Why Typicality Does Not Explain the Approach to Equilibrium

A gas that is confined to the left half of a container uniformly spreads over the entire available space as soon as the confining wall is removed. Yet we never observe the reverse process of a uniformly distributed gas suddenly concentrating in the left half of the container. Such irreversible behaviour is characteristic of many processes and is enshrined in the so-called Second Law of thermodynamics, which, roughly, states that entropy cannot decrease in isolated systems. Statistical mechanics (SM) aims to explain irreversible behaviour in terms of the dynamical laws governing the individual molecules of which the gas is made up. What is it about molecules and their motions that leads them to spread out when the wall is removed? And crucially, what accounts for the fact that the reverse process never happens? An important answer to these questions was suggested by Boltzmann (1877), and variants of it are currently regarded by many as the most promising option among the innumerable of approaches to statistical mechanics. An important contemporary version of the Boltzmannian approach, originating in the work of Joel Lebowitz (1993a, b), differs from traditional approaches in that it explains irreversibility in terms of the notion of ‘typicality’. Intuitively, something is typical if it happens in the ‘vast majority’ of cases: typical lottery tickets are blanks, typical olympic athletes are well trained, and in a typical series of a thousand coin tosses the ratio of the number of heads and the number of tails is approximately one. The leading idea of a typicality-based approach to SM is to show that thermodynamic behaviour is typical; that is, that the entropy in a system typically increases. This approach has grown increasingly popular in recent years and has been advocated by a number of authors (references will be given below). The problem with understanding this approach is that it comes in different versions, which are, however, not recognised as such, much less clearly distinguished. We often find different