Ensemble Method in the Theory of Irreversibility

TLDR

The paper proposes a new formulation of Van Hove and Prigogine’s irreversibility methods to simplify understanding and application. The authors employ projection operators in the Hilbert space of Gibbsian ensemble densities to separate relevant and irrelevant parts, avoid diagram summation, and illustrate the approach by deriving the Prigogine‑Brout master equation for a weakly interacting classical system. They show that the relevant part obeys a kinetic equation generalizing Van Hove’s master equation to arbitrary order.

Abstract

We describe a new formulation of methods introduced in the theory of irreversibility by Van Hove and Prigogine, with the purpose of making their ideas easier to understand and to apply. The main tool in this reformulation is the use of projection operators in the Hilbert space of Gibbsian ensemble densities. Projection operators are used to separate an ensemble density into a ``relevant'' part, needed for the calculation of mean values of specified observables, and the remaining ``irrelevant'' part. The relevant part is shown to satisfy a kinetic equation which is a generalization of Van Hove's ``master equation to general order.''' Diagram summation methods are not used. The formalism is illustrated by a new derivation of the Prigogine-Brout master equation for a classical weakly interacting system.