S-operator (matrix) formalism for the Liouville and von Neumann equations and the approach to equilibrium

Abstract

A formalism is obtained for classical systems which is the counterpart of the S-operator formalism of quantum theory. It is an extension of the formalism of N-particle kinetic equations (generalized master equations) in statistical mechanics, and depends on the application of orthogonal projection operators to the Liouville equation in interaction picture. It is applied to systems of particles with finite-range, central-force interactions in the absence of external forces. A pair of coupled integral equations for the projected parts of the density operator is obtained, which can be iterated to produce series expansions in which propagation alternates between complementary parts of a dyadic Hilbert space. In first order the N-particle kinetic equation is recovered. A Møller wave operator and an S-operator are defined, in which the Liouville operator replaces the hamiltonian of quantum scattering theory. Lippmann-Schwinger equations and Born iterative series are also obtained for the eigenfunctions and propagators of the Liouville operator. The alternation of propagation in these series is analogous to that of the Faddeev equations. The approach to equilibrium is described in Schrödinger picture. The density operator is written as the sum of a time-independent equilibrium component of unit norm, and a time-dependent component whose norm is zero at all times; conservation of probability is maintained by the equilibrium component. Correspondingly, the expectation of an observable is the sum of its equilibrium expectation value and a time-dependent term which approaches zero as time becomes infinite. The initial ensemble can be regarded as regressing, after infinite time, to the same equilibrium state as the one, at an infinitely earlier time, from which it evolved. The formalism is extended to the von Neumann equation for quantum systems in the dyadic (super-operator) space.