Time's Arrow and Lanford’s Theorem

Abstract

It has been a longstanding problem to show how the irreversible behaviour of macroscopic systems can be reconciled with the time-reversal invariance of these same systems when considered from a microscopic point of view. A result by Lanford (1975, 1976, 1981) shows that, under certain conditions, the famous Boltzmann equa- tion, describing the irreversible behaviour of a dilute gas, can be obtained from the time-reversal invariant Hamiltonian equations of motion for the hard spheres model. Here, we examine how and in what sense Lanford’s theorem succeeds in deriving this remarkable result. Many authors have expressed different views on the question which of the ingredients in Lanford’s theorem is responsible for the emergence of irreversibil- ity. We claim that the true culprit for the emergence of irreversibility lies in a point that has hitherto not been sufficiently emphasized, i.e. in the choice of incoming, rather than outgoing, configurations for collision points. We argue that this choice ought to be recognized clearly as an explicit assumption in the theorem, and discuss its implications for the question in what sense irreversible behaviour follows from Lanford’s theorem.