**3. Discussion of the Markov representation derived from Hamiltonian dynamics, and estimation of the uniformization time**

The previous results show that the coarse grained mesoscopic dynamics can eventually be represented by a Master Equation, because the memory of this dynamics is gradually lost over time. However, they do not provide the time scale of this fading. In order to estimate its order of magnitude simply, we make an intuitive remark: the conditional probability to jump from some mesostate *i* to another one can be evaluated without knowing the past history of the system if one knows the initial microscopic distribution over *i.* The only unbiaised initial distribution is the uniform one. Thus, one can consider that the system has a memory limited to one time step if uniformity is approximately realized in each mesoscopic cell: this is the basis of the elementary Markov models of mesoscopic evolution. Let *T* be the average time

needed to reach uniformity in a mesoscopic scale, starting from strong inhomogeneity. In a first approximation it is reasonable to use this uniformization time *T* to characterize the time scale over which a Markov evolution can describe the system.

which is far larger than the age of universe (now estimated to be about

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

work of purely Hamiltonian systems, the microscopic distribution within a

**3.2 An elementary, empirical approach of mesoscopic systems**

Although these calculations are very rudimentary, it is clear that, in the frame-

mesocopic cell remains far from uniformity during any realistic time if it is initially

More generally, it is clear that the uniformization time *T* should be of the order of Poincaré time [32–36] in a mesoscopic cell, which is known to be extraordinarily

The practical relevance of Markov processes to model a large class of physical systems is supported by a vast literature. We have seen that the progressive erasure of its memory over time allows one to justify the use of a Markov process to represents the evolution of the coarse-grained system. However, such representation can also stem from random disturbances due to the measurements or other sources of stochasticity: then, one has to renounce to a purely deterministic microscopic dynamics, as formerly proposed by many authors, even without adopting the formalism of Quantum Mechanics. It is interesting to compare the time scales of the

Suppose now that the measure process does not induce any significant change in the average molecules energy - so, their average velocity remains unchanged – but that it causes a random reorientation of their velocity. A rudimentary, one dimensional model of such a randomization could be to assume that each time a molecule is about to pass to a neighboring cell, it will go indifferently to one of the neighboring microscopic cell. In a one dimensional version of the model, a molecule perform a random walk on the *η* = *l*/*λ* = 10<sup>2</sup> points representing the microscopic cells contained in the mesoscopic cell, and we adopt periodic conditions at the boundaries of the mesocopic cell. The *η* � *η* transition matrix of the process is a circulant matrix which, in its simplest version, has transition probabilities ½ to jump from any state to one of its neighbors, and it is known that its eigenvalues *λ<sup>k</sup>* are *λ<sup>k</sup>* = cos (2π*k*/*η*), *k* = 0, 1, … [*η*/2]. The number of jumps necessary for relaxing to the

<sup>1</sup>*=*ð Þ � ln *<sup>λ</sup>*<sup>1</sup> <sup>∝</sup>2 2ð Þ <sup>π</sup>*=*<sup>η</sup> �<sup>2</sup> <sup>≈</sup>500 s*:*

which correspond to a relaxation time of 500. *λ*/*v* ≈ 10�<sup>8</sup> s, which is very short for current measurements, but comparable with (or even larger than) the time scale of fast modern experiments. Considering a 3-dim model would not change this time scale significantly. It is conceivable that he molecules are not necessarily reoriented each time they leave a microscopic cell. Even if the proportion of reoriented mole-

many simple measures. In this case the Markov representation can be justified.

In analogy with the previous randomized system, we can introduce a new source of stochasticity in the coarse-grained deterministc systems considered in Sections 2 and 3. This could be done by assuming that a particle cannot be described by a

, the relaxation time is of order 10�<sup>2</sup> s, which is insignificant in

relaxation to equilibrium in both approaches with an elementary example.

*3.2.1 Uniformization induced by randomization*

uniform, asymptotic distribution is of the order

cules is as low as 10�<sup>6</sup>

**97**

*3.2.2 Semi-classical Hamiltonian systems*

<sup>14</sup> � <sup>10</sup><sup>9</sup> years, or 4.4 � <sup>10</sup><sup>17</sup> s)!

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

fairly inhomogeneous.

long [9, 37].
