On the General Theory of the Approach to Equilibrium. I. Interacting Normal Modes

Publisher Website
Google Scholar
Abstract
A general method which permits the derivation of the equations which describe the approach to equilibrium correct to an arbitrary finite order in the coupling constant is presented. This method is applied in the present paper to normal modes interacting through three-phonon processes. In a subsequent paper the method will be applied to interacting particles. The distribution function is first Fourier-analyzed with respect to the angle variables. All Fourier components, except the distribution function of action variables, describe correlations among the normal modes. The formal solution for the Fourier components is studied in the limiting case of a very large number of degrees of freedom N → ∞, and for large times by means of a diagram technique. Each component ρ3n can be split into 2 parts: ρ3n′ and ρ3n″; one (ρ′) due to ``scattering'' of the normal modes satisfies diagonal differential equations. The other (ρ3n″) contains the direct interaction between the normal modes involved in the corresponding correlation. It is completely determined by the functions ρ3n′. The study of this set of equations enables us to study the approach to equilibrium.