and phase averages, and von This is a time average of the values of f Thedifferencebetweenthetwotheorems Neumann writes that Andre Weil had made over a part of the path of the solution of is that Birkhoff proved that the convergence the very same suggestion to him. von Neu- the equations with value x at time t = 0. of the functions of x ontheleftsideispoint- mann seized on the notion of metric transi- Theargumentthatitisreasonabletoassume wise (the limit in general tivity, introduced, somewhat ironically, by

1908 | www.pnas.org/cgi/doi/10.1073/pnas.1421798112 Moore Downloaded by guest on September 26, 2021 Birkhoff and Smith (11) a few years earlier that is a good approximation to the inte- the mechanical system, say for gas dynamics, PERSPECTIVE in 1928, and proved his mean ergodic theo- gral of f over the space and that the more starting from any point, under time evolution rem under the hypothesis of metric transi- iterates one includes in the average, the Pt, would eventually pass through every state tivity. See the article by Mackey (12) for better the approximation. Therefore, it is on the energy surface. Maxwell and his fol- more details. like a numerical integration scheme. lowers in England called this concept the According to Birkhoff and Koopman, von Finally, we need to define metric transitiv- continuity of path (3). It is clear that under Neumann communicated his result person- ity, a concept, as previously noted, that was this assumption, time averages are equal to ally to both of them on October 22, 1931, and introduced by Birkhoff and Smith (11). A phase averages, but it is also equally clear to pointed out to them that his result raised the one-parameter group of measure preserving us today that a system could be ergodic in important question of whether a pointwise transformation PtðxÞ (or a single transforma- this sense only if phase space were one di- result might be valid. Birkhoff then went to tion P)onameasurespaceM is metrically mensional. Plancherel (14) and Rosenthal work and, by different methods, quickly es- transitive provided that any μ measurable set (15) published proofs of this, and earlier, tablished his pointwise ergodic theorem. He invariant under Pt for all t (or P)musthave Poincare (16) had expressed doubts about submitted his paper to PNAS on December 1, zero measure or its must have Boltzmann’s ergodic hypothesis. Certainly 1931, for appearance in the December 1931 zero measure. This means that the flow is part of the problem Maxwell and Boltzmann issue. One presumes that he sent copies to indecomposable or irreducible in the sense faced was that the mathematics necessary for Koopman and von Neumann, who would that one cannot decompose it into a a proper discussion of the foundations of sta- have noticed that Birkhoff had not given von of two disjoint subflows. It also means that tistical mechanics, such as the measure the- Neumann adequate credit and recognition there are no measurable functions invariant ory of Borel and Lebesgue, and elements of for his result. von Neumann evidently undertheflow(orthetransformationP). modern topology had not been discovered planned to include his ergodic theorem and It is heuristically reasonable to argue, until the first decade of the 20th century and its proof in a much longer paper he was owing to the molecular chaos in gas dynam- were hence unavailable to them. writing for the Annals of Mathematics, but ics, that there are no nonconstant continuous In their influential 1911 article, Ehrenfest he then apparently quickly drafted a short invariants or so-called first integrals of the and Ehrenfest (17) summarized and dis- paper for PNAS with his proof of the mean motion. However, more is required for metric cussed problems with the ergodic hypothesis ergodic theorem and submitted it to PNAS transitivity—namely no nonconstant measur- and then proposed instead the quasi-ergodic on December 10, 1931. It appeared in the able invariants of the motion. In the example hypothesis as a replacement. This hypothesis January 1932 issue. One suspects that these from gas dynamics, the total energy is clearly states that some orbit of the flow will pass events led Koopman and Birkhoff to write an invariant of the motion, but we have re- arbitrarily close to every point of phase space, and publish their paper in PNAS 2 months stricted the flow to a surface of constant en- or in other words this orbit is topologically later, which set matters straight and clearly ergy. In addition, total is nor- dense in the phase space. This hypothesis is acknowledged von Neumann’s priority. It mally preserved, but although momentum a far more plausible one than the old ergodic should also be noted that E. Hopf (13) pre- is preserved in collisions between molecules, hypothesis, and it does imply that any con- sented a slightly different proof of the mean collisions with the walls do not preserve mo- tinuous function invariant under the flow ergodic theorem and some improvements mentum, so this possible invariant of the mo- is constant. Some authors [von Plato (18)] on the Birkhoff theorem in a paper, which tion disappears. Although the term metric have argued that, despite what Boltzmann appeared in the January 1932 issue of transitivity is still in use, current terminology, had written down most of the time in his PNAS. For whatever reason, the Birkhoff due to von Neumann, is that any flow or articles about the ergodic hypothesis, that paper and its result has over time become single transformation with this property is heprobablyreallymeantsomethinglikewhat the better known of the two papers, but simply called ergodic. was later termed the quasi-ergodic hypothe- in light of these historical details, the It is worth observing that metric transitiv- sis. However, the quasi-ergodic