Publication 15-CNA-023
Bose-Einstein condensation in a hyperbolic model for the Kompaneets equation
Joshua Ballew
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
jballew@andrew.cmu.edu
Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu
Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@cmu.edu
Abstract: In low-density or high-temperature plasmas, Compton scattering is the
dominant
process responsible for energy transport. Kompaneets in 1957 derived a
non-linear degenerate parabolic equation for the photon energy distribution.
In this paper we consider a simplified model obtained by neglecting
diffusion
of the photon number density in a particular way. We obtain a non-linear
hyperbolic PDE with a position-dependent flux, which permits a
one-parameter
family of stationary entropy solutions to exist. We completely describe the
long-time dynamics of each non-zero solution, showing that it approaches
some
non-zero stationary solution. While the total number of photons is formally
conserved, if initially large enough it necessarily decreases after
finite time
through an out-flux of photons with zero energy. This corresponds to
formation
of a Bose-Einstein condensate, whose mass we show can only increase with
time.
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