Nonextensive statistics of Landsberg-Vedral entropy

Abstract. The general formalism for the nonextensive statistics based on the Landsberg-Vedral parametric entropy in the framework of the microcanonical, canonical and grand canonical ensembles was derived. The formulas for the first law of thermodynamics and the thermodynamic quantities in the terms of ensemble averages were obtained in a general form. It was found that under the transformation q → 2 − q the probabilities of microstates of the nonextensive statistics based on the Landsberg-Vedral entropy with the standard expectation values formally resemble the corresponding probabilities of the Tsallis statistics with the generalized expectation values.


Introduction
Initially, the equilibrium statistical mechanics was implemented on the basis of the Boltzmann entropy, that established the connection between the macroscopic variable, entropy, and the number of microstates of a system [1,2] (see also [3]) and on the basis of the more general Gibbs formula, S = −k B i p i ln p i , that is the entropy or "uncertainty" of the probability distribution for the microstates of a system [4]. Lately, the Shannon expression [5] for the information entropy of the discrete probability distribution allowed E.T. Jaynes to give another formulation of the equilibrium statistical mechanics based on the maximum entropy principle or the second law of thermodynamics [6]. The Rényi alternative method of defining the information entropy [7,8] opened a way for the introduction of new definitions of statistical entropies in the information theory (see, for instance, [3]). They had a significant impact on the development of the statistical models beyond the Boltzmann-Gibbs equilibrium statistical mechanics. The paper of C. Tsallis [9] established a new direction in science showing that on the basis of the Havrda-Charvát-Daroczy-Tsallis parametric entropy [3,10,11] it is possible to construct the stationary statistical mechanics.
The Tsallis nonextensive statistics [9,12] has received a wide recognition since it is confirmed by the experiment especially in high energy physics [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. At the same time, the Boltzmann-Gibbs statistics faces difficulties in describing real phenomena, for example, in nuclear processes at high energies since equilibrium statistical mechanics is a simplified ideal limiting case of the behavior of reality [6]. The equilibrium theory neglects complex processes and phenomena which take place in real conditions. However, nonextensive stationary statistics such as the Tsallis statistics covers these complex phenomena and describes them parametrically due to the fact that a theory like that goes beyond the framework of * e-mail: parvan@theor.jinr.ru,parvan@theory.nipne.ro the idealized Boltzmann-Gibbs statistics. The nonextensive parametrization may indicate the presence of complex processes in real phenomena such as, for example, nontrivial macromotions, particle transformation reactions, jets, internal currents in the system and etc. Nevertheless, the Tsallis statistics may have some indeterminacy in its formulation related to the generalized expectation values and the normalization condition for the probabilities of microstates [12,[28][29][30]. The main aim of this paper is to introduce the nonextensive statistics based on the Landsberg-Vedral entropy [31] which may be formulated on the basis of the standard linear expectation values consistent with the normalization condition for the probabilities of microstates. The Landsberg-Vedral entropy is defined as [31] where p i is the probability of the i-th microscopic state of the system. The main important properties of the Landsberg-Vedral entropy that can be mentioned are: positivity S ≥ 0, concavity in {p i }, the Gibbs limit lim q→1 S = −k B i p i ln p i , and the peculiar nonextensive additivity rule [31], , for two independent events A and B.

Microcanonical ensemble
The thermodynamic potential of the microcanonical ensemble is the entropy. Let us consider the Landsberg-Vedral entropy (1) which was introduced in [31]. The probabilities of microstates are constrained by the additional function and are obtained from the constrained local extrema of the thermodynamic potential (1) by the method of the Lagrange multipliers [32], Φ = S − λϕ, and where λ is an arbitrary real constant. Substituting Eqs. (1) and (2) into Φ and using Eqs. (3) and (2), we obtain Note that here and throughout the paper we use the system of natural units = c = k B = 1. Let us suppose that z is a variable of state of the system. Then, we obtain dS = dz z (S − W 1/z ln W) + W 1/z d ln W. The statistical weight W = W(E, V, N). Therefore, we have , The first partial derivatives of the thermodynamic potential S with respect to the variables of state in Eqs. (6)- (7) can also be verified from Eq. (4). Equation (5) represents the first law of thermodynamics. In the limit q → 1 we obtain all the relations of the Boltzmann-Gibbs statistics.

