Revising Statistical Mechanics: Probability, Typicality and Closure Time

Abstract

Standard statistical mechanics routinely assumes that the probable behavior of a system is determined by the phase-space volume of its present macrostate. In this context, typicality is merely the latest language in which one expresses this presumed relation between phase-space measure and probability. I argue that such a connection cannot hold in general, as we cannot, for example, reconstruct without further information the history of a system found to be in equilibrium now. Even if the system is not in equilibrium, we cannot in general know its history, unless it has been closed for an extremely long time, in which case, its present state most likely arose from equilibrium. As a consequence, there is no way in general to relate probabilities to phase-space measure. The standard exposition of statistical mechanics cannot be expected to adequately cover non-equilibrium behavior, therefore. I show that the past hypothesis requires an incredible degree of fine-tuning to explain this behavior, one that is as hard to explain as the observed behavior itself. Finally, an analysis of the diffusion equation suggests that the problem is independent of microscopic time-reversibility, and lies instead with the loss of microscopic information entailed in the very definition of macrostates.