**4. Conclusion**

We have studied the mesoscopic, stochastic process derived from a deterministic dynamics applied to the cells determined by measure inaccuracies. The stationary process, which arises when the microscopic initial state is distributed according to a time invariant measure, was studied by Kolmogorov and further authors: we extended their methods and some of their results, and considered the nonstationary process which stems from a noninvariant initial measure. We have shown that, according to Jaynes' principle, the "exact" mesocopic process can be approximately replaced by the Markov process which, at any time *n*, reproduces the one-time probability of each mesostate and the transition probabilities from it. This Markov process maximizes the trajectory entropy up to time *n*, as well as the entropy at time *n*, conditioned by prior events. The Jaynes' principle, however, does not control the accuracy of this estimate: this was our next concern.

So, a sequence of successive approximations has been defined for the stationary mesoscopic process, based on one of our main results: the probability of any mesostate state conditioned by all past events, can be approximated by its probability conditioned by the *n* last past events only, the integer *n* being determined by the maximum distance allowed between these probabilities, as small as it may be. This property entails that the nonstationary mesoscopic process can be approximated by a *n*-times Markov process or even, after a time coarse-graining, by an ordinary one-time Markov process. These approximations require certain conditions which should be fulfilled by "normal" physical systems, with possible exceptions for slowly relaxing systems. If they are satisfied, the existence of a thermodynamic equilibrium is derived for a coarse-grained system obeying a measure-preserving deterministic dynamics, in particular an Hamiltonian dynamics, without introducing *ad-hoc* external noises. However, very rough estimations of the relaxation time show that for reasonable values of the parameters this time is extraordinarily long and completely unrealistic.

We conclude that, although the basic hypotheses of thermodynamics can be justified from a Hamiltonian or deterministic microscopic dynamics applied to the mesoscopic cells, the observed time scales of the relaxation to equilibrium cannot be explained without going beyond pure Hamilton mechanics, by introducing additional random effects, in particular due to the intrinsic imprecision of the particles localization.
