# Phase Space Cell in Nonextensive Classical Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}in the MB extensive phase space Ω

_{0}and in the deformed phase space Ω, we derive the dimensions of both elementary cells ∆Ω

_{0}and ∆Ω. The cell ∆Ω results to be smaller than ∆Ω

_{0}(if q < 1) so that, in this case, the third power of Planck constant is not suitable as the value of ∆Ω

_{0}because h

^{3}is the smallest cell admissible due to the Heisenberg principle. Otherwise ∆Ω is larger than, ∆Ω

_{0}if q > 1 (h is expressed in units [energy length]).

## 2. Phase Space Volume and Elementary Cell

_{r}] can be divided into small cells of volume ∆Ω, so that coordinates do not vary sensibly within them.

_{1}, n

_{2}, · · ·, n

_{t}) = [n

_{r}] and is given by:

_{0}(the symbol ∂ means variation). After using the Lagrange method with the usual constraints and with negligible interactions the following is obtained:

_{0}is of course

_{0}), defined below in (2.24).

_{0}. The constraints are

_{0}and Ω are different in size. ∆Ω

_{0}and ∆Ω are also different. We realize that the differences could be considered negligible, while their effects seem to be quite important for the evaluation of several physical quantities. Let us remark that the number of microstates or discrete events does not change from one space to the other one. Therefore, we set the equation W = W

_{0}, because we want to count the same number of microstates both in Ω and Ω

_{0}. Using the relations reported above in this Section, we can explicitate the expressions of Ω and ∆Ω after simple calculations. After using the following relations:

_{0}and ∆Ω

_{0}, respectively as follows:

_{0}. Therefore, we are not allowed in their classical extensive description to take ∆Ω

_{0}= h

^{3,}because h

^{3}should be the smallest elementary permissible cell. On the other hand, we shall verify that ∆Ω

_{0}must always be much larger than h

^{3}. It must be a macrocell. The equation of state calculated by means of the distribution (2.9) is given by

^{1-q}.

_{0}imposes that

_{0}and that the equation of state for NETS classical systems be correctly expressed also in the small deviations limit imply that the standard phase space elementary cell be given by the expression (2.25) (we send to Sect. 3 for some more details).

^{3}, usually taken as elementary cell. However, this requirement is not a problem, because of the uncertainty of the classical elementary cell and because Darwin and Fowler [3] showed that macrocells should be used to satisfy Boltzmann requirements of a great average number of particles in each cell.

_{0}of Eq. (2.25) means to have one particle in each macrocell and, posing ∆Ω

_{0}= X h

^{3}, to have 1/X particles in each microcell h

^{3}, i.e. one particle in thousands of cells, where

^{3}does not allow the conservation of the value of the number of permissible microstates, nor the subdivision in a fixed number of micro cells independent on β and q.

_{0}and Ω and to preserve the correct form of the equation of state).

## 3. Application to Classical Ideal Gas: Entropy, Chemical Potential, Free Energy

^{*}previously introduced, associated with the constraint in NETS, is defined by

_{T}is a constant depending on q which becomes the Boltzmann constant k for q → 1 [18,28] and T

_{phys}, the physical temperature, is

_{T}and not to T

_{phys}or β

^{*}). The partition function Z

_{q}(β) is defined by

_{T}is the square bracket factor of Tsallis distribution [Eq.(2.10)]. Two other quantities that are useful for NETS classical ideal gas calculations are: the coefficient

_{0}as

_{1}≈(∆Ω

_{0})

^{-N}and C

_{1}and U

_{1}do not depend on ∆Ω

_{0}. The above functions enter into the calculation of the equation of state, which can be derived by means of the usual thermodynamic relations. As we have already discussed in the previous Section, by considering a nonextensive classical ideal gas with distribution function ni of Eq. (2.9) (small deviations from MB distribution), we have calculated that the equation of state is given by P V = N k T (1 - 5 δ + 46 δ

^{2}) i.e., in the limit of small deviations we must have $\beta $* = $\beta $/C

_{δ}as it can be easily verified.

_{δ}if the elementary cell (macrocell) ∆Ω

_{0}of Eq. (2.25) is assumed.

_{0}given by (2.25) we obtain that Zq, Cq and Uq do not explicitly depend on the elementary cell. Zq and Cq do not depend on $\beta $ either:

_{0}and Ω depend on the elementary cells ∆Ω

_{0}and ∆Ω, respectively. If ∆Ω

_{0}is a constant, like h

^{3}, Ω

_{0}does not depend on β, but only on N and Ω depends on N and q. Instead, if ∆Ω

_{0}has the expression of Eq. (2.25) and ∆Ω is given by Eq. (2.28), then Ω

_{0}is also a function of β and Ω is also a function of β and q.

^{3/2}/∆Ω

_{0}, is the single particle number of cells.

_{0}given by Eq. (2.25), we obtain S

^{B}= 5N k/2 because the single particle number of cells equals N and the value of the constant is zero.

_{0}has been used and therefore

^{2}with large N. The chemical potential µ of a classical ideal gas is an intensive quantity, defined by

_{0}is the elementary cell taken usually equal to h

^{3}. The quantity µ can be negative or positive and is equal to zero only at the particular value of temperature

_{0}the definition of Eq. (2.25). We find that for an extensive ideal classical gas we have µ = 0. This result can be explained because adding one particle at constant energy and volume the elementary cells decrease their single volumes and the total work done by the system to diminish the spatial volume of N + 1 cells is k T. Therefore,from Eq.(3.14) we have

_{0}given by Eq. (2.25) at constant energy and total volume, the function F and its variation due to the addition of one particle to the system are given by the following relations:

_{B}= –N k T ,

## 4. Interpretation of the Parameter q

## 5. Conclusions

_{0}and ∆Ω

_{0}because we require that the number W

_{0}of available microstates in the MB extensive phase space Ω

_{0}equals the number W in the deformed (nonextensive) space: W

_{0}= W. This condition imposes particular analytical expressions of ∆Ω

_{0}and ∆Ω. The number of elementary cells results equal to the number of particles N.

^{3}or left undetermined.

## Acknowledgments

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Quarati, F.; Quarati, P. Phase Space Cell in Nonextensive Classical Systems. *Entropy* **2003**, *5*, 239-251.
https://doi.org/10.3390/e5020239

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Quarati F, Quarati P. Phase Space Cell in Nonextensive Classical Systems. *Entropy*. 2003; 5(2):239-251.
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Quarati, Francesco, and Piero Quarati. 2003. "Phase Space Cell in Nonextensive Classical Systems" *Entropy* 5, no. 2: 239-251.
https://doi.org/10.3390/e5020239