The H-theorem and reversibility

Exactly integrable analogue of a one-dimensional gravitating system

“…Evidence of the approach to thermal equilibrium has only been recently established for a two component system [6,7]. For an extensive review of the history of the relevant investigations see [1,7]. In this paper we offer a possible explanation for the incredibly slow exploration of phase space these systems exhibit.…”

“…One-dimensional models of gases have been studied extensively in statistical mechanics and one-dimensional lattice models have been used in solid state physics to study metal alloys, magnetic spin systems, glasses, phonon propagation, electronic bands and a host of other systems. One dimensional gravitational systems have been used to investigate relaxation to equilibrium, phase transitions, the gravothermal catastrophe, and cosmology [1]. An important example is the system consisting of a one-dimensional chain of coupled, nonlinear, oscillators.…”

“…where Θ(x)is the usual step function [1,7]. This system has been studied in the literature extensively and is known to have an infinite number of stationary solutions [1,7,9]. In particular, the maximum entropy solution is f ∼ exp(−βv 2 /2) cosh −2 (2πM 2 Gx/2E).…”