Explaining Thermodynamic-Like Behaviour In Terms of Epsilon-Ergodicity

Macroscopic systems such as gases reach equilibrium when left to themselves. Why do they behave in this way? The canonical answer to this question, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer has been criticised on different grounds and is now widely regarded as flawed. In this paper we argue that some of the main arguments against Boltzmann's answer, in particular, arguments based on the KAM-theorem and the Markus-Meyer theorem, are beside the point or inconclusive. We then argue that something close to Boltzmann's original proposal is true: systems approach equilibrium if they are epsilon ergodic, i.e. ergodic on the entire accessible phase space except for a small region of measure epsilon. This answer is promising because there are good reasons to believe that relevant systems in statistical mechanics are epsilon-ergodic.