Equilibrium statistical mechanics for incomplete nonextensive statistics

Abstract

The incomplete nonextensive statistics in the canonical and microcanonical ensembles is explored in the general case and in a particular case for the ideal gas. By exact analytical results for the ideal gas it is shown that taking the thermodynamic limit, with $z=q/(1-q)$ being an extensive variable of state, the incomplete nonextensive statistics satisfies the requirements of equilibrium thermodynamics. The thermodynamical potential of the statistical ensemble is a homogeneous function of the first degree of the extensive variables of state. In this case, the incomplete nonextensive statistics is equivalent to the usual Tsallis statistics. If $z$