Quantum kinetic theory: modelling and numerics for Bose-Einstein condensation

Abstract

In this work we review some modelling and numerical aspects in quantum kinetic theory for a gas of interacting bosons and we will try to explain what makes Bose-Einstein condensation in a dilute gas mathematically interesting and numerically challenging. A particular care is devoted to the development of efficient numerical schemes for the quantum Boltzmann equation that preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and generalized Bose-Einstein distributions as steady states. These properties are essential in order to develop numerical methods that are able to capture the challenging phenomenon of bosons condensation. We also show that the resulting schemes can be evaluated with the use of fast algorithms. In order to study the evolution of the condensate wave function the Gross-Pitaevskii equation is also presented together with some schemes for its efficient numerical solution.