Progress of Theoretical Physics Supplement No. 116, 1994 Ergodicity and Time-Scales for Statistical Equilibrium in Classical Dynamical Systems Giancarlo BENETTIN Dipartimento di Matematica Pura e Applicata, Universita di Padova Via G. Belzoni 7, 35121 Padova, Italy Part I. The ergodic problem, and FPU -like results § 1. Introduction 207 Ergodic theory is, nowadays, a highly developed branch of mathematics, in fact a relevant part of dynamical systems theory. Its physical motivation, as is well known, is the so-called ergodic problem, namely Boltzmann's problem of the dynamical foundation of classical statistical mechanics. The aim of this lecture is to reconsider the physical ergodic problem, and suitably revisit it in the light of some recent results from the theory of dynamical systems. We shall discuss in particular a rather crucial question, namely the time scale one needs in order a dynamical system possibly exhibit an ergodic-like behavior. This point of view is not so common: ordinary macroscopic times (those, say, of the usual experiments) are so large, with respect to the times of microscopic dynamics, that they are usually considered as practically infinite (exactly in the same way a length of, say, one centimeter is commonly treated, in statistical mechanics, as infinite). Nowadays, however, it is rather well established that weakly nonlinear Hamiltonian systems typically evolve only on very long time scales, say e 11 ' (or e 11 'a, O< a< 1), where c is the coupling, or the nonlinearity, in the system; in the same way, for systems containing high frequency oscillators one finds time scales growing exponentially with the frequency, T erw. It is then conceivable that classical dynamical systems, including many particle systems which are relevant for statistical mechanics, although possibly ergodic on a really infinite time interval, nevertheless exhibit a definitely non-ergodic behavior for an extremely long time scale, possibly of the same order, or even larger, than the experimental times. As is not much known, this point of view is not at all new, but was clearly expressed almost one century ago by Boltzmann 1 > (1895) and Jeans 2 >,s> (1903, 1905), as a possible way to explain dynamically, as a non-equilibrium phenomenon, the so- called "freezing" of energy in high frequency degrees of freedom, which later on was recognized as a quantum effect. In spite of the success of the quantum interpretation, it is worthwhile to reconsider these old ideas from a modern point of view. This can be helpful, in particular, for a deeper understanding of the relation between classical and quantum behavior of statistical systems. Downloaded from https://academic.oup.com/ptps/article-abstract/doi/10.1143/PTP.116.207/1938672 by guest on 29 July 2018