The Einstein-Smoluchowski promeasure and the stochastic Boltzmann-Peierls equation

Abstract

The essential properties of a spatially homogeneous gas consisting of quasiparticles (phonons, magnons, rotons, etc.) are usually described in terms of the one-point distribution function f. This is the last in a series of papers (Physica A 180 (1992) 309, 336), the overall objective of which is the construction of the Einstein-Smoluchowski promeasure μ ɛ in order to formulate and develop a theory dealing with the equilibrium fluctuations of f. The basic idea was the inversion of Boltzman's relation connecting entropy and probability. As a consequence, the derivation of μ ɛ started by considering an infinite-dimensional analog of Einstein's thermodynamic theory of equilibrium fluctuations. Now, investigating a situation in which the time evolution of f is governed by the linear stochastic differential Boltzman-Peierls equation, the object here is to find the conditions under which the stationary measure for the aforesaid equation can be obtained from the Einstein-Smoluchowski promeasure μ ɛ . In agreement with such a general program, the fluctuation-dissipation theorem is analysed and some evidence of the necessity for an algebraic approach to the stochastic kinetic equations is also presented. And finally, the paper discusses the time-correlation function formalism and shows that this formalism is consistent with the universal ideas of Einstein. To summarize, the present paper demonstrates how the Einstein-Smoluchowski promeasure μ ɛ can be used to study some problems in the extended context of the time-dependent theory of equilibrium fluctuations.