Fast solvers for the quantum Boltzmann equation and the Bose-Einstein condensation

Bose-Einstein condensate
The interest in the quantum framework of the Boltzmann equation has in­creased dramatically in the recent years. Although the quantum Boltzmann equation (QBE) for a single specie of particles is valid for a gas of fermions as well as for a gas of bosons, blow up of the solution in finite time may occur only in the latter case. In particular this equation has been successfully used for computing non­equilibrium situations where Bose­Einstein condensate occurs.
At very low but finite temperature a large fraction of the atoms would go into the lowest energy quantum state. This phenomenon is now called Bose­-Einstein condensation (see [link]). Experimentally this has been achieved thanks to strong advancements in trapping and cooling techniques for neutral atoms leading to the 2001 Nobel prize in physics by E.A.Cornell, W.Ketterle and C.E.Wiemann ([here] is a movie of BEC in Rb-87).
In the sequel we report some animated movies of numerical simulations in quantum kinetic theory. Simulations have been computed using the fast method developed in [1] and refers both to the case of a gas of bosons and of a gas of fermions. In particular in the second movie numerical evidence of the Bose-Einstein condensation is given. We refer to [1,2] for the simulations details.


video

1. Subcritical regime, regular Bose-Einstein stationary state

video

2. Critical regime, Bose-Einstein condenstaion

video

3.The case of Fermions, Fermi-Dirac steady state. Same initial data as in 1.

References
  1. P.Markowich, L.Pareschi, Fast, conservative and entropic numerical methods for the Bosonic Boltzmann equation, Numerische Math. 99 (2005), pp.509--532.
  2. P.Markowich, L.Pareschi, W.Bao, Quantum kinetic theory: modelling and numerics for Bose-Einstein condensation, Chapter 10, Modeling and computational methods for kinetic equations, Series: Modeling and Simulation in Science, Engineering and Technology, Birkhauser (2004), pp.287-320.