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Approach to Equilibrium for the Stochastic NLS

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Abstract

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 Schrödinger 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 uniform in the frequency truncation N. The limit N →∞ is discussed.

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Authors and Affiliations

  1. Departments of Mathematics and Physics, Rutgers University, Piscataway, NJ, 08854-8019, USA

    J. L. Lebowitz

  2. Centre de Physique Théorique, UMR 7644 CNRS, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Ph. Mounaix

  3. CNRS and Department of Mathematics, Université Cergy-Pontoise, 95302, Cergy-Pontoise Cedex, France

    W.-M. Wang

Authors
  1. J. L. Lebowitz
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  2. Ph. Mounaix
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  3. W.-M. Wang
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Correspondence to J. L. Lebowitz.

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Communicated by G. Gallavotti

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Lebowitz, J.L., Mounaix, P. & Wang, WM. Approach to Equilibrium for the Stochastic NLS. Commun. Math. Phys. 321, 69–84 (2013). https://doi.org/10.1007/s00220-012-1632-7

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  • DOI: https://doi.org/10.1007/s00220-012-1632-7

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