**1. Introduction**

It is generally admitted that Thermodynamics and Statistical Physics could be deduced from an exact classical or quantum Hamiltonian dynamics, so that the various paradoxes related to irreversibility could also be explained, and nonequilibrium situations could be rigorously studied as well. These questions have been and still are discussed by many authors (see, for instance Refs. [1–4] and many classical textbooks, for instance [5–10]), who have introduced various plausible hypotheses [7–14], related to the ergodic principle [8–11], to solve them. It seems that there are two major kinds of problems. First, to justify that physical systems can reach an equilibrium state when they are isolated, or in contact with a thermal bath (which remains to be defined). Secondly, to justify various types of reduced stochastic dynamics, depending on the phenomena to be described: Boltzmann equations, Brownian motions, fluid dynamics, Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchies, etc..: see for instance Refs [1–11, 15, 16]. Concerning the first type of problems (reaching an equilibrium, if any) very rough estimations

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 and references).

properties, covering many realistic cases. These conclusions can be extended to a large class of deterministic systems generalizing classical Hamiltonian systems,

Consider a deterministic system S. Its states *x*, belonging to a state space *X*, will

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

Assume that the exact microstate *x*<sup>0</sup> is unknown at time 0, but is distributed according to the probability measure *μ* on the phase space *X.* The microscopic

for any measurable subset *A* of *X*. If *μ* is stationary, it is preserved by the dynamics: *μ<sup>t</sup>* (*A*) = *μ* (*A*). This condition, however, it not necessarily satisfied, in

b. the absolutely continuous case: *X* ⊂*R<sup>n</sup>*, where (*i) R* is the set of real numbers and *n* is an integer (usually very large, and even in the case of Hamiltonian dynamics), and (*ii*) the measure *μ* is absolutely continuous with respect to the

*μ<sup>t</sup>* ð Þ¼ *A μ φ*ð Þ �*<sup>t</sup> A :* (1)

: these exists an integrable probability density *p*(*x*)

vol*A* ¼ vol ð Þ *φ*�*tA :* (3)

*p x*ð Þ¼ , *t p*ð Þ� 0, *φ*�*tx p*ð Þ *φ*�*tx :* (4)

*p x*ð Þ*dω*ð Þ *x :* (2)

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.

we will not use such properties in this chapter.

probability distribution *μ<sup>t</sup>* at time *t* is given by

We will focus on two important cases:

Lebesgue measure *ω* on *Rn*

*<sup>V</sup>* <sup>¼</sup> volð Þ� *<sup>X</sup>* <sup>Ð</sup>

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particular for physical systems during their evolution.

such that for any measurable subset *A* of *X*

dynamics for any *t* and any measurable subset *A* of *X.*:

a. the finite case: *X* is finite and consists in *N* microstates.

*μ*ð Þ¼ *A*

ð *A*

Furthermore, we assume that (*iii*) the Lebesgue measure of *X* (or volume of *X*)

The last two assumptions obviously generalize basic properties of Hamiltonian

dynamics in a finite volume of phase space. Thus, by (1)–(3), the probability density is conserved along any trajectory: at time *t* the probability density is

*<sup>X</sup>dω*ð Þ *x* is finite, and (*iv*) the Lebesgue measure *ω* is preserved by the

which we now describe. We first specify our hypotheses and notations.

*Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov…*

*2.1.1 Deterministic microdynamics*

*DOI: http://dx.doi.org/10.5772/intechopen.95903*

*2.1.2 Microscopic distribution*

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

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 to the stationary distribution, as expected.

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 dynamics.

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 time coarse-grained mesoscopic distribution [18].

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 Quantum Mechanics formalism.
