Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature

TL;DR

This paper analyzes a linearized quantum Boltzmann equation for bosons at low temperature, proving existence, uniqueness, and algebraic convergence to equilibrium despite unbounded collision frequency.

Contribution

It introduces an approximation of the linearized quantum Boltzmann equation for bosons at low temperature and proves convergence to equilibrium with algebraic rate.

Findings

Existence and uniqueness of solutions with conserved energy

Solutions converge algebraically to stationary state

Collision frequency is unbounded from below and above

Abstract

We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasiparticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the existence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity.