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arXiv · 0911.2587

Nonequilibrium stationary state of a truncated stochastic NLSE: I. Formulation and mean field approximation

Philippe Mounaix·Pierre Collet·Joel L. Lebowitz
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Abstract

We investigate the stationary state of a model system evolving according to a modified focusing truncated nonlinear Schr\"odinger equation (NLSE) 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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Philippe Mounaix, Pierre Collet, Joel L. Lebowitz. 2009-11-13. Nonequilibrium stationary state of a truncated stochastic NLSE: I. Formulation and mean field approximation. https://doi.org/10.1103/physreve.81.031109

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