Thermodynamic relations and fluctuations in the Tsallis statistics

The statistics which show power-like distributions have been interested in many branches of science. One of them is the Tsallis statistics which is an possible extension of the Boltzmann-Gibbs statistics, and the statistics has been applied in various fields TsallisBook ; Tsallis:Entropy:2019 . The entropy called Tsallis entropy and the escort average are employed in this statistics, and the probability distribution is obtained in the maximum entropy principle. The relations between thermodynamic quantities, such as internal energy and entropy, have been discussed. The statistics may describe the phenomena which show power-like distributions.

The physical temperature and the physical pressure were introduced with the Tsallis entropy Kalyana:2000 ; Abe-PLA:2001 ; S.Abe:physicaA:2001 ; Aragao:2003 ; Ruthotto:2003 ; Toral:2003 ; Suyari:2006 . In the Boltzmann-Gibbs statistics, the inverse temperature is given by the partial derivative of the entropy with respect to the internal energy. In the Tsallis statistics, the physical temperature was introduced in the similar way, though the inverse temperature-like parameter appears as a Lagrange multiplier in the maximum entropy principle. The physical temperature seems to be an appropriate variable to describe the system Ishihara:phi4 ; Ishihara:free-field ; Ishihara:2023 .

An entropic variable as a function of the Tsallis entropy was introduced by considering the Legendre transform structure in the Tsallis statistics S.Abe:physicaA:2001 . It is considered that the Legendre transform structure is an essential ingredient plastino1997 . In contrast, it is rarely noted that the Legendre transform structure may be unnecessary in the unconventional statistics C-Yepes . It was also shown that the Legendre transform structure is robust against the choice of entropy and the definition of mean value plastino1997 ; yamano2000 . The variables can not be determined uniquely by the Legendre transform structure, though the structure determines the conjugate variable for a given variable S.Abe:physicaA:2001 . Therefore, it may be better to introduce the entropic variable without using the Legendre transform structure explicitly when the variables can be determined by another consideration, though the structure is desirable. The entropic variable is useful in the description of the thermodynamics, because a temperature-like parameter is defined by using the entropy and because the Legendre transform structure is supported by the relation between the entropic variable and the temperature-like parameter.

It is considered that Tsallis-type distributions are related to fluctuations. The Tsallis-type distributions are often used to describe the phenomena, such as the momentum distributions Cleymans2012 ; Marques2015 ; Azmi2015 ; Thakur2016 ; Khuntia2017 ; Si2017 ; Bhattacharyya2018 ; Parvan2020 and fluctuations Osada:Isihara:2018 at high energies. The distribution was obtained by assuming that the heat capacity of the environment is exactly constant Watanabe2004 . The entropic parameter $q$ of the Tsallis statistics is related to the heat capacity which is connected with the fluctuation of the inverse temperature Wilk2009 . The relation between the fluctuation and the entropic parameter was discussed in the study of the time dependence of the entropic parameter Wilk2021 . This distribution was obtained for the system with fluctuation Wilk2000 ; Saha2021 . The distribution was derived in the studies of critical end point Ayala2020 , quantum entanglement Ourabah2017 ; Castano2021 , entropy exchange Castano2022 , and so on. The parameter $q$ is also related to the number of the configurations of intensive quantity Castano2021 . The property of the parameter $q$ should be important in the Tsallis statistics Parvan2006 .

The fluctuations of the thermodynamic quantities are significant in the Tsallis statistics. The fluctuation of the energy in the canonical ensemble was obtained Liyan:2008 by solving the differential equation for $0<q<1$ in the optimal Lagrange multiplier formalism Martinez2000 . The fluctuations were also calculated by maximizing the entropy that is constructed from probabilities with the deviation parameter from the equilibrium value Vives:2002 . The calculations of the physical quantities and the calculations of the Tsallis quantities are required to clarify the effects of the statistics.

