A stochastic thermalization of the Discrete Nonlinear Schr\"odinger Equation

TL;DR

This paper introduces a stochastic perturbation to the discrete nonlinear Schrödinger equation, demonstrating that the system's invariant measure is the canonical Gibbs distribution and showing convergence to steady waves under certain conditions.

Contribution

It presents a novel stochastic model conserving mass, proves the uniqueness of the invariant Gibbs measure, and establishes convergence to energy-minimizing steady states in specific regimes.

Findings

Canonical Gibbs distribution is the unique invariant measure.

Solutions converge to steady waves minimizing energy at low temperature.

Model captures thermalization process in discrete nonlinear Schrödinger systems.

Abstract

We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schr\"odinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique invariant measure. In the one-dimensional cubic focusing case on the torus, we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass.