Analytic solution of relaxation in a system with exponential transition probabilities. III. Macroscopic disequilibrium

Abstract

The exponential transition probability, in the version that permits an analytical solution of the relaxation problem, is used to compute a number of macroscopic (bulk) observables for a model system based on multiphoton excitation of SF6 coupled to a rare-gas heat bath. Two extreme cases are considered: Initial excitation as a delta function, or as a Poisson distribution. It turns out that regardless of initial conditions, all macroscopic observables are functions of time, including the relaxation time, so that the system does not undergo simple exponential decay. This is because the first moment of the exponential transition probability does not satisfy the linear ‘‘sum rule.’’ The exponential transition probability causes the overall (or bulk) average of energy transferred (〈〈ΔE〉〉) to be constrained to a maximum which is independent of the nature and level of initial excitation, thus producing a bottleneck in the macroscopic relaxation process when excitation is sufficiently high. The consequence is that the initially more highly excited system takes longer to reach steady state, with a relaxation time that is initially nearly proportional to initial excitation and which decreases as the system approaches steady state. It is only in the immediate vicinity of steady state that simple exponential relaxation takes place, with the shortest relaxation time. Several consequences of this, particularly the population distribution as a function of time, are illustrated and discussed.