Analytic solution of relaxation in a system with exponential transition probabilities. IV. Decay at early times

Abstract

Analytic solution of the master equation using the exponential transition probability has been obtained previously in part III [J. Chem. Phys. 80, 2504 (1984)] in the form of an infinite series eigenfunction expansion for c(x,t), the population distribution. While the number of terms that effectively contribute to the sum is only one at equilibrium, it increases to infinity at time zero. Thus such eigenfunction expansion is not useful for describing the bulk properties [i.e., averages over c(x,t)] of the relaxing system at early times. It is nevertheless possible to solve the relaxation problem at early times by noting that the final (postcollision) energy distribution resulting from the nth collision is in fact the initial energy distribution for the next [(n+1)th] collision. It is shown that in this way simple analytical expressions can be obtained for various bulk properties of the relaxing system from the first collision onward—but not all the way to equilibrium—if the initial (at time zero) energy distribution is a delta function. It turns out that for the first several hundred collisions or so the (bulk-) average energy 〈〈y〉〉 decays linearly with time, and as a result the average energy transferred per collision is an energy-independent constant. The relaxation time decreases linearly with time and after only a few collisions c(x,t) becomes a Gaussian. The limitations of this approach are noted and discussed.