In this paper, we consider thermodynamic relations, and attempt to find the expressions of the fluctuations in the Tsallis statistics. In section 2, we consider the thermodynamic relations with physical quantities, such as physical temperature and physical pressure. In section 3, the fluctuations are discussed in the Tsallis statistics with the introduced entropic variable. The fluctuations of the physical quantities and the fluctuations of the Tsallis quantities are obtained. The last section is assigned for conclusions.

We treat a system and an environment. The system and the environment are labeled with the superscripts $(S)$ and $(E)$, respectively. The total system constructed from the system and the environment is labeled by the superscript $(S+E)$.

We attempt to find the relations among the internal energy $U_{q}$, the physical temperature $T_{\mathrm{ph}}$, the entropic variable $X_{q}$, the physical pressure $P_{\mathrm{ph}}$, and the volume $V$. The entropic variable $X_{q}$ was already introduced in the reference S.Abe:physicaA:2001 . This variable $X_{q}$ is given below in this paper. The following discussion is based on the discussion given in the references Abe-PLA:2001 and S.Abe:physicaA:2001 .

The Tsallis entropy $S_{q}(U_{q},V)$ with the entropic parameter $q$ satisfies the following relation:

$\displaystyle S_{q}^{(S+E)}=S_{q}^{(S)}+S_{q}^{(E)}+(1-q)S_{q}^{(S)}S_{q}^{(E)}.$

(1)

The additivity of the internal energy is assumed:

$\displaystyle U_{q}^{(S+E)}=U_{q}^{(S)}+U_{q}^{(E)}.$

(2)

The total volume $V^{(S+E)}$ is the sum of the volumes, $V^{(S)}$ and $V^{(E)}$:

$\displaystyle V^{(S+E)}=V^{(S)}+V^{(E)}.$

(3)

The maximum entropy principle requires $\delta S_{q}^{(S+E)}=0$, and the total internal energy and the total volume satisfy $\delta U_{q}^{(S+E)}=0$ and $\delta V^{(S+E)}=0$. With these requirements, we define the physical temperature $T_{\mathrm{ph}}$ and the physical pressure $P_{\mathrm{ph}}$:

	$\displaystyle\frac{1}{T_{\mathrm{ph}}^{(S)}}=\frac{1}{1+(1-q)S_{q}^{(S)}}\left(\frac{\partial S_{q}^{(S)}}{\partial U_{q}^{(S)}}\right)_{V^{(S)}},$		(4a)
	$\displaystyle\frac{1}{T_{\mathrm{ph}}^{(E)}}=\frac{1}{1+(1-q)S_{q}^{(E)}}\left(\frac{\partial S_{q}^{(E)}}{\partial U_{q}^{(E)}}\right)_{V^{(E)}},$		(4b)
	$\displaystyle\frac{P_{\mathrm{ph}}^{(S)}}{T_{\mathrm{ph}}^{(S)}}=\frac{1}{1+(1-q)S_{q}^{(S)}}\left(\frac{\partial S_{q}^{(S)}}{\partial V^{(S)}}\right)_{U_{q}^{(S)}},$		(4c)
	$\displaystyle\frac{P_{\mathrm{ph}}^{(E)}}{T_{\mathrm{ph}}^{(E)}}=\frac{1}{1+(1-q)S_{q}^{(E)}}\left(\frac{\partial S_{q}^{(E)}}{\partial V^{(E)}}\right)_{U_{q}^{(E)}}.$		(4d)

We have the relations $T_{\mathrm{ph}}^{(S)}=T_{\mathrm{ph}}^{(E)}$ and $P_{\mathrm{ph}}^{(S)}=P_{\mathrm{ph}}^{(E)}$ from Eqs.(1), (2), and (3) with these definitions. These equations $T_{\mathrm{ph}}^{(S)}=T_{\mathrm{ph}}^{(E)}$ and $P_{\mathrm{ph}}^{(S)}=P_{\mathrm{ph}}^{(E)}$ indicate that the physical temperature and the physical pressure characterize the equilibrium. We use the names, physical temperature and physical pressure, in this paper, though names, equilibrium temperature and equilibrium pressure, might be adequate for the above introduced temperature and pressure.

