Spectral theory of thermal relaxation
Résumé
We review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system S, with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperature T. The zeroth law of thermodynamics postulates that, independently of its initial configuration, the system S approaches a unique stationary state as t-->infinity. By definition, this limiting state is the equilibrium state of S at temperature T. Statistical mechanics further identifies this state with the Gibbs canonical ensemble associated with S. For simple models we prove that the above picture is correct, provided the equilibrium state of the environment R is itself given by its canonical ensemble. In the quantum case we also obtain an exact formula for the thermal relaxation time.