Relaxation of a Gas of Harmonic Oscillators

Abstract

The temporal evolution of the vibrational distribution function for a gas of harmonic oscillators undergoing binary collisions, in which they can exchange vibrational quanta among themselves as well as transfer energy between the vibrational and translational degrees of freedom, is determined exactly. The solution, given in terms of a generating function, involves only a double integral and is valid for transition probabilities due to an interaction potential linear in the oscillator coordinate. The relaxation toward ``local equilibrium'' of the vibrational distribution is found to be at least twice as fast as the relaxation of the average vibrational energy to the final equilibrium value. The solution is valid for an arbitrary initial vibrational distribution and for arbitrary ``dilution'' of the oscillators by inert collision partners.