Title: Convergence, and lack of convergence, of phase space distributions to microcanonical equilibrium

Abstract: It is well known that phase space distribution functions obeying Liouville's theorem can converge weakly, but not strongly, to the microcanonical distribution. Similarly, coarse-grained distributions can converge pointwise, although unlike weak convergence, coarse-graining involves an arbitrary partition of phase space. We prove a theorem that strengthens the result that strong convergence cannot occur. Specifically, we show that time evolution is an isometry for a broad class of metrics of statistical distance on the space of phase space distributions. We also compare the physical significance of both weak and strong convergence. Next, we generalize the definition of coarse-graining to remove the need for a partition of phase space. This generalized coarse-graining is similar to E.T. Jaynes' maximum entropy distribution, but with an important difference. Much of the literature on the dynamics of phase space distributions is focused on providing a rigorous description of thermalization and the 2nd law of thermodynamics; meanwhile, a separate body of literature questions the relevance of weak convergence to thermalization. We aim to clarify this disagreement by specifying more clearly the scenarios in which weak convergence is, and is not, relevant.