## 1. Introduction

In modern physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity in general serves as a primarily problem. The equilibrium thermodynamics is a phenomenological theory defined on the class of homogeneous functions of the zero and first order [15].

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index

The aims of this study are to establish the connection between the Tsallis statistics, i.e., the statistical mechanics based on the Tsallis statistical entropy, and the equilibrium thermodynamics and to prove the zero law of thermodynamics.

The structure of the chapter is as follows. In Section 2, we review the basic postulates of the equilibrium thermodynamics. The equilibrium statistical mechanics based on generalized entropy is formulated in a general form in Section 3. In Section 4, we describe the Tsallis statistics and analyze its possible connection with the equilibrium thermodynamics. The main conclusions are summarized in the final section.

## 2. Equilibrium thermodynamics

### 2.1. Thermodynamic potentials

In the equilibrium thermodynamics, the physical properties of the system are fully identified by the fundamental thermodynamic potential

where the vector

where

If the function

for every nonzero vector

The first thermodynamic potential

Solving this system of equations, we obtain

Substituting Eq. (6) into the fundamental thermodynamic potential

This Legendre transform is always well defined when the fundamental thermodynamic potential

The first differential and the first partial derivatives of the first thermodynamic potential

The second differential and the second partial derivatives of the first thermodynamic potential

The symmetry conditions for the matrix elements,

If the function

for every nonzero vector

The Legendre transform (Eq. (7)) is involutive [19], i.e., if under the Legendre transformation

where

This Legendre transform is well defined when the function

The second thermodynamic potential

where, now,

Then the first differential and the first partial derivatives of the second thermodynamic potential

The second differential and the second partial derivatives of the second thermodynamic potential

The symmetry conditions for the matrix elements,

### 2.2. Principle of additivity

In the equilibrium thermodynamics, all thermodynamic quantities belong to the class of homogeneous functions of zero and first order, which imposes the additional constraints on the thermodynamic system. The homogeneous function of

and the Euler theorem[1] -

where

Let us divide the system into two subsystems:

Then

The homogeneous function of the first degree

and the homogeneous function of the zero degree

Note that the zero law of thermodynamics is expressed by Eqs. (21) and (25) when the temperature

## 3. Equilibrium statistical mechanics

In comparison with the equilibrium thermodynamics, the system in the equilibrium statistical mechanics is described by two additional elements: the microstates of the system and the probabilities of these microstates. As in the equilibrium thermodynamics, the macrostates of the system are fixed by the set of independent variables of state. The thermodynamic potential is a universal function that depends not only on the macroscopic state variables of the system but also on the microstates of the system and their probabilities. The extensive thermodynamic quantities are calculated as averages over the ensemble of microstates. However, the intensive thermodynamic quantities are defined in terms of the first derivatives of the thermodynamic potential with respect to the extensive variables of state.

Let us formulate the main statements of the equilibrium statistical mechanics. Let the thermodynamic potential be a function

The first thermodynamic potential

Here and in the following, the first thermodynamic potential will be considered only for the statistical ensembles for which

The second thermodynamic potential

Let

where

The ensemble averages for the extensive dynamical variables

where

The first and the second thermodynamic potentials, which are the extensive functions of the variables of state, can also be written as (28)

where

In the equilibrium statistical mechanics, the unknown probabilities of microstates

where

Substituting Eq. (30) into Eq. (34) and using Eq. (27), we obtain the probabilities related to the first thermodynamic potential

where

Substituting Eq. (31) into Eq. (34) and using Eq. (27), we obtain the probabilities related to the second thermodynamic potential

where

Let us consider the first thermodynamic potential. Substituting Eq. (35) into Eq. (26) and using Eq. (36), we obtain the expression for the first thermodynamic potential as

Then the partial derivatives of the first thermodynamic potential (Eq. (39)) with respect to the variables of state can be written as

Here we used the conditions

The fundamental thermodynamic potential can be written as

Let us consider the second thermodynamic potential. Substituting Eqs. (37), (38), and (31) into Eq. (29), we obtain the expression for the second thermodynamic potential

where

where

Finally, it should be mentioned that the equilibrium statistical mechanics is thermodynamically self-consistent if the statistical variables

## 4. Tsallis statistical mechanics

The Tsallis statistical mechanics is based on the generalized entropy which is a function of the entropic parameter

where

### 4.1. Canonical ensemble

The thermodynamic potential of the canonical ensemble, the Helmholtz free energy, is the first thermodynamic potential

