Chaos and the approach to equilibrium in a discrete sine-Gordon equation

Abstract

In this paper we study the long time dynamics of a Hamiltonian system with many degrees of freedom. We numerically investigate the system's approach to equipartation of energy when the intial energy is confined to one or a small set of Fourier modes. We find that there is a transition from a low energy regime in which energy does not spread appreciably among the modes to a high energy regime in which the system rapidly approaches equipartition, and that this transition coincides with the onset of chaos in the system, as evidenced by a sharp rise in the largest Lyapunov exponent. For low frequency initial conditions, the critical parameter for this transition is the scaled energy E 1 = ( L/ N) 2 E. Using a generalization of the traditional Chirikov resonance overlap calculation on a three-mode subset of the full system, we predict the onset of widespread chaos and the transition to equipartition on relatively short timescales.