A connection between deterministic motion and the steady-state probability distribution

We start with a given deterministic equation of motion for a set of macrovariables. Suppose that we can construct a corresponding master equation which has the following properties. In the limit of large systems the deterministic equation can be derived from the master equation for not too long times and the steady-state probability distribution is the exponential of an extensive quantity. Then, using a theorem concerning the solutions of master equations, we can show that the solutions of the deterministic equations evolve into the direction of a nondecreasing steady-state probability.