Asymptotic time behaviour of nonlinear classical field equations

Abstract

Hamiltonian systems with many degrees of freedom, like large assemblies of interacting particles in a box, are described by Gibbs-Boltzmann statistics, as far as their average properties are concerned. This does not hold for the long-time behaviour of classical nonlinear field equations, as has been already noticed by Jeans, because of the infinite heat capacity of this field. Thus, nonlinear (and nonintegrable) classical fields cannot relax for long times towards an ill-defined thermal equilibrium. The author considers an example of this relaxation problem: the long-time evolution of solutions of the nonlinear Schrodinger equation, in the defocusing case. Under some assumptions that for long times there is a cascade towards smaller and smaller scales, he introduces a kind of dissipation in a system that is formally reversible, and he gives the scaling laws for this.