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 ei,i=1,2,…,I, is described by the system of the nonlinear rate equations dpi(t)/dt=− ln pi(t)+a(t)+eib(t)+pi(t0)(i=1,2,…,I) with two constraints ∑ipi(t)=1 and ∑ieipi(t)=E, where pi(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 pi(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 pi occur and the behavior of the remainder ones is monotonic, or all pi 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.