A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases

¹ Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI 53706-1388, USA.
² Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, USA.

^* Corresponding author: minhbinht@mail.smu.edu

Received: 20 October 2020
Accepted: 7 July 2021

Abstract

When the temperature of a trapped Bose gas is below the Bose-Einstein transition temperature and above absolute zero, the gas is composed of two distinct components: the Bose-Einstein condensate and the cloud of thermal excitations. The dynamics of the excitations can be described by quantum Boltzmann models. We establish a connection between quantum Boltzmann models and chemical reaction networks. We prove that the discrete differential equations for these quantum Boltzmann models converge to an equilibrium point. Moreover, this point is unique for all initial conditions that satisfy the same conservation laws. In the proof, we then employ a toric dynamical system approach, similar to the one used to prove the global attractor conjecture, to study the convergence to equilibrium of quantum kinetic equations.