  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 [F. Filbet, J.
Hu, and S. Jin, ESAIM: Math. Model. Numer. Anal., 2011], 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.
