Adiabatic Invariants and Time Scales for Energy Sharing in Models of Classical Gases

Abstract

The purpose of this paper is twofold. On one hand I would like to illustrate an application of the theory of adiabatic invariants — essentially, Nekhoroshev-like exponential laws — to classical statistical mechanics. On the other hand, I would like to revisit some ideas of distinguished physicists, like Jeans and Landau, who long time ago, much before modern perturbation theory, introduced exponential laws in connection with problems of adiabatic invariance (Jeans,^1,2 1903 and 1905; Landau and Teller,³ 1936). These authors worked only heuristically, as physicists often do, but their ideas are definitely deep, and as we shall see, at least in some cases their heuristic arguments can be turned into effective proofs.