Convergence to Equilibrium for Many Particle Systems

Abstract

The goal of this paper is to give a short review of recent results of the authors concerning classical Hamiltonian many particle systems. We hope that these results support the new possible formulation of Boltzmann's ergodicity hypothesis which sounds as follows. For almost all potentials, the minimal contact with external world, through only one particle of N , is sufficient for ergodicity. But only if this contact has no memory. Also new results for quantum case are presented. Thermodynamic limit 21Part I 3. note that there are N N deterministic maps, among them N ! one-to-one maps, and among the latter only (N − 1)! maps with unique cycle. Thus, Cesaro ergodicity is also a rare event but not so rare as strong ergodicity.If the set X is not finite, for example a smooth manifold, the situation becomes enormously more complicated. Ludwig Boltzmann did not give exact mathematical formulations. Later on, various formulations of the problem appeared. For some history of ergodicity theory we refer to [1] and references therein. What could be the ways to avoid extreme complexity ? First of all, one must find wider and possibly alternative exact formulations of the problem.2 Finite quantum dynamics with random time switchingHere we consider the situation of the section 0.3.2, and assume both groups U t i to be unitary evolutions in C N . Examples could be quantum walks on finite lattices.