*2.1.2 Microscopic distribution*

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

$$
\mu\_t\left(A\right) = \mu(\rho\_{-t}\left|A\right). \tag{1}
$$

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 particular for physical systems during their evolution.

We will focus on two important cases:

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

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 Lebesgue measure *ω* on *Rn* : these exists an integrable probability density *p*(*x*) such that for any measurable subset *A* of *X*

$$
\mu(A) = \int\_A p(\mathbf{x}) d\alpha(\mathbf{x}).\tag{2}
$$

Furthermore, we assume that (*iii*) the Lebesgue measure of *X* (or volume of *X*) *<sup>V</sup>* <sup>¼</sup> volð Þ� *<sup>X</sup>* <sup>Ð</sup> *<sup>X</sup>dω*ð Þ *x* is finite, and (*iv*) the Lebesgue measure *ω* is preserved by the dynamics for any *t* and any measurable subset *A* of *X.*:

$$\text{vol}A \, = \text{vol} \, (\varrho\_{-t} A). \tag{3}$$

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

$$p(\mathbf{x}, t) \; := p\left(\mathbf{0}, \boldsymbol{\varrho}\_{-t} \mathbf{x}\right) \equiv p\left(\boldsymbol{\varrho}\_{-t} \mathbf{x}\right). \tag{4}$$

#### *2.1.3 Initial microscopic distribution: The stationary situation*

Suppose that S is an isolated physical system and no observation was made on S at time 0 nor before 0. Then, in the absence of any knowledge on S, we admit that at the initial time S is distributed according to the only unbiased probability law, which is the uniform law. This is clearly justified in the finite case, according to the physical meaning traditionally given to probability: in fact, attributing different probabilities for two distinct microstates of *X* would imply that some measurement would allow one to distinguish them objectively, which is not the case at time 0.

transposition to the continuous case is generally obvious, although the complete

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

If time 0 is the beginning of all observations and actions, we assume that the initial microscopic distribution *μ* is uniform and stationary, as discussed previously, and the probability to find system S in the mesostate *i*<sup>0</sup> at time 0 is *p*(*i*0, 0) = *μ*(*i*0,). The probability to be in *<sup>i</sup>* at time *<sup>t</sup>* is *<sup>p</sup>*0ð Þ¼ *<sup>i</sup>*, *<sup>t</sup> <sup>μ</sup>t*ðÞ¼ *<sup>i</sup> μ φ*�*<sup>t</sup>* ð Þ ð Þ*<sup>i</sup>* . The stationary joint

and the conditional probability of finding S in the *i* at time *t*, knowing that it was

*<sup>p</sup>*<sup>0</sup>ð Þ *<sup>i</sup>*0, 0 <sup>¼</sup> *μ φ*�*<sup>t</sup>* ð Þ *<sup>i</sup>*∩*i*<sup>0</sup>

Similarly, the stationary *n*-times joint probability and related conditional proba-

with, for any *<sup>t</sup>*: *<sup>p</sup>*<sup>0</sup>ð*i*0, *<sup>t</sup>*; *<sup>i</sup>*1, *<sup>t</sup>*<sup>1</sup> <sup>þ</sup> *<sup>t</sup>*; … *in*�1, *tn*�<sup>1</sup> <sup>þ</sup> *<sup>t</sup>*Þ ¼ *<sup>p</sup>*<sup>0</sup>ð Þ *<sup>i</sup>*0, 0; *<sup>i</sup>*1, *<sup>t</sup>*1; … *in*�1, *tn*�<sup>1</sup> . For the sake of simplicity, we will discretize the times 0 < *t*<sup>1</sup> < *t*<sup>2</sup> … , and write *ti* = *kiτ*, *ki* being a nonnegative integer and *τ* a constant time step, which will be

If S is a physical system, interactions may exist before or at time 0, so that the S can be constrained to lie in a certain subset *A* of *X* at time 0. However, since it is not possible to distinguish two microstates corresponding to the same mesostate, *A* should be a union of mesostates, or at least one mesostate. If it is known that at time 0 the microsate *x* of the system belongs to the mesostate *i*, we should assume that the initial microscopic distribution is uniform over *i*, since no available observation can give further information on *x*: so, in the discrete case, if *n*(*i*) is the number of

*n i*ð Þ *<sup>χ</sup><sup>i</sup>*

*<sup>p</sup>*0ð ÞÞ ¼ *<sup>i</sup>*0, 0; *<sup>i</sup>*, *<sup>t</sup> μ φ*�*<sup>t</sup>* ð Þ¼ *<sup>i</sup>*∩*i*<sup>0</sup> *<sup>μ</sup> <sup>i</sup>*∩*φ<sup>t</sup>* ð Þ *<sup>i</sup>*<sup>0</sup> (5)

*μ*ð Þ *i*<sup>0</sup>

*n*�1 *it n*�1

<sup>¼</sup> *<sup>μ</sup> <sup>i</sup>*∩*φ<sup>t</sup>* ð Þ *<sup>i</sup>*<sup>0</sup> *μ*ð Þ *i*<sup>0</sup>

∩ … ∩*i*<sup>0</sup> � � *:* (7)

ð Þ *x* (8)

*χi*ð Þ *x*

*<sup>N</sup>* (9)

(6)

derivations may be more difficult.

