Approach to Equilibrium for the Stochastic NLS
| Autor: | Joel L. Lebowitz, Ph. Mounaix, W.-M. Wang |
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| Rok vydání: | 2013 |
| Předmět: |
Physics
Probability (math.PR) Mathematical analysis FOS: Physical sciences Statistical and Nonlinear Physics Torus Mathematical Physics (math-ph) Exponential function 82-XX symbols.namesake Mathematics - Analysis of PDEs Frequency domain FOS: Mathematics symbols Spectral gap Boundary value problem Gibbs measure Hamiltonian (quantum mechanics) Nonlinear Schrödinger equation Mathematics - Probability Mathematical Physics Analysis of PDEs (math.AP) |
| Zdroj: | Communications in Mathematical Physics. 321:69-84 |
| ISSN: | 1432-0916 0010-3616 |
| DOI: | 10.1007/s00220-012-1632-7 |
| Popis: | 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. Comment: 15 pp |
| Databáze: | OpenAIRE |
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