Relaxation of the String Oscillator

Abstract

As was pointed out by Herzfeld, a single collision between an oscillator and an impinging particle usually consists of a multiplicity of impacts. If the oscillator is a particle on an inextensible string (mechanically equivalent to a particle in a box), and if it interacts impulsively with particles of identical mass incident along its line of oscillation, then each collision either consists of a single impact, in which case momentum is exchanged, or of a double impact, in which case there is no net momentum exchange. The transition probability is calculated exactly, and the resulting integro‐differential equation for the oscillator distribution function is solved explicitly. It is shown that the approach to equilibrium is not in general exponential in the time, even in its last stages, but that it may be made so with a special initial distribution. The effect of impact multiplicity on the time of relaxation is discussed.