The Einstein-Smoluchowski promeasure for (quasiparticle) gases

Abstract

Considering an ideal Bose-Einstein gas composed of quasiparticles (phonons, magnons, rotons, etc.), the object of this paper is the development of a tractable generalization of Einstein's theory of equilibrium fluctuations. It is assumed that the state of the gaseous system may be characterized by the distribution function f(k) which determines the number of quasiparticles with quasimomentum ħk. It allows one to embark on a program in which the inversion of Boltzmann's relation connecting entropy and probability can be used as an organizing principle for the development of the theory dealing with the equilibrium fluctuations of f(k). In the present approach, Einstein's distribution law in the Gaussian approximation is identified with the equilibrium promeasure μϵ defined on the suitably chosen, infinite-dimensional Hilbert space H. This promeasure gives the possibility of calculating the fluctuations in the value of f(k) and its moments. Although μϵ cannot be extended to a measure on H, what one discovers is that, in this case, one obtains a manifestly infinite-dimensional analog of Einstein's theory of equilibrium fluctuations. The generalization of the method to the case of classical and relativistic gases is straightforward.