On deterministic numerical methods for the quantum Boltzmann-Nordheim equation. I. Spectrally accurate approximations, Bose-Einstein condensation, Fermi-Dirac saturation

Spectral methods, thanks to their high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the collisional kinetic equations of Boltzmann type, such as the Boltzmann-Nordheim equation. This equation, modeled on the seminal Boltzmann equation, describes using a statistical physics formalism the time evolution of a gas composed of bosons or fermions. Using the spectral-Galerkin algorithm introduced in Filbet et al. (2012) [11], together with some novel parallelization techniques, we investigate some of the conjectured properties of the large time behavior of the solutions to this equation. In particular, we are able to observe numerically both Bose-Einstein condensation and Fermi-Dirac relaxation. & COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

Spectral methods are an effective way to approximate collisional kinetic equations, and they are characterized by high accuracy and fast algorithms. In this study, the spectral-Galerkin algorithm introduced by Filbet et al. (2012) is employed, along with new parallelization techniques, to investigate conjectured properties of the solutions to the Boltzmann-Nordheim equation. Numerical observations demonstrate both Bose-Einstein condensation and Fermi-Dirac relaxation.