Stability of Equilibria with a Condensate

We consider a quantum system composed of a small part, having finitely many degrees of freedom, interacting with a free, spatially infinitely extended Bose gas. An equilibrium state for the uncoupled system is given by the product state where the small part is in the Gibbs state at some temperature T > 0, and the Bose gas is in a state where a spatially homogeneous Bose-Einstein Condensate is immersed in black body radiation at the same temperature T . An interaction between the two subsystems is specified by a coupled dynamics. The interaction strength is measured by the size of a coupling constant. We show that the equilibrium state of the uncoupled system is stable: any initial state close to it, evolving according to the interacting dynamics, converges to it, in the successive limits of large time and small coupling constant. We deduce the stability result from properties of structure and regularity of eigenvectors of the generator of the dynamics, called the Liouville operator. Among our technical results is a Virial Theorem for Liouville type operators which has new applications to systems with and without a condensate. ∗Supported by a CRM-ISM postdoctoral fellowship in conjunction with McGill University; [email protected]; http://www.math.mcgill.ca/∼merkli/