Dynamics of mean-field bosons at positive temperature

TL;DR

This paper investigates the evolution of a weakly interacting Bose gas at positive temperature after trap removal, demonstrating stability of the initial state structure through a combination of Hartree and free evolutions, with a novel use of Hartree-Fock-Bogoliubov equations.

Contribution

It establishes the stability of the Gibbs state structure under many-body dynamics using Hartree-Fock-Bogoliubov equations for fluctuations.

Findings

The one-particle density matrix remains close to the ideal gas form.

The condensate wave function evolves according to the time-dependent Hartree equation.

Thermally excited particles follow free evolution.

Abstract

We study the time evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in arXiv:2009.00992 that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in $N$, as $N \to \infty$, by the one of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree-Fock-Bogoliubov equations to define a fluctuation dynamics.