Quantum Systems: A Mathematical Journey from Few to Many Particles

On a model of the Kompaneets equation and Bose-Einstein condensation

Hailiang Liu

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 R. Pego.