A stochastic thermalization of the Discrete Nonlinear Schrodinger Equation

We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schrodinger 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.

Automated summary

We present a stochastic perturbation of the discrete nonlinear Schrodinger equation that conserves mass and models the effect of a heat bath at a specific temperature. We prove that the corresponding canonical Gibbs distribution is the only invariant measure. In the one-dimensional cubic focusing case, we demonstrate that as time approaches infinity, with continuous approximation and low temperature, the solution converges to the steady wave of the continuous equation that minimizes energy for a given mass.