Analytic solution of relaxation in a system with exponential transition probabilities

Abstract

In a thermal system, the simple exponential (SE) model of transition probabilities provides a fairly simple analytical expression for the nonequilibrium rate constant (ko) for thermal dissociation of polyatomic gas. However, the SE model is poorly normalized, and because of some mathematical shortcuts it yields no relaxation times. The present work investigates two properly normalized versions of the exponential model, called Model A and Model B, and gives the analytic solution of the relaxation problem. In the case of Model A, which is the most realistic physically but difficult mathematically, the relaxation problem could be solved only for an infinite reaction threshold energy, yielding ko=0 together with a partly discrete and partly continuous spectrum of relaxation times. However, the mean first-passage time approach yields a nontrivial expression for ko that is valid under most conditions likely to be encountered in practice. Model B is physically less realistic but mathematically simpler, and yields ko essentially equivalent to that of the SE model, but also provides some indication of how the relaxation times depend on the threshold energy. It is shown that the SE model yields a reliable ko only when energy transfer is inefficient and the threshold to reaction is high.