In this talk, I will discuss mechanisms by which (nonlinear) quantum systems effectively 'thermalize' into long time stationary distributions.
Focusing on the paradigmatic example of the Dicke model (a large spin coupled to a boson mode), I will demonstrate how a constructive description of the thermalization process is
facilitated by the Glauber Q or Husimi function, for which the evolution equation turns out to be of Fokker-Planck type. The equation describes a competition of classical instabilities
and quantum diffusion. By this mechanism the system follows a "quantum smoothened" approach to equilibrium, which avoids the notorious singularities inherent to classical chaotic flows.
I will touch upon the relevance of the above picture to recent experiments in cold atom physics.