hypothesis von Neumann paper deserves at least ity is a necessary and sufficient condition for does not imply metric transitivity. For in- equal billing. the validity of the ergodic theorem. To see stance, it is not even true that a minimal flow There are also ergodic theorems for a single this, assume the ergodic theorem holds and (every orbit is dense) with an invariant mea- measure-preserving P and its iterates Pn. then apply the statement of the theorem to sure is metrically transitive [see Furstenberg They assert the existence of the time limit the indicator function f of a supposed invari- (19)]. for examples. Therefore, although the A— f below and that it converges to the phase av- ant measurable set that is, is equal 1 on original ergodic hypothesis was too strong to A A erage if P is metrically transitive; that is and 0 on the complement of ;theleftside be plausible, the quasi-ergodic hypothesis was of time averages is always equal to f,butthe too weak to establish equality of time and XN Z f − n right side is a constant function. Hence, is phase averages. Further mathematical, prog- lim ðN + 1Þ 1 f ðP xÞ = f ðvÞdμðvÞ: N→∞ a constant function, and the alleged invariant ress had to await the concept of metric tran- n= 0 M set is of measure zero or its complement is of sitivity and the ergodic theorems of 1931 and measure zero. 1932. For more details, see the survey article The convergence is pointwise for almost all It is interesting to look back at the early of Mackey (20). x for integrable f,andinthemeanforf history of statistical mechanics to see how the One reaction to the Birkhoff and von square summable. One way to conceive of founders of the subject handled the topic of Neumann ergodic theorems might be that metric transitivity and the ergodic theorem time averages and space averages. Boltzmann they do not really solve the problem of for a single transformation is that for almost (4) coined the terms ergoden or ergodische equating time average and phase averages all points x,then iterates under P of x is (which we translate as ergodic) from the but only reduce it to a possibly equally dif- distributed in some sense evenly through- Greek eργoν (energy) and oδoσ (path) or ficult problem of proving metric transitiv- out the space so that taking the average of energy path. He put forth what he called ity. For instance, how can one prove that afunctionf over these points gives a result the ergodic hypothesis, which postulated that a one-parameter flow is metrically transitive

Moore PNAS | February 17, 2015 | vol. 112 | no. 7 | 1909 Downloaded by guest on September 26, 2021 and indeed how do you know metrically series and use the invariance of A under Pt so T is a 3D compact manifold. transitive systems exist at all. At this point, let to see that all Fourier coefficients except for flow Pt on T flows a point ðx; vÞ along the us transfer to current terminology and simply the constant term vanish. Therefore, f is geodesic curve starting at x with tangent call metrically transitive transformations er- constant, and this establishes . vector v adistancet with Ptðx; vÞ = ðy; wÞ, godic, as von Neumann suggested. An important set of examples for the where y is the endpoint of the geodesic of As to the existence of ergodic transfor- subsequent development of ergodic theory length t and w is its tangent vector at y. mations, Oxtoby and Ulam (21) showed that is the shift transformations. Let F be a finite Hedlund (24) and Hopf (25) independently on a compact polyhedron M equipped with set of n elements and assign a probability established that such flows were ergodic a finite Lebesgue–Stieljes measure, the set measure to F; that is nonnegative num- and also later extended the result to higher of all ergodic measure preserving homeo- bers p1; ...; pn,whosesumis1.Then dimensional manifolds with negative cur- morphismsisadenseGδ, or a residual set, form the infinite product M of the mea- vature. A key part of the reasoning was among all measure preserving homeomor- sure space F with itself using the integers the fact that negative curvature makes phisms. Hence, not only do ergodic maps as the index set. Thus, M consists of dou- nearby diverge exponentially from exist, but almost all measure preserving bly infinite sequences of points of F, s = each other so that the flow has sensitive homeomorphisms are ergodic in a topologi- ½...; sðn − 1Þ; sð0Þ; sð1Þ; ...,withsðiÞ in dependence on initial conditions in that, if cal sense. However, there is a different an- F. The shift transformation is defined by u and v are very close together, PtðuÞ and swer for Hamitonian dynamical systems. P½sðnÞ = sðn + 1Þ. P has the effect of shift- PtðvÞ will diverge away from each other Here almost all systems are nonergodic, in ing a sequence s one place. P also preserves exponentially in t. This property is called the same sense as above (22). The space and the product measure on M,anditiseasily hyperbolicity, and it is known to be key in the topology are different in these two cases seen to be ergodic. Indeed any invariant many proofs showing that flows or single so there is no contradiction. set in the language of probability theory transformations are ergodic. It is also a von Neumann in his Annals of Mathemat- is statistically independent from the family property that a system of colliding gas ics paper (23) provides an intriguing and of of F that are defined by any molecules will have. If one perturbs very powerful answer to the existence problem. finite set of indices. Hence, it is a so-called slightly the position and momentum of He shows that any one-parameter flow or tail event, and by the Borel–Cantelli Lemma agasmolecule,theninitsnextcollision,it anysingletransformationonafinitemeasure of probability theory, it has probability zero will bounce off