Simplified Derivation of the Non-Equilibrium Probability Distribution

Abstract

A simple and transparent derivation of the formally exact probability distribution for classical non-equilibrium systems is given. The corresponding stochastic, dissipative equations of motion are also derived.

Quantum Statistical Mechanics. IV. Non-Equilibrium Probability Operator and Stochastic, Dissipative Schrodinger Equation

“…the discussion in the conclusion of Ref. [4]). Finally, the point entropy derived above is based upon the average trajectory (more precisely, the average of the square of the stochastic propagator), and its gradient gives the difference in this average value.…”

“…(20) of Ref. [4]). This is only constant as far as the excited sub-space is concerned, since |ψ(t) = P0 (t) |ψ .…”

“…The Green-Kubo relations, [42][43][44] which express the transport coefficients as an integral of certain equilibrium time correlation functions, come directly from this dynamic part. 1,3,4…”

“…This paper develops the formal theory of nonequilibrium quantum statistical mechanics in terms analogous to the author's theory for the classical nonequilibrium case, [1][2][3][4] which was itself built on an earlier approach to classical equilibrium theory. 5,6 The present work uses recent analysis of equilibrium quantum statistical mechanics, [7][8][9] which in general this reproduces conventional text book results, [10][11][12] but with the theory formulated in terms of the wave function and wave space.…”

“…The present quantum derivation follows the recently simplified classical version. 4 A second contribution of the present paper to nonequilibrium quantum statistical mechanics is the derivation of the stochastic, dissipative Schrödinger equation for open non-equilibrium systems. It turns out that this derivation involves non-linear quantum operators.…”