The stochastic discrete nonlinear Schr\"odinger equation: microscopic derivation and finite-temperature phase transition

TL;DR

This paper introduces a stochastic version of the 1D discrete nonlinear Schrödinger equation derived from first principles, revealing a finite-temperature phase transition with implications for experimental realizations.

Contribution

It provides a microscopic derivation of the stochastic DNSE and demonstrates its phase transition behavior, linking theoretical predictions with potential experimental setups.

Findings

Disordered and localized dynamics observed in the stochastic DNSE.

Finite-temperature phase transition captured by mean-field approach.

Unexpected noise-dependent coarsening dynamics resembling stochastic resonance.

Abstract

We study a stochastic version of the one-dimensional discrete nonlinear Schr{\"o}dinger equation (DNSE), which is derived from first principles, and thus possesses all the properties required by statistical mechanics, such as detailed balance and the H-theorem. The stochastic version shows disordered and localised dynamics, and displays a corresponding phase transition at a finite temperature value. The phase transition can be captured in a quantitative way by a mean-field type approach. The corresponding coarsening dynamics shows an unexpected dependence on the noise strength, which is reminiscent of stochastic resonance. The phase transition is linked with negative temperature phase transitions, which have been reported recently for the Hamiltonian dynamics of the DNSE. Our approach gives a clue to how these negative temperature phase transitions can be implemented in experimental setups, which are inevitably coupled to a positive temperature heat bath.