Probabilistic issues in statistical mechanics

Abstract

Beyond the syntax of statistical mechanics, the focus here is on those of its semantic aspects that depend on the foundations of probability theory. The first problem is to confront the practitioners’ neglect of the logical precautions—recursive function theory—required by a strict adherence to the limiting relative frequencies approach of von Mises. The second problem is to reconcile de Finetti's professed finitism—and Boltzmann's—with de Finetti's so-called subjective approach and the implementation of the conditions under which the celebrated de Finetti exchangeability theorem makes pragmatic sense. The third problem—or collection of problems—to be met is the passage from Clausius’ thermodynamical entropy to Shannon's information, or the justification of the use of maximum principles to derive canonical equilibrium distributions. Each of these three problems has classical and quantum versions; depending on the semantic options one takes, the extent to which these two realms differ may appear to be fundamental, merely convenient, or ultimately specious.