Consider a gas confined to the left half of a container. Then remove the
wall separating the two parts. The gas will start spreading and soon be evenly
distributed over the entire available space. The gas has approached equilibrium.
Why does the gas behave in this way? The canonical answer to this question,
originally proffered by Boltzmann, is that the system has to be ergodic for the
approach to equilibrium to take place. This answer has been criticised on different
grounds and is now widely regarded as flawed. In this paper we argue that these
criticisms have dismissed Boltzmann’s answer too quickly and that something
almost like Boltzmann’s answer is true: the approach to equilibrium takes place if
the system is epsilon-ergodic, i.e. ergodic on the entire accessible phase space
except for a small region of measure epsilon. We introduce epsilon-ergodicity and
argue that relevant systems in statistical mechanics are indeed espsilon-ergodic.