On the General Theory of the Approach to Equilibrium. II. Interacting Particles

Abstract

The general method described in a recent paper by Prigogine and Henin [J. Math. Phys. 1, 349 (1960), hereafter referred to as I] is applied to a system of interacting particles. A full use is made of the diagram technique due to Prigogine and Balescu. The distribution function is Fourier analyzed and each Fourier coefficient is decomposed into two parts: one (ρ′) whose evolution results from scattering processes and which obeys a diagonal differential equation; the second one (ρ″), whose evolution is due to direct mechanical interactions which build the correlation described by the Fourier coefficient. The ρ″ can be expressed in terms of functions ρ′ corresponding to lower correlations. We study first the velocity distribution function. Only scattering processes contribute to the evolution of this function. The equations obtained ensure evolution of this function to the correct equilibrium value at any order in the concentration C and the coupling constant λ. We then study the asymptotic behavior of the Fourier coefficients which describe correlations among the particles. The part ρ′ of these coefficients corresponding to scattering processes vanishes for large time. In other words, scattering processes play a fundamental role in the establishment of the correct velocity distribution function, but once this is achieved they become ineffective. The correct equilibrium correlations, which in equilibrium theory are described in terms of cluster diagrams, are built by direct mechanical interactions among the particles involved in the correlation. We give a detailed proof up to order C². The extension to higher orders does not introduce any new feature.