title: Nonequilibrium stationary state of a truncated stochastic nonlinear Schrödinger equation: formulation and mean-field approximation.
abstract: We investigate the stationary state of a model system evolving according to a modified focusing truncated nonlinear Schrödinger equation used to describe the envelope of Langmuir waves in a plasma. We restrict the system to have a finite number of normal modes each of which is in contact with a Langevin heat bath at temperature T . Arbitrarily large realizations of the field are prevented by restricting each mode to a maximum amplitude. We consider a simple modeling of wave breaking in which each mode is set equal to zero when it reaches its maximum amplitude. Without wave breaking the stationary state is given by a Gibbs measure. With wave breaking the system attains a nonequilibrium stationary state which is the unique invariant measure of the time evolution. A mean-field analysis shows that the system exhibits a transition from a regime of low-field values at small |lambda| , to a regime of higher-field values at large |lambda| , where lambda<0 specifies the strength of the nonlinearity in the focusing case. Field values at large |lambda| are significantly smaller with wave breaking than without wave breaking.
authors.affiliation: Centre de Physique Théorique, UMR 7644 du CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France. mounaix@cpht.polytechnique.fr
authors.forename: Philippe
authors.initials: P
authors.lastname: Mounaix
authors.forename: Pierre
authors.initials: P
authors.lastname: Collet
authors.forename: Joel L
authors.initials: JL
authors.lastname: Lebowitz
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D008956:Models, Chemical; D017711:Nonlinear Dynamics; D011789:Quantum Theory; D013269:Stochastic Processes