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Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise

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Abstract

We construct generalized grand-canonical- and canonical Gibbs measures for a Hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear Schrödinger equation with a focusing nonlinearity of order p < 6. Key properties of these Gibbs measures, in particular absence of “phase transitions” and regularity properties of field samples, are established. We then study a time evolution of this system given by the Hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature. The noise is of Ornstein–Uhlenbeck type for the Fourier modes of the field, with the strength of the noise decaying to zero, as the frequency of the mode tends to ∞. We prove exponential approach of the state of the system to a grand-canonical Gibbs measure at a temperature and “chemical potential” determined by the stochastic noise term.

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Authors and Affiliations

  1. Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Eric A. Carlen

  2. Institut für Theoretische Physik, ETH Zürich, Zürich, Switzerland

    Jürg Fröhlich

  3. Departments of Mathematics and Physics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Joel Lebowitz

Authors
  1. Eric A. Carlen
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  2. Jürg Fröhlich
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  3. Joel Lebowitz
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Correspondence to Joel Lebowitz.

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Communicated by H. Spohn

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Carlen, E.A., Fröhlich, J. & Lebowitz, J. Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise. Commun. Math. Phys. 342, 303–332 (2016). https://doi.org/10.1007/s00220-015-2511-9

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  • DOI: https://doi.org/10.1007/s00220-015-2511-9

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