Canonical ensemble
The thermodynamic potential of the canonical ensemble, the free energy F, is obtained from the fundamental thermodynamic potential, the energy E, by the Legendre transform. Using Eq. (1), we have where E = i p i E i . Substituting Eqs. (8) and (2) into the Lagrange function of the canonical ensemble [34], Φ = F − λϕ, and using Eqs. (3) and (2), we obtain where χ ≡ i p q i , λ is an arbitrary real constant and Λ = λ − T/(q − 1). Substituting p i into χ and using Eq. (9), we find an equation for Λ as Substituting p i into Eqs. (1) and (8), and using Eq. (2), we have In the canonical ensemble the entropy S = S (T, V, z, N). Thus, using Eq. (1) and the definition of z given in Eq. (4), we obtain Substituting p i into Eq. (13) and using i d p i = 0 and Eq. (9), we can write The differential of the energy E can be written as where ∂E i /∂T = ∂E i /∂z = 0. Then, substituting Eq. (15) into Eq. (14), we obtain the first law of thermodynamics (cf. [35] for the Tsallis statistics) Then, from Eq. (16) and the Legendre transform (8) the differential thermodynamic relation for the thermodynamic potential F is dF = −S dT − pdV − Xdz + µdN.
Taking the first derivative of the thermodynamic potential (8) with respect to temperature T , volume V, number of particles N and variable z, and using Eq. (9), and the relations we obtain the entropy S , the pressure p, the chemical potential µ and the conjugate force X, respectively, as Applying the Legendre back-transformation to the function F and using Eqs. (8) and (18), we obtain the mean energy of the system as Let us rewrite the probabilities of microstates (9) in another representation. Substituting Eq. (9) into the function χ and using the relation for Z c and χ, we obtain 1 + (q − 1) Λ T = q χ + (q − 1) E T . Substituting this equation into Eq. (9), we have where χ = q/c. It can be observed that under the transformation q → 2 − q the probability of microstates (21) formally recovers the probability of microstates of the Tsallis statistics with escort probabilities [12,36]. Note that in the Gibbs limit q → 1 we obtain all the relations of the Boltzmann-Gibbs statistics in the canonical ensemble and the parameter X vanishes.

Grand canonical ensemble
The thermodynamic potential of the grand canonical ensemble is related to the fundamental thermodynamic potential, energy E, by the Legendre transform. Using Eq. (1), we have where E = i p i E i and N = i p i N i . Substituting Eqs. (22) and (2) into the Lagrange function of the grand canonical ensemble [30], Φ = Ω − λϕ, and using Eqs. (3) and (2), we obtain where Z q−1 Ω = q/χ 2 , λ is an arbitrary real constant and χ and Λ are defined below Eq. (9). Substituting p i into χ and using Z Ω and the relation between Z Ω and χ, we found an equation for Λ as Substituting Eq. (23) into Eqs. (1) and (22), and using Eq. (2), we have In the grand canonical ensemble the entropy S = S (T, V, z, µ). Thus, using Eq. (1) and the definition of z given in Eq. (4), we obtain exactly Eq. (13). Substituting Eq. (23) into Eq. (13) and using i d p i = 0 and the relation between Z Ω and χ, we obtain where X is the same as in Eq. (17). The differential of the mean energy E and the mean number of particles N can be written as where Then, from Eq. (29) and the Legendre transform (22) we have dΩ = −S dT − pdV −Xdz−Ndµ.
Taking the first derivative of the thermodynamic potential (22) with respect to temperature T , volume V, chemical potential µ and variable z, and using Eq. (23) and the relation between Z Ω and χ, and ∂E i /∂T = ∂N i /∂T = 0, i ∂p i /∂T = 0, we obtain the entropy S , pressure p, mean number of particles N and conjugate force X, respectively, as Applying the Legendre back-transformation to the function Ω and using Eqs. (22) and (30), we obtain the mean energy of the system as In the Gibbs limit q → 1 we obtain all the relations of the Boltzmann-Gibbs statistics in the grand canonical ensemble and the parameter X vanishes, X = 0.

Conclusions
In conclusion, we have introduced new nonextensive statistics based on the Landsberg-Vedral entropy. In its formulation we have used the standard linear expectation values constrained with the standard normalization condition for the probabilities of microscopic states of the system. The three ensembles were considered: microcanonical, canonical and grand canonical. The probabilities of microstates were obtained from the principle of maximum entropy using the method of the Lagrange multipliers. In each of these three ensembles we have derived exactly the first law of thermodynamics. We have also obtained the exact relations between the thermodynamic definitions of the thermodynamic quantities and their ensemble averages. The ensemble averages for the pressure, the chemical potential, the mean number of particles and the mean energy in both canonical and grand canonical ensembles are the same as those of the Boltzmann-Gibbs statistics with the exception of the form of the probability of microstates. In this formalism the Legendre transform is preserved. We have found that under the transformation q → 2 − q the probabilities of microstates for the nonextensive statistics based on the Landsberg-Vedral entropy formally recover the probabilities of microstates of the Tsallis statistics with escort probabilities. However, the nonextensive statistics based on the Landsberg-Vedral entropy does not require introduction of the complicated escort probabilities and generalized expectation values which lie in the definition of the Tsallis statistics with escort probabilities.