Collision Integrals in the Kinetic Equations of dilute Bose-Einstein Condensates

TL;DR

This paper derives a kinetic equation for Bogoliubov excitations in a Bose-Einstein condensate, introducing a new collision integral that accounts for processes changing bogolon number, which significantly impacts the system's relaxation dynamics.

Contribution

It introduces a third collision integral in the kinetic equation for bogolons, capturing processes where one bogolon becomes three or vice versa, a novel addition to existing models.

Findings

The ${ m G}^{31}$ collision integral significantly affects bogolon dynamics.

Processes changing bogolon number are important for relaxation.

The derived kinetic equation improves understanding of BEC excitations.

Abstract

We derive the mean field kinetic equation for the momentum distribution of Bogoliubov excitations (bogolons) in a spatially uniform Bose-Einstein condensate (BEC), with a focus on the collision integrals. We use the method of Peletminksii and Yatsenko rather than the standard non-equilibrium Green's function formalism. This method produces three collision integrals ${\cal G}^{12}$, ${\cal G}^{22}$ and ${\cal G}^{31}$. Only ${\cal G}^{12}$ and ${\cal G}^{22}$ have been considered by previous authors. The third collision integral ${\cal G}^{31}$ contains the effects of processes where one bogolon becomes three and vice versa. These processes are allowed because the total number of bogolons is not conserved. Since ${\cal G}^{31}$ is of the same order in the interaction strength as ${\cal G}^{22}$, we predict that it will significantly influence the dynamics of the bogolon gas, especially the relaxation of the total number of bogolons to its equilibrium value.