**Appendix A: Approximating the** *n***-times conditional probability**

With the notations of Section 4.2, we consider approximation (55), which is the basis of the *n*-times Markov approximation both in the stationary and nonstationary situations. Repeating approximation (51) we can write

*Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov… DOI: http://dx.doi.org/10.5772/intechopen.95903*

$$p\left(i\_{N+n-1}, N+n-1; \ldots; i\_N, N\right| i\_{N-1}, N-1; \ldots; i\_{N-n}, N-n; \ldots; i\_0, 0\right) = \tag{5}$$

$$= p\left(i\_{N+n-1}, N+n-1\middle|i\_{N+n-2}, N+n-2; \ldots; i\_0, 0\right) \ldots p\left(i\_N, N\middle|i\_{N-1}, N-1; \ldots; i\_0, 0\right)$$

$$\approx p^{(n)}\left(i\_{N+n-1}, N+n-1\middle|i\_{N+n-2}; \ldots i\_{N-1}, N-1\right) \ldots p^{(n)}\left(i\_N, N\middle|i\_{N-1}, N-1; \ldots; i\_{N-n}, N-n\right). \tag{6}$$

The last line of (49) is *<sup>p</sup>*ð Þ *<sup>n</sup> iN*þ*n*�1, *<sup>N</sup>* <sup>þ</sup> *<sup>n</sup>* � 1; … ; *iN*, *N i*<sup>j</sup> *<sup>N</sup>*�1, *<sup>N</sup>* � 1; … ; *<sup>i</sup>*0, 0<sup>Þ</sup> � . We write

$$p\left(i\_{N+n-1}, N+n-1; \ldots; i\_N, N\left|i\_{N-1}, N-1; \ldots; i\_{N-n}, N-n; \ldots; i\_0, 0\right) \equiv \right)$$

$$p^{(n)}\left(i\_{N+n-1}, N+n-1; \ldots; i\_N, N\left|i\_{N-1}, N-1; \ldots; i\_{N-n}, N-n; \ldots; i\_0, 0\right) \mid Q\_N^{(n)}.\tag{60}$$

and for *k* > *n* we define *lk*, using the abbreviations (32)

$$d\_k(i\_0, \ldots i\_{k-1}) \equiv \ln \frac{\Pi\_k(i\_k)}{\Pi\_k^{(n)}(i\_k)}.\tag{61}$$

We have by (24)

point, but by a probability density centered on the point that would represent it classically: such a description borrows one, but not all, of the axioms of wave mechanics, and it can be qualified as a "semi-quantical" description. A similar assumption can be introduced without referring to quantum mechanics, by noticing that a particle cannot be localized in a given mesoscopic cell with complete certainty, because of its finite size: if it is mainly attributed to a given cell, there exists a small probability that it also belongs to a neighboring cell. Even without formalizing these possibilities, one can presume that such random effects shorten drastically the memory of the mesoscopic process, and make it short with respect to ordinary measure times: then the Markov approximation described in Section 2 can correctly

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

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

So, a sequence of successive approximations has been defined for the stationary

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.

With the notations of Section 4.2, we consider approximation (55), which is the basis of the *n*-times Markov approximation both in the stationary and nonstationary

**Appendix A: Approximating the** *n***-times conditional probability**

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

represent the evolution of the observed coarse-grained process.

accuracy of this estimate: this was our next concern.

extraordinarily long and completely unrealistic.

situations. Repeating approximation (51) we can write

**98**

**4. Conclusion**

$$s\_{\pi}(p) - s\_{k}(p) = \sum\_{i\_{0}, \dots, i\_{k}} p\_{N}(i\_{0}, 0; \dots; i\_{k}, k) \text{ } \ln \frac{\Pi\_{k}(i\_{k})}{\Pi\_{k}^{(n)}(i\_{k})} = \left\langle l\_{k}(i\_{0}, \dots i\_{k-1}) \right\rangle \equiv \sigma\_{k}(n). \tag{62}$$

(Note that *σ<sup>k</sup>* is positive, although this not necessarily true for *lk*). By (24) for any positive *ε*

$$2\left\langle d^2(p\_k, p\_k^{(n)}) \right\rangle \le \left\langle \sigma\_k(i\_0, \dots i\_{k-1}) \right\rangle \\
< \varepsilon \text{ if } n \text{ is large enough.} \tag{63}$$

Averaging the logarithm of Eq. (60) we have

$$\left\langle L\_N^{(n)} \right\rangle \equiv \left\langle \ln Q\_N^{(n)} \right\rangle = \sum\_{k=0}^{n-1} [s\_n(p) - s\_{N+k}(p)] \propto n \left[ s\_n(p) - s\_{\ast\ast}(p) \right] \tag{64} = n\delta(n) \tag{64}$$

*δs n*ð Þ� *sn* ð Þ� *p s*∞ð Þ *p* can be interpreted as an entropy fluctuation with respect to its equilibrium thermodynamic value. If such a fluctuation relaxes exponentially to 0 with time, as usual, the last term of (54) tends to 0 when *n* ! ∞. Then, the *n*-times Markov approximations 4.2 and 5.1 are justified. Although exponential relaxation can be considered as a characteristic of "normal" physical systems, slower relaxations can occur: in this case the Markov approximation may be invalid.
