On the stochastic representation and Markov approximation of Hamiltonian systems

We study the coarse-grained distribution of a Hamiltonian system on the space partition determined by the initial measurement inaccuracies. Using methods of coding theory, introduced by Shannon and further researchers, Kolmogorov treated the stationary case for a discretized time, when the microscopic system is initially uniformly distributed. Following his work, we consider the non-stationary mesoscopic process induced by the Hamiltonian evolution from an inhomogeneous initial distribution. In general, this process has an infinite memory, but we show that its memory fades out with time: with any finite accuracy a, it can be approximated by a process with a memory limited to the n past events, n depending only on a. As a result, under suitable hypotheses, the mesoscopic process obeys an approximate Markov equation on groups of n successive states. More roughly, one obtains an ordinary Markov system by introducing a time coarse-graining on n successive elementary time steps. So, in a generic case, the system eventually tends to equilibrium for any initial mesoscopic distribution.