Study of the collision operator, ψ(z), for some exactly soluble models

Abstract

The collision operator, ψ( z), introduced in the study of non-equilibrium statistical mechanics by Prigogine, Balescu, and Résibois is calculated exactly for some simple quantum-mechanical models. The calculations are done first for the discrete-energy spectrum case. Then, the behavior of the solutions is studies in the limit: l (extension of the system) → ∞, where continua appear in the spectra. It is found that the exact collision operators ψ( z, l) posses the property that lim z →0 ψ ( z, l) is either zero or underfined for finite l, while lim z →10+ lim l →∞ ψ ( z, l) is a nonzero operator.