Typicality and the Role of the Lebesgue Measure in Statistical Mechanics

Abstract

Consider a finite collection of marbles. The statement “half the marbles are white” is about counting and not about the probability of drawing a white marble from the collection. The question is whether non-probabilistic counting notions such as half, or vast majority can make sense, and preserve their meaning when extended to the realm of the continuum. In this paper we argue that the Lebesgue measure provides the proper non-probabilistic extension, which is in a sense uniquely forced, and is as natural as the extension of the concept of cardinal number to infinite sets by Cantor. To accomplish this a different way of constructing the Lebesgue measure is applied. One important example of a non-probabilistic counting concept is typicality, introduced into statistical physics to explain the approach to equilibrium. A typical property is shared by a vast majority of cases. Typicality is not probabilistic, at least in the sense that it is robust and not dependent on any precise assumptions about the probability distribution. A few dynamical assumptions together with the extended counting concepts do explain the approach to equilibrium. The explanation though is a weak one, and in itself allows for no specific predictions about the behavior of a system within a reasonably bounded time interval. It is also argued that typicality is too weak a concept and one should stick with the fully fledged Lebesgue measure. We show that typicality is not a logically closed concept. For example, knowing that two ideally infinite data sequences are typical does not guarantee that they make a typical pair of sequences whose correlation is well defined. Thus, to explain basic statistical regularities we need an independent concept of typical pair, which cannot be defined without going back to a construction of the Lebesgue measure on the set of pairs. To prevent this and other problems we should hold on to the Lebesgue measure itself as the basic construction.