Wave turbulence and Bose-Einstein condensates

TL;DR

This paper investigates the asymptotic behavior of nonlinear Schrödinger equations related to Bose-Einstein condensates, presenting a statistical approach that predicts equilibrium states and matches numerical simulations.

Contribution

It introduces a maximum entropy-based statistical framework for analyzing stationary states in discretized nonlinear Schrödinger systems, including Bose-Einstein condensates.

Findings

Quantitative agreement between theory and numerical simulations.

Particle spectral density follows a 1/k^2 law at large times.

Transient dynamics exhibit rapid oscillations of the condensate.

Abstract

Asymptotic behavior of a class of nonlinear Schr\"odinger equations are studied. Particular cases of 1D weakly focusing and Bose-Einstein condensates are considered. A statistical approach is presented to describe the stationary probability density of a discretized finite system. Using a maximum entropy argument, the theory predicts that the statistical equilibrium is described by energy equivalued fluctuation modes around the coherent structure minimizing the Hamiltonian of the system. Good quantitative agreement is found with numerical simulations. In particular, the particle number spectral density follows an effective $1/k^2$ law for the asymptotic large time averaged solutions. Transient dynamics from a given initial condition to the statistically steady regime shows rapid oscillation of the condensate.