Liouville equation and Markov chains: epistemological and ontological probabilities

Abstract

The greatest difficulty of a probabilistic approach to the foundations of Statistical Mechanics lies in the fact that for a system ruled by classical or quantum mechanics a basic description exists, whose evolution is deterministic. For such a system any kind of irreversibility is impossible in principle. The probability used in this approach is epistemological. On the contrary for irreducible aperiodic Markov chains the invariant measure is reached with probability one whatever the initial conditions. Almost surely the uniform distributions, on which the equilibrium treatment of quantum and classical perfect gases is based, are reached when time goes by. The transition probability for binary collision, deduced by the Ehrenfest‐Brillouin model, points out an irreducible aperiodic Markov chain and thus an equilibrium distribution. This means that we are describing the temporal probabilistic evolution of the system. The probability involved in this evolution is ontological.