Quantum Systems: A Mathematical Journey from Few to Many Particles
On a model of the Kompaneets equation and Bose-Einstein condensation
Iowa State University
We study the Kompaneets equation with an aim to understand a
phenomenon related to Bose-Einstein condensation which appears during the
process of relaxation to equilibrium for the photon energy distribution.
This talk is concerned with a model for the Kompaneets equation with the
spontaneous scattering term dropped, posed on a bounded interval. Uniqueness
of a large class of weak solutions with initial data of finite moments is
established, together with a proven moment stability and comparison
principle. Existence of the global weak solution is proved via a regularized
problem subject to an additional linear boundary condition at origin.
Bose-Einstein condensation is shown to form in finite time when initial
photon number is above a threshold value. Below this threshold a condensate
will develop for some initial data, but not for others. Large time
convergence to equilibrium solutions is established, with convergences rates
obtained for some special data.
This is a joint work with D. Levermore and