the other molecule at a quite space can be written as a possibly continuous or one. Hence, shifts are ergodic. different direction, a situation that is repeated sum of ergodic ones. To make this precise, let Specializing to the case n = 2and in further collisions, resulting in rapid diver- ðM; μÞ be a finite measure space and form p1 = p2 = 1=2, one sees that M is the proba- gence of its path from the path of the un- the product of M with the unit interval I with bility model of doubly infinite sequences of perturbed molecule. p Lebesgue measure I × M = M .Letp be the the results of tossing a fair coin. Letting There is a substantial and fascinating projection to the first factor I and denote the f ðsÞ = sð Þ = f ðsÞ = amount of papers in ergodic theory pur- p r 1if 0 1 (heads) and 0if part of M that lies over a point r in I by M . sð0Þ = 0(tails), the time average of f in the suing a program using hyperbolicity to Pr Then suppose that t is an on ergodic theorem is help prove that various approximations Mr. Then assuming some regularity in r,one to a system of a volume of gas molecules r can piece together these Pt into a measure XN confined in a container are ergodic. The p p ð + NÞ−1 f ½PnðsÞ; preserving flow Pt on M by setting 1 program has not succeeded as yet in proving p r 0 Pt ðr; mÞ = ½r; Pt ðmÞ. This flow is clearly that an actual model of a gas is ergodic, but nonergodic as it has many invariant sets, the results are getting close. The first paper namely J × M for any measurable subset J which is simply the proportion of heads was the path breaking paper of Sinai (26). N + of I, but it is displayed as a continuous in the first 1 coin tosses. The ergodic Sinai redefined the problem from one sum of ergodic ones, the Pr.vonNeumann’s theorem says that this converges for almost where the gas molecules are contained in t s f M theorem states that any general measure all to the integral of over , which is a cubical enclosure with reflecting walls to p = preserving flow has the structure like Pt 1 2. This is, of course, just the strong law a model where the move in a cube on Mp in which it is displayed as the (pos- of large numbers, a fundamental theorem with periodic boundary conditions. The sibly) continuous sum of its ergodic com- in probability theory, and, therefore from molecules collide with each other but not r r ponents Pt on M . the perspective of probability theory, the with the walls. This is not physically re- Irrational rotations on a provide ergodic theorem emerges as a far-reach- alistic,butitcanshedlightontheoriginal important examples of ergodic flows. We ing generalization of the strong law of problem by analogy. Much progress has represent points on a 2D torus T by pairs large numbers. been made on this model, especially by of complex numbers ðw; zÞ each of abso- Another set of examples of significance Simanyi and Szasz (27) and Simanyi (28), lute value one, and we consider the flow for ergodic theory as well as statistical where, using hyperbolocity as indicated defined by mechanics are geodesic flows, in particular above, a nearly complete resolution of geodesic flows on compact Riemannian ergodicity issues is established for such Ptðw; zÞ = ½expðitÞw; expðitbÞz; manifolds of negative curvature. First, con- Sinai flows [see also the survey article by sider the 2D case of a surface. Geodesic Szasz (29)]. where b is irrational. All orbits are dense so flow take place on the unit tangent bundle Going back to the more realistic case of this flow is quasi-ergodic, and it is also ergo- T of such a surface, which consists of pairs molecules in a container with reflecting dic. To see this, assume there is an invariant ðx; vÞ,wherex is a point of the surface M walls, one should begin in dimension two measurable set A and let f be the indicator and v is a unit tangent vector at x.ThenT with a single ball—a billiard flow in a pla- function of this set (1 on A and 0 on its consists of points of M with a circle above nar region or table. If the table is a rectan- complement). Expand f in a double Fourier each point consisting of unit tangent vectors, gle, the momenta of the ball over time can

1910 | www.pnas.org/cgi/doi/10.1073/pnas.1421798112 Moore Downloaded by guest on September 26, 2021 take on only a finite number of values so π. The authors point out an interesting cor- contained in a cube of any dimension at PERSPECTIVE the flow on a phase space of fixed energy ollary of this result, which is that a mechan- least two bouncing off each other and off cannot be ergodic. However, if one has ical system of two particles of masses m1 the hard walls is ergodic on any surface a billiard table of more complex , and m2 moving along a finite track without of constant energy. This appears to be a rig- the situation becomes more interesting. For friction, bouncing off each other elastically orous ergodicity result in a situation that instance, if the billiard table is a polygon, and bouncing off the fixed end points is comes closest to an actual gas. However, this then Kerckhoff et al. (30) show that for ergodic on a phase space of constant energy whole array of theorems, of which we have topologically almost all polygons, the bil- for topologically almost all values of the mentioned only a few, suggest that the hy- liard flow on a phase space of constant en- ratio m1=m2. This is sort of like a one- pothesis of ergodicity (or metric transitivity) ergy is ergodic. In particular, one has to dimensional gas. for a physical system like that of gas dynamics stay away from rational polygons where Finally, Simanyi (31) has established that mentioned at the outset of this essay is very all of the angles are rational multiples of a system consisting of two hard spheres plausible.

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