Markov evolution and H-theorem under finite coarse graining in conservative dynamical systems

The necessary and sufficient conditions in order that a coarse graining with respect to a partition with finite number of cells leads to a Markovian evolution with a monotonically increasing entropy in abstract conservative dynamical systems are derived. These conditions are in particular verified when the symbolic dynamics associated to the partition induces a Markov chain. In this case the irreversible approach to equilibrium entails that the system should have a Bernoulli factor.