Energy Transients in Harmonic Oscillator Systems

Abstract

The approach to energy equilibration is discussed for a system of N isolated classical oscillators, coupled or uncoupled. Exact solutions can be given for the kinetic and potential energies vs time, for a simple boundary condition which distributes the potential energy equally between all the normal modes at zero time and which gives zero kinetic energy to all the normal-mode oscillators at zero time. Most solutions are ergodic but a special class is nonergodic, i.e., the potential energy vs time shows exact recursions. The half-life for approaching the equilibrium value of potential or kinetic energy is of the order of the reciprocal maximum frequency of the oscillators. The fluctuations from equilibrium are treated by an extension of the method of Rayleigh's problem of random walks. The results are of interest in connection with the ergodic theorem of statistical mechanics.