Long-time behaviour of macroscopic quantum systems (Vol. 41, No. 6)

The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann's 1929 article on the quantum ergodic theorem (QET). We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, to be much misunderstood and very relevant to the recent discussion on the general and abstract reasons why, and the exact sense in which, an isolated macroscopic quantum system will approach thermal equilibrium from (more or less) any initial state. In his paper, von Neumann studied the long-time behaviour of macroscopic quantum systems. His main result, the QET, expresses so-called "normal typicality": for a typical finite family of commuting macroscopic observables, every initial wave function ψ(0) from a micro-canonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ψ(t) is close to their micro-canonical distribution.

In our commentary, we provide a gentle introduction to the QET and discuss its relevance to the approach to thermal equilibrium. There is, in fact, no consensus about the definition of thermal equilibrium for a quantum (or even a classical) system in microscopic terms; the main divide in the literature lies between the "ensemblists" who regard thermal equilibrium as a property of an ensemble (or a mixed state) and the "individualists" who regard thermal equilibrium as a property of an individual system (in a pure state). As we explain, von Neumann's concept of equilibrium is influenced by both views but mainly based on the individualist view, a view that has gained ground recently.