The differential of the Tsallis entropy is

$\displaystyle dS_{q}^{(S)}=\left(\frac{\partial S_{q}^{(S)}}{\partial U_{q}^{(S)}}\right)_{V^{(S)}}dU_{q}^{(S)}+\left(\frac{\partial S_{q}^{(S)}}{\partial V^{(S)}}\right)_{U_{q}^{(S)}}dV^{(S)}.$

(5)

We have the following relation by using Eqs. (4a) and (4c):

$\displaystyle dU_{q}^{(S)}=\Bigg{(}\frac{T_{\mathrm{ph}}^{(S)}}{1+(1-q)S_{q}^{(S)}}\Bigg{)}dS_{q}^{(S)}-P_{\mathrm{ph}}dV^{(S)}.$

(6)

This is the first law of the thermodynamics in the Tsallis statistics. We introduce an entropic variable $X_{q}^{(S)}$ by requiring that Eq. (6) has the following form:

$\displaystyle dU_{q}^{(S)}=T_{\mathrm{ph}}^{(S)}dX_{q}^{(S)}-P_{\mathrm{ph}}^{(S)}dV^{(S)}.$

(7)

This requirement is satisfied by defining $X_{q}^{(S)}$ as

$\displaystyle X_{q}^{(S)}=\frac{1}{1-q}\ln(1+(1-q)S_{q}^{(S)}).$

(8)

The entropic variables $X_{q}^{(E)}$ and $X_{q}^{(S+E)}$ are defined in the same way. From Eq. (7), it is natural to define the heat transfer $Q^{(S)}$ as follows:

$\displaystyle Q^{(S)}=T_{\mathrm{ph}}^{(S)}dX_{q}^{(S)}.$

(9)

The alternative definition of heat transfer is given as $T^{(S)}dS_{q}^{(S)}$ by using the temperature $T^{(S)}$ which is the inverse of the Lagrange multiplier C.Tsallis1998 . The temperature $T^{(S)}$ is called the Tsallis temperature in this paper. It may be worth to mention that the relation, $T_{\mathrm{ph}}^{(S)}dX_{q}^{(S)}=T^{(S)}dS_{q}^{(S)}$, is easily shown Ishihara:EPJB:95 . Similar relation between the heat transfer in the incomplete non-extensive statistics and that in the Rényi statistics was shown in the reference Parvan2004 .

As pointed by some researchers S.Abe:physicaA:2001 ; Wang:prepri:2003 , the introduced entropic variables, $X_{q}^{(S)}$ and $X_{q}^{(E)}$, are additive:

$\displaystyle X_{q}^{(S+E)}=X_{q}^{(S)}+X_{q}^{(E)}.$

(10)

This property is easily shown from the pseudo-additivity of the Tsallis entropy. The pseudo-additivity of the Tsallis entropy $S_{q}$ is mapped to the additivity of the entropic variable $X_{q}$.

Equation (7) indicates that the variables $T_{\mathrm{ph}}^{(S)}$ and $X_{q}^{(S)}$ are a Legendre pair. Therefore, the free energy $F_{q}^{(S)}$ in terms of $T_{\mathrm{ph}}^{(S)}$ is naturally introduced by using the Legendre transformation of $U_{q}^{(S)}$ Abe-PLA:2001 ; Ishihara:EPJB:95 :

$\displaystyle F_{q}^{(S)}=U_{q}^{(S)}-T_{\mathrm{ph}}^{(S)}X_{q}^{(S)}.$

(11)

There is another definition of the free energy $\tilde{F}^{(S)}$ C.Tsallis1998 ; Ishihara:EPJB:95 which is given by $\tilde{F}^{(S)}=U_{q}^{(S)}-T^{(S)}S_{q}^{(S)}$.

