Exactly Solvable Nonlinear Relaxation Processes. Systems of Coupled Harmonic Oscillators

Abstract

A class of nonlinear relaxation processes is discussed which involves the interaction of two finite systems characterized by special forms of the transition probabilities. For these particular sets of transition probabilities it is possible to reduce the initial set of coupled nonlinear kinetic equations to a set of linear equations with time‐dependent coefficients which are amenable to exact analytical solutions. This reduction is effected through the use of summational invariants expressed in terms of the appropriate combinations of the moments of the distribution functions of the two systems. The vibrational—vibrational relaxation of two interacting systems of harmonic oscillators A and B with identical frequencies has been worked out as a specific example. For the type of relaxation processes discussed here it is found that the relaxation of System A is independent of the form of the initial distribution of System B (and thus of its time history) and vice versa.