Ergodicity, ensembles, irreversibility in Boltzmann and beyond

Abstract

Abstract:the implications of the original misunderstanding of the etymology of the word "ergodic" are discussed, and the contents of a not too well known paper by Boltzmann are critically examined. The connection with the modern theory of Ruelle is attempted. §1The etymology of the word "ergodic" and the heat theorems.Trying to find the meaning of the word "ergodic" one is led to a 1884 paper by Boltzmann, [B84].1 This paper by Boltzmann is seldom quoted 2 and no english translation is available yet. But I think that this is one of the most interesting papers of Boltzmann: it is a precursor of the work of Gibbs, [G], on the ensembles, containing it almost entirely (if one recalls that the equivalence of the canonical and microcanonical ensembles was already established (elsewhere) by Boltzmann himself, at least in the free case [B66],[B68]), and I will try to motivate such statement. The paper stems from the fundamental, not too well known, work of Helmholtz, [He1], [He2], who noted that monocyclic systems 3 could be used to provide models of thermodynamics in a sense that Boltzmann undertakes to extend to a major generalization. After an introduction, whose relative obscurity has been probably responsible for the little attention this paper has received, Boltzmann introduces the notion of "stationary" probability distribution on the phase space of N interacting particles enclosed in a vessel with volume V . He calls a family E of such probabilities a monode, generalizing an "analogous" concept on monocyclic systems. 4In fact the orbits of a monocyclic system can be regarded as endowed with a probability distribution giving an arc length a probability proportional to the time spent on it by the motion: hence their family forms a family of stationary probability distributions. Etymologically this undoubtedly 5 means a family of stationary distributions with a "unique nature", (each ! expanded and revised version of a conference read at the celebration of the 150 th -anniversary of the birth of Boltzmann, Vienna, 24 february, 1994. * Dipartimento di Fisica, Università di Roma, P.le Moro 2, 00185 Roma. E-mail 40221::gallavotti. This paper is archived in mp arc@math.utexas.edu, #94-66 updated copies (in Postscript) can also be obtained by sending request to the author, by E-mail. 1 see the footnote of S. Brush in his edition, [Bo2], of the Lectures on Gas Theory, on p. 297 ( §32): here the Boltzmann's paper is quoted as the first place where the word is introduced, although the etymology is taken from the Erhenfests' paper, which is incorrect on this point: see [EE], note #93, p.89, (where also the first appearance of the word is incorrectly dated and quoted). 3 this is what we call today a system whose phase space contains only periodic orbits, or cycles: i.e. essentially a one dimensional conservative system. 4 in fact Boltzmann first calls a monode just a single stationary distribution regarded as an ensemble. But sometimes later he implicitly, or explicitly, thinks of a monode as a collection of stationary distr...

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