  We derive novel results on the ergodic theory of irreducible, aperiodic
Markov chains. We show how to optimally steer the network flow to a stationary
distribution over a finite or infinite time horizon. Optimality is with respect
to an entropic distance between distributions on feasible paths. When the prior
is reversible, it shown that solutions to this discrete time and space steering
problem are reversible as well. A notion of temperature is defined for
Boltzmann distributions on networks, and problems analogous to cooling (in this
case, for evolutions in discrete space and time) are discussed.
