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Updated: Jun 14, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Nonequilibrium stationary state of a truncated stochastic nonlinear Schrödinger equation: formulation and mean-field
Philippe Mounaix1, Pierre Collet, Joel L Lebowitz
1Centre de Physique Théorique, UMR 7644 du CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France. mounaix@cpht.polytechnique.fr
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.
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