*2.1.1 Deterministic microdynamics*

show [17] that the time scales to reach equilibrium, using only Hamiltonian dynamics and measure inaccuracies, are extremely large, contrarily to everyday experience, and quantum estimations are even worse [17]. Essentially, these times scale as Poincaré recurrence times and they increase as an exponential of the number of degrees of freedom (see Section 3 of this chapter for a brief discussion

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

dynamics, which, by Liouville theorem, include Hamiltonian dynamics.

to the stationary distribution, as expected.

time coarse-grained mesoscopic distribution [18].

**2.1 Microscopic dynamics: Definitions and notations**

Quantum Mechanics formalism.

Here we concentrate on the second type of problems: is it possible to derive a stochastic Markovian process from an "exact" deterministic dynamics, just by coarse graining the microscopic state space? We generalize and complete the formalism recently presented [18] for Hamiltonian systems. Our framework is now more general and applies to all deterministic systems with a measure preserving

Following Kolmogorov, we start with a measure space with a discrete time dynamics given by the successive iterations of a measure preserving mapping. The Kolmogorov entropy, or trajectory entropy, has been defined by Kolmogorov as an invariant of stationary dynamical systems (see Arnold and Avez book [19] for a pedagogical presentation). We follow his work and generalize part of his results. We also use martingale theory [20–23] to show that the stationary coarse-grained process almost surely tends to a Markov process on partial histories including *n* successive times, when *n* tends to infinity. From this result, we show that in the nonstationary situation, the probability distribution of such partial histories approximately satisfies a Master equation. Its probability transitions can be computed from the stationary distribution, expressed in terms of the invariant measure. It follows that, with relevant hypotheses, the mesoscopic distribution indeed tends

Our next step is to coarse grain time also. The new, coarse-grained time step is now *n τ*, *τ* being the elementary time step of the microscopic description, and *n* being the number of elementary steps necessary to approximately "erase" the memory with a given accuracy. The microscopic dynamics induces new dynamics on partial histories of length *n.* We show that it is approximately Markovian if *n* is large enough. This idea is a generalization of the Brownian concept: a particle in a fluid is submitted to a white noise force which is the result of the coarse-graining of many collisions, and the time step is thus the coarse-graining of many microscopic

time steps [8, 24]. The Brownian motion emerges as a time coarse-grained

In Section 2, we recall various mathematical concepts (Kolmogorov entropy, martingale theory) and use them to derive the approximate Markov property of the partial histories, and eventually to obtain an approximate Master Equation for the

In Section 3, we briefly consider the problem of relaxation times and recall very rough estimations showing that an exact Hamiltonian dynamics predicts unrealistic, excessively large relaxation times [17], unless the description is completed by introducing other sources of randomness than the measure inaccuracies leading to space coarse-graining. Note that, following Kolmogorov [19], we do not address the

**2. Microscopic and mesoscopic processes in deterministic dynamics**

approximated by Markov processes provided that they satisfy reasonable

It has been shown recently [18] that coarse-grained Hamiltonian systems can be

and references).

dynamics.

**84**

Consider a deterministic system S. Its states *x*, belonging to a state space *X*, will be called "microstates", in agreement with the usual vocabulary of Statistical Physics. The deterministic trajectory due to the microscopic dynamics transfers the microstate *x*<sup>0</sup> at time 0 to the microstate *xt* = *φt*ð Þ *x*<sup>0</sup> at time *t*. The evolution function *φ<sup>t</sup>* satisfies the current properties of dynamic systems: *φ<sup>t</sup> φ<sup>s</sup>* ¼ *φt*þ*s*, *φ*<sup>0</sup> ¼ *I*, *t* and s being real numbers and *I* being the identical function.

The dynamics is often invariant by time reversion, as assumed in many works on Statistical Physics: we refer to classical textbooks on the subject for details [5–8], but we will not use such properties in this chapter.
