Explaining Thermodynamic-Like Behaviour In Terms of Epsilon-Ergodicity

Abstract

Why do gases reach equilibrium when left to themselves? The canonical answer, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer is now widely regarded as flawed. We argue that some of the main objections, in particular, arguments based on the KAM-theorem and the Markus-Meyer theorem, are beside the point. We then argue that something close to Boltzmann's proposal is true: gases behave thermodynamic-like if they are epsilon-ergodic, i.e., ergodic on the phase space except for a small region of measure epsilon. This answer is promising because there is evidence that relevant systems are epsilon-ergodic.