  This work explores the manner in which classical phase space distribution
functions converge to the microcanonical distribution. We first prove a theorem
about the lack of convergence, then define a generalization of the
coarse-graining procedure that leads to convergence. We prove that the time
evolution of phase space distributions is an isometry for a broad class of
statistical distance metrics, implying that ensembles do not get any closer to
(or farther from) equilibrium, according to these metrics. This extends the
known result that strong convergence of phase space distributions to the
microcanonical distribution does not occur. However, it has long been known
that weak convergence can occur, such that coarse-grained
distributions--defined by partitioning phase space into a finite number of
cells--converge pointwise to the microcanonical distribution. We define a
generalization of coarse-graining that removes the need for partitioning phase
space into cells. We prove that our generalized coarse-grained distribution
converges pointwise to the microcanonical distribution if the dynamics are
strong mixing. As an example, we study an ensemble of triangular billiard
systems.