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

bilities are readily obtained from

probability to find S in *i*<sup>0</sup> at time 0 and in *i* at time *t* is

*<sup>p</sup>*<sup>0</sup>ð Þ¼ *<sup>i</sup>*, *t i*<sup>j</sup> 0, 0 *<sup>p</sup>*<sup>0</sup>ð Þ *<sup>i</sup>*, *<sup>t</sup>*; *<sup>i</sup>*0, 0

*<sup>p</sup>*<sup>0</sup>ð*i*0, 0; *<sup>i</sup>*1, *<sup>t</sup>*1; … *in*�1, *tn*�<sup>1</sup>Þ ¼ *μ φ*�*<sup>t</sup>*

microstates included in *i* and *χ<sup>i</sup>* (*x*) the characteristic function of *i*

case, with obvious adaptations to the continuous case.

*i* 1

*p x*ð Þ¼ , 0 <sup>X</sup>

*p x*ð Þ¼ , 0j*x*∈*<sup>i</sup>* <sup>1</sup>

the same way, replacing the number of microscopic states contained in the mesostate *i* by its volume *v*(*i*). For simplicity, we follow considering the discrete

In the absolutely continuous case, the similar conditional density is obtained in

If one only knows the mesoscopic initial distribution *p*(*i*,0) that at time 0 the system belongs to *i*, for each mesostate *i* of M, the initial microscopic distribution

*n i*ð Þ *p i*ð Þ , 0 *<sup>χ</sup>i*ð Þ¼ *<sup>x</sup>* <sup>X</sup>

*i p i*ð Þ , 0 *μ*ð Þ*i*

*2.2.2 The stationary situation*

in *i*<sup>0</sup> at time 0 is

taken as time unit.

becomes

**87**

*2.2.3 Non stationary situation*

In the absolutely continuous case, initial uniformity is less obvious: it amounts to assuming that the system should be found with equal probability in two regions of the state space with equal volumes if no information allows one to give preference to any of these regions. This is of course a subjective assertion, but for Hamiltonian systems it agrees with the semi-quantum principle which asserts that, in canonical coordinates, equal volumes of the phase space correspond to equal numbers of quantum states.

Another way for choosing the initial probability distribution is to make use of Jaynes' principle [25], which is to maximize the Shannon entropy of the distribution under the known constraints over this distribution: in the present case of an isolated system which has not been previously observed, this principle also leads to the uniform law. It is not really better founded than the previous, elementary reasoning, but it may be more satisfying and it can be safely used in more complex situations. We refer to most textbooks on statistical mechanics for discussing these well-known, basic questions.

The uniform distribution in a finite space, either discrete or absolutely continuous, is clearly stationary. In addition to the previous hypotheses, we will assume that the space *X* is indecomposable [26]: the only subsets of *X* which are preserved by the evolution function *φ<sup>t</sup>* are the empty set ∅ and *X* itself. Then, the stationary probability distribution is unique [18].

For simplicity, we will henceforth assume that the phase space *X* is finite.

**Initial, nonstationary situation.** In certain situations, the system can be prepared by submitting it to specific constraints before the initial time 0. Then it may not be distributed uniformly in *X* at *t* = 0. We will consider this case in the next paragraph.

#### **2.2 Mesoscopic distributions**

#### *2.2.1 Mesoscopic states*

Because of the imprecision of the physical observations, it is impossible to determine exactly the microstate of the system, but it is currently admitted that the available measure instruments allow one to define a finite partition of *X* into subsets *i*∈ M � (*i k* ), *k* = 1, 2, … *M*, such that it is impossible to distinguish two microstates belonging to the same subset *i*. So, in practice the best possible description of the system consists in specifying the subset *i* where its microstate *x* lies: *i* can be called the *mesostate* of the system. The probability for the system to be in the mesostate *i* at time *t* will be denoted *p*(*i*,*t*). It is not sure, however, that two microstates belonging two different mesostates can always be distinguished: this point will be considered in Section 3.2.2.

*Remark:* for convenience, we use the same letter *p* to denote the probability in a countable state space, as well as the probability density in the continuous case. This creates no confusion when the variable type is explicitly mentioned. This is the case now since, as mentioned previously, we assume that the space *X* is discrete. The

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

transposition to the continuous case is generally obvious, although the complete derivations may be more difficult.