The first thermodynamic potential (Eqs. (26) and (29)),

Here the constraint (Eq. (27)) in the canonical ensemble is in the form

Applying the method of Lagrange multipliers (Eqs. (32)-(34)) with the Lagrange function

and [7]

where

where

where

Making use of Eqs. (40), (50), and (54), we can write the entropy of the system as

Here we have used the conditions that the derivative of the constraint (Eq. (53)) with respect to

Using Eqs. (41), (50), and (54), we obtain the pressure,

Here we have used the conditions that the derivatives of the constraint (Eq. (53)) with respect to the variables of state

Substituting Eqs. (50) and(54) into Eq. (41), we obtain the variable

where we have used the conditions that the derivative of the constraint (53) with respect to

Thus, from the results given in Eqs. (62) and (65)-(67), we see that the differential of the thermodynamic potential (Eq. (50)) satisfies [7, 15]

Using Eqs. (50) and (69), we obtain the fundamental equation of thermodynamics [7, 14, 15]

To prove the homogeneity properties of the thermodynamic quantities and the Euler theorem for the Tsallis statistics in the canonical ensemble, we will consider, as an example, the exact analytical results for the nonrelativistic ideal gas.

#### 4.1.1. Nonrelativistic ideal gas: canonical ensemble

Let us investigate the nonrelativistic ideal gas of identical particles governed by the classical Maxwell-Boltzmann statistics in the framework of the Tsallis and Boltzmann-Gibbs statistical mechanics.

It is convenient to obtain the exact results for the ideal gas in the Tsallis statistics by means of the integral representation for the Gamma function (see [9] and reference therein):

Thus, solving Eqs. (57) and (58) for the ideal gas in the canonical ensemble, the norm function

and

The energy (Eq. (61)) and the thermodynamic potentials (Eqs. (59) and (60)) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics can be written as [7]

and

The entropies (Eqs. (63) and (64)) and the pressure (Eq. (65)) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics can be written as [7]

and

The chemical potential (Eq. (66)) and the variable (Eqs. (67) and (68)) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics become [7]

and

#### 4.1.2. Nonrelativistic ideal gas in the thermodynamic limit: canonical ensemble

Let us try to express the thermodynamic quantities of the nonrelativistic ideal gas directly in terms of the thermodynamic limit when the entropic parameter

Note first that the canonical partition function (Eq. (75)) for the nonrelativistic ideal gas for the Boltzmann-Gibbs statistics can be rewritten as

The norm functions (Eqs. (73) and (74)) for the ideal gas in the thermodynamic limit in the Tsallis and Boltzmann-Gibbs statistics can be rewritten as [7]

where Eq. (84) is restricted by the conditions

In the thermodynamic limit (Eq. (82)), the energy of the system (Eq. (76)) and the thermodynamic potential (Eq. (77)) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics become [7]

and [7]

respectively. The entropy (78) and the pressure (79) for the ideal gas in the Tsallis and Boltzmann-Gibbs statistics in the thermodynamic limit can be written as [7]

and [7]

where

In the thermodynamic limit (82), the chemical potential (80) and the variable (81) for the ideal gas in the canonical ensemble for the Tsallis and Boltzmann-Gibbs statistics are [7]

and [7]

Thus, from the results for the Tsallis statistics given in Eqs. (86), (90), (92), (94), and (96), we see that the Euler theorem (Eq. (22)) is satisfied [7]

Moreover, the thermodynamic quantities (86), (88), (90), (92), (94) and (96) satisfy the relation for the thermodynamic potential

Next we shall verify that, when the entropic parameter

However, the temperature and the specific variables of state (Eq. (82)) are the same in each subsystem

Considering Eqs. (83), (100), and (101), we can verify that the Tsallis thermodynamic potential (Eq. (88)) and the entropy (Eq. (90)) of the canonical ensemble are homogeneous functions of the first order, i.e.,

The Tsallis pressure (Eq. (92)), the chemical potential (Eq. (94)), and the variable (Eq. (96)) are the homogeneous functions of the zero order, i.e.,

Thus, the principle of additivity (Eqs. (21), (24), and (25)) is totally satisfied by the Tsallis statistics. Equations (101) and (103) prove the zero law of thermodynamics for the canonical ensemble.

