Boltzmann-type collision operators for Bogoliubov excitations of Bose-Einstein condensates: A unified framework

TL;DR

This paper develops a unified kinetic framework for Bogoliubov excitations in Bose-Einstein condensates, deriving collision operators that include various interaction processes without approximations.

Contribution

It introduces a comprehensive derivation of collision operators for Bogoliubov excitations, capturing all relevant interactions in a unified manner.

Findings

Identifies three key collisional processes involving condensate and excitations.

Derives explicit forms of collision operators for different interaction types.

Provides a foundation for more accurate kinetic modeling of Bose-Einstein condensates.

Abstract

Starting from the Bogoliubov diagonalization for the Hamiltonian of a weakly interacting Bose gas under the presence of a Bose-Enstein Condensate, we derive the kinetic equation for the Bogoliubov excitations. Without dropping any of the commutators, we find three collisional processes. One of them describes the $1\leftrightarrow2$ interactions between the condensate and the excited atoms. The other two describe the $2\leftrightarrow2$ and $1\leftrightarrow3$ interactions between the excited atoms themselves.