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

TL;DR

This paper links quantum Boltzmann models for Bose gases to chemical reaction networks and proves their convergence to a unique equilibrium using toric dynamical systems, advancing understanding of quantum kinetic equations.

Contribution

It introduces a novel connection between quantum Boltzmann equations and chemical reaction network theory, proving convergence to equilibrium with a new mathematical approach.

Findings

Quantum Boltzmann models converge to a unique equilibrium.

The equilibrium is independent of initial conditions, given conservation laws.

The proof employs toric dynamical system techniques similar to the global attractor conjecture.

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, derived in [49,50].