Kinetic equations for classical and quantum Brownian particles and eigenfunction expansions as generalized functions

Abstract

We study momentum relaxation processes of a classical and a quantum Brownian particle by considering the eigenvalue problem of the collision operators in the kinetic equations. The collision operators are anti-Hermitian with an appropriate inner product defined by an integral with a weight factor given by the inverse of the equilibrium distribution function. Owing to the weight factor, the norm of a momentum distribution function is infinite, if the distribution is characterized by a temperature higher than a threshold temperature determined by the environmental temperature. Although the eigenfunction expansion of a given distribution function with an infinite norm does not converge to a function in the Hilbert space, it has a legitimate meaning as a generalized function and defines a linear functional. We introduce an H-function through the norm which directly reflect the spectral properties of the collision operators. When the norm of the momentum distribution function diverges, the H-function reduces to a simple form characterized by only the value at the accumulation point of the spectrum of the collision operator.