The entropic variable $X_{q}$ is valid for the physical temperature $T_{\mathrm{ph}}$ because of the relation among $T_{\mathrm{ph}}$, $X_{q}$, and $U_{q}$: $1/T_{\mathrm{ph}}=(\partial X_{q}/\partial U_{q})_{V}$. The first law and the heat transfer are described with $T_{\mathrm{ph}}$ and $X_{q}$, and the free energy is defined by using Legendre transformation with $T_{\mathrm{ph}}$ and $X_{q}$, as shown above.

In this paper, we considered the thermodynamic relations in the Tsallis statistics by assuming the form of the first law with physical quantities. We also calculated the fluctuations of thermodynamic quantities by using the entropic variable defined with the Tsallis entropy.

The first law is naturally described with the physical temperature $T_{\mathrm{ph}}$ and the physical pressure $P_{\mathrm{ph}}$ by introducing the entropic variable $X_{q}$ conjugate to the physical temperature. The requirement that the first law has the form $dU_{q}=T_{\mathrm{ph}}dX_{q}-P_{\mathrm{ph}}dV$ suggests the form of $X_{q}$ as a function of the Tsallis entropy $S_{q}$, where $U_{q}$ is the energy and $V$ is the volume.

We obtained the mean squares of the fluctuations of the physical quantities: the physical temperature $T_{\mathrm{ph}}$, the physical pressure $P_{\mathrm{ph}}$, the volume $V$, and the entropic variable $X_{q}$. The mean squares of the fluctuations of physical quantities in the Tsallis statistics are the same as those in the conventional statistics. We also calculated the mean square of the fluctuation of the energy. The mean square of the relative fluctuation of the energy for ideal gas has the well-known expression.

The mean square of the fluctuation of the Tsallis entropy $S_{q}$ and the mean square of the fluctuation of the Tsallis temperature $T$ are given with physical quantities. The mean square of the fluctuation of $S_{q}$ is represented with the heat capacity at constant (physical) pressure $C_{qP}$ and the entropic variable $X_{q}$. The mean square of the fluctuation of $T$ is also represented with the physical temperature, heat capacities, and the entropic variable $X_{q}$. The mean square of the relative fluctuation of the Tsallis entropy $\langle(\Delta S_{q}/S_{q})^{2}\rangle$ with $q\neq 1$ is given by $(q-1)^{2}C_{qP}$ for large $S_{q}$. The mean square of the relative fluctuation of the Tsallis temperature $\langle(\Delta T^{(S)}/T^{(S)})^{2}\rangle$ is represented with the deviation $(q-1)$ and heat capacities. The fluctuations of the Tsallis quantities in the Tsallis statistics approach the fluctuations in the Boltzmann-Gibbs statistics as the entropic parameter $q$ approaches one.

It is possible to compare the results in this study with the results in other studies. The thermodynamic quantities in the Rényi statistics, in which the standard average is employed, were investigated, and it was shown that the thermodynamic quantities in the Rényi statistics are the same as those in the Boltzmann-Gibbs statistics Parvan:2010 . It was also shown that the physical quantities are $q$-independent within the framework of Tsallis statistics Toral:2003 . This property is shown in the present study: the mean squares of the fluctuations of the physical quantities, $T_{\mathrm{ph}}$, $V$, $X_{q}$, and $P_{\mathrm{ph}}$, are the same as those in the Boltzmann-Gibbs statistics. The fluctuations of the physical quantities do not contain the entropic parameter $q$ explicitly: the deviation $q-1$ does not appear. In contrast, the fluctuation of the Tsallis entropy $S_{q}$ and the fluctuation of the Tsallis temperature $T$ contain the entropic parameter $q$ explicitly: the deviation $q-1$ appears in the fluctuations.