On the rate of approach to steady state for some Lindblad evolutions equations arising in statistical mechanics

This lecture concerns some many-particle models arising in statistical mechanics where the dynamics id governed by a Lindblad type equations. We will discuss models with and without detailed balance, the latter case corresponding to a non-equilibrium steady state, with a special emphasis on fermion systems. Although the relative entropy is strictly decaying along the evolution with or without detailed balance, fewer methods are presently available for accurately estimating the rate of decrease of relative entropy, or even understanding the nature of steady sates, in the absence of detailed balance. Some recent results in this direction will be presented, some of which are from joint work with Joel Lebowitz and David Huse.