Analytic solution of the master equation using the exponential transition probability has been obtained previously in part III lsqb;J. Chem. Phys.80, 2504 (1984)rsqb; in the form of an infinite series eigenfunction expansion forc(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 lsqb;i.e., averages overc(x,t)rsqb; 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 thenth collision is in fact the initial energy distribution for the next lsqb;(n+1)thrsqb; 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 onwardmdash;but not all the way to equilibriummdash;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 (bulkhyphen;) average energy lang;lang;yrang;rang; decays linearly with time, and as a result the average energy transferred per collision is an energyhyphen;independent constant. The relaxation time decreases linearly with time and after only a few collisionsc(x,t) becomes a Gaussian. The limitations of this approach are noted and discussed.
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