Approach to equilibrium for the stochastic NLS

Details Approach to equilibrium for the stochastic NLS Approach to equilibrium for the stochastic NLS

2012-02-08

arxiv.org/abs/1202.1642v1

Physics

Mathematical Physics

15 pp

Scientific paper

We study the approach to equilibrium, described by a Gibbs measure, for a system on a $d$-dimensional torus evolving according to a stochastic nonlinear Schr\"odinger equation (SNLS) with a high frequency truncation. We prove exponential approach to the truncated Gibbs measure both for the focusing and defocusing cases when the dynamics is constrained via suitable boundary conditions to regions of the Fourier space where the Hamiltonian is convex. Our method is based on establishing a spectral gap for the non self-adjoint Fokker-Planck operator governing the time evolution of the measure, which is {\it uniform} in the frequency truncation $N$. The limit $N\to\infty$ is discussed.

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