A Hilbert space derivation of the classical generalized master equation

Abstract

The Liouville equation of classical mechanics in angle-action variables has been formulated in a Hilbert space of L-2 functions. If angle-action variables for the total system do not exist, but if the Hamiltonian can be written as H(w, J)=H 0(J)+ΔH(w, J), in angle-action variables ( w, J) for the unperturbed system with Hamiltonian H 0( J), an exact solution of the perturbed Liouville equation is obtained by treating the perturbation as the inhomogeneous term of an inhomogeneous Liouville equation for the unperturbed system. The Hilbert space formalism is used to derive a generalized master equation for the reduced distribution function of the action variables, following the method of Zwanzig without perturbation theory. In the weak coupling case for homogeneous systems the master equation is the same as that of Brout and Prigogine. In this limit the inhomogeneity is determined by ρ in′( t, w, J ), the asymptotic solution of the Liouville equation at t=−∞. It is shown that the rate of change of entropy consists of two terms, one depending upon ρ in′( t, w, J ) and vanishing when ρ in′( t, w, J ) is independent of w , and another non-negative term giving the rate of dissipation. These results apply equally to finite systems or to infinite systems, that is, whether the spectrum of the Liouville operator is discrete or continuous. There is no restriction in the Hamiltonian to two-body interactions.