Ergodicity, Irreversibility and Approach to Equilibrium

Abstract

The density distribution function or simply distribution function in phase space is considered. It can be shown that if the distribution function tends to a constant (uniform distribution) after a long time, an aged system is attained, where ergodicity holds. The problem lies in the interpretation how the distribution function has a limit and tends to a constant. Ergodicity on tori can be derived, provided that we take the limit as the long-time average of the distribution function. In non-integrable hamiltonian systems the distribution function can be assumed to tend to a uniform distribution in the meaning of weak convergence. The aged system thus attained has the properties of ergodicity, mixing and increase of entropy representing irreversibility. The almost same arguments can be shown to hold for quantal systems as well, where density matrix, Wigner and Husimi functions play the main role. It is suggested that ergodicity and irreversibility in quantal systems are observed in the limit \(\hbar \rightarrow 0\).