### 4.2. Microcanonical ensemble

The thermodynamic potential of the microcanonical ensemble, the entropy, is the second thermodynamic potential

The entropy

where

Applying the method of Lagrange multipliers (Eqs. (32)-(34)) with the Lagrange function

where

Then the first derivative (Eq. (44)) of the thermodynamic potential

The first derivative (Eq. (45)) of the thermodynamic potential

The first derivative (Eq. (45)) of the thermodynamic potential

The first derivative (Eq. (45)) of the thermodynamic potential

where

#### 4.2.1. Nonrelativistic ideal gas: microcanonical ensemble

Let us consider the nonrelativistic ideal gas of

where

The temperatures (Eqs. (114) and (115)) for the ideal gas for the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble correspond to [6]

The pressures (Eqs. (116) and (117)) and the chemical potentials (Eqs. (118) and (119)) for the ideal gas for the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble can be written as [6]

The variable (Eq. (120)) for the ideal gas for the Tsallis statistics in the microcanonical ensemble is [6]

However, the variable (Eq. (121)) for the Boltzmann-Gibbs statistics vanishes,

#### 4.2.2. Nonrelativistic ideal gas in the thermodynamic limit: microcanonical ensemble

Let us rewrite the thermodynamic quantities of the nonrelativistic ideal gas in the microcanonical ensemble in the terms of the thermodynamic limit when the entropic parameter

Then in the thermodynamic limit (Eq. (128)), the statistical weight (Eq. (122)) for the nonrelativistic ideal gas can be rewritten as [6]

Substituting Eq. (129) into Eqs. (112) and (113) and using Eq. (128), we obtain the entropy as [6]

The temperatures (Eqs. (123) and (124)) for the nonrelativistic ideal gas in the thermodynamic limit (128) can be rewritten as [6]

The pressure (125) and the chemical potential (126) for the nonrelativistic ideal gas in the thermodynamic limit (128) become [6]

The variable (Eq. (127)) for the Tsallis statistics in the thermodynamic limit (Eq. (128)) corresponds to [6]

For the Boltzmann-Gibbs statistics, we have

Let us verify the principle of additivity for the nonrelativistic ideal gas in the microcanonical ensemble in the thermodynamic limit when the entropic parameter

However, the specific variables of state (Eq. (128)) are the same in each subsystem (intensive)

Considering Eqs. (129), (137), and (138), we can verify that the Tsallis thermodynamic potential (Eq. (130)) of the microcanonical ensemble is a homogeneous function of the first order, i.e.,

Now, considering Eqs. (129), (137), and (138), we find that the Tsallis temperature (Eq. (132)), the pressure (Eq. (134)), the chemical potential (Eq. (135)), and the variable (Eq. (136)) are the homogeneous functions of the zero order, i.e.,

Thus, the principle of additivity (Eqs. (21), (24), and (25)) is totally satisfied by the Tsallis statistics in the microcanonical ensemble. Equation (140) proves the zero law of thermodynamics for the microcanonical ensemble [6].

### 4.3. Equivalence of canonical and microcanonical ensembles

We can now easily prove the equivalence of the canonical and microcanonical ensembles for the Tsallis statistics in the thermodynamic limits (Eqs. (82) and (128)). Using Eqs. (83) and (129), it is easy to verify that Eq. (132) for the temperature of the microcanonical ensemble and Eq. (86) for the energy of canonical ensemble are identical. Comparing Eqs. (83) and (129) and using Eq. (86), we have

Substituting Eq. (144) into Eq. (130) for the entropy of the microcanonical ensemble, we obtain the entropy of the canonical ensemble (Eq. (90)). Equation (134) for the pressure of the microcanonical ensemble is identical to Eq. (92) for the pressure of the canonical ensemble. Substituting Eqs. (144) and (86) into Eq. (135) for the chemical potential of the microcanonical ensemble, we obtain Eq. (94) for the chemical potential of the canonical ensemble. Moreover, substituting Eqs. (144) and (86) into Eq. (136) for the variable

## 5. Conclusions

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium statistical mechanics, the thermodynamic potential is a composed function that can depend on the set of independent variables of state explicitly and implicitly through the probabilities of microstates. The probabilities of microstates are determined from the second part of the second law of thermodynamics, i.e., the maximum entropy principle. The equilibrium probability distributions are found from the constrained extremum of the thermodynamic potential as a function of a multidimensional set of probabilities considering that the statistical entropy is defined. The equilibrium thermodynamics and statistical mechanics are defined only on the class of homogeneous functions, i.e., all thermodynamic quantities describing the thermodynamic system should belong to the class of homogeneous functions of the first or zero orders.

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index