ASYMPTOTIC TIME BEHAVIOR OF NONLINEAR CLASSICAL FIELD-EQUATIONS

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 non-integrable) classical fields cannot relax for long times towards an ill-defined thermal equilibrium. I consider 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, I introduce a kind of dissipation in a system that is formally reversible, and I give the scaling laws for this.