Statistical Mechanics on Stochastic Phase Spaces

Abstract

Classical statistical mechanics, as well as quantum statistical mechanics, are both acknowledgedly statistical theories even if one subscribes to the conventional point of view about the intrinsically disparate natures of classical and quantum mechanics, respectively. Yet, the conventional mathematical frameworks for these two theories are remarkably dissimilar even at the most fundamental level of description of states of ensembles: in the classical case states are described by distribution functions ƒ(q, p) on phase space, whereas in the quantum case they are given by density operators ρ.

“I maintain that the mathematical concept of a point in a continuum has no direct physical significance…. This attitude is generally accepted in quantum mechanics. But it has actually a more fundamental significance and is only indirectly connected to the special features characteristic of quantum mechanics. It ought to be applied to classical mechanics as well.… It is misleading to compare quantum mechanics with deterministically formulated classical mechanics; instead, one should first reformulate the classical theory, even for a single particle, in an indeterministic, statistical manner. Then some of the distinctions between the two theories disappear, others emerge with great clarity. Amongst the first is the feature of quantum mechanics, that each measurement interrupts the automatic flow of events and introduces new initial conditions (so-called ‘reduction of probability’); this is true just as well for a statistically formulated classical theory.” Born (1955C, pp. 3–4, 25–26)