Justifying typicality measures of Boltzmannian statistical mechanics and dynamical systems

Abstract

A popular view in contemporary Boltzmannian statistical mechanics is to interpret the measures as typicality measures. In measuretheoretic dynamical systems theory measures can similarly be interpreted as typicality measures. However, a justification why these measures are a good choice of typicality measures is missing, and the paper attempts to fill this gap. The paper first argues that Pitowsky's (2012) justification of typicality measures does not fit the bill. Then a first proposal of how to justify typicality measures is presented. The main premises are that typicality measures are invariant and are related to the initial probability distribution of interest (which are translationcontinuous or translation-close). The conclusion are two theorems which show that the standard measures of statistical mechanics and dynamical systems are typicality measures. There may be other typicality measures, but they agree about judgements of typicality. Finally, it is proven that if systems are ergodic or epsilon-ergodic, there are uniqueness results about typicality measures.

When do Gibbsian phase averages and Boltzmannian equilibrium values agree?

“…Presentations of the conception of equilibrium in Boltzmannian statistical mechanics (BSM) often begin with what is now known as the combinatorial argument, and then present the result of these combinatorial considerations as a definition of equilibrium. However, it is now recognised that combinatorial considerations do not provide a general definition of equilibrium (see Uffink, 2007, andWerndl andFrigg, 2015a, for discussions). We therefore work with the time-average conception of equilibrium, which has recently been proposed by Werndl and Frigg (2015a, 2015b, 2017b.…”

Foundation of statistical mechanics: Mechanics by itself

“…The conceivability that the chosen measure could turn out to be not the Lebesgue measure, but some other measure, emphasizes an important feature of our Demonic procedure: In that procedure, we have described the probabilities of magnitudes in our actual universe, not proven them from first mechanical principles. Attempts at such proofs sometimes turn to arguing that the Lebesgue (or “standard”) measure is “natural.” Some of the arguments for the “naturalness” of the Lebesgue measure are based on its roles in theorems of mechanics (see, e.g., Werndl, ). The Lebesgue measure is conserved by the dynamics, according to Liouville's theorem, and is uniquely conserved if the dynamics is ergodic (in the sense of the Von Neumann–Birkhoff ergodicity theorem).…”

Foundation of statistical mechanics: The auxiliary hypotheses

“…Despite the aesthetic appeal of such a move, its justification is unclear. Another option might be that the Lebesgue measure has certain mechanical properties that make it “natural” (see e.g., Werndl, ). The Lebesgue measure is conserved by the dynamics, according to Liouville's theorem, and is uniquely conserved if the dynamics is ergodic (in the sense of the Von Neumann–Birkhoff ergodicity theorem).…”