A nonlinear model for relaxation in excited closed physical systems

The relaxation process of a perturbed isolated physical system consisting of entities, that occupy the energy levels e(i), i = 1, 2,..., I, is described by the system of the nonlinear rate equations dp(i)(t)/dt = - ln p(i)(t) + a(t) + e(i)b(t) + p(i) (t(o)) (i = 1, 2,..., I) with two constraints Sigma(i) p(i) (t) = 1 and Sigma(i) e(i) p(i) (t) = E, where p(i) (t) is the time dependent probability distribution and E the mean energy. Those equations are derived and heurestically justified by Englman in Appendix A. The behavior of the probabilities p(i) (t) during the course of the time-evolution process was investigated. Our numerical results brought out the approach to the Boltzmann distribution in equilibrium. We found that the probabilities during the course of the relaxation behave in the following manner: either in the first onset after a perturbation, local extrema of some p(i) occur and the behavior of the remainder ones is monotonic, or all p(i) are monotonic functions. Of special interest is the power law dependence of extrema time with the number of energy levels. The entropy behaves in good agreement with the entropy principle. Numerical results illustrate the model. (C) 2002 Elsevier Science B.V. All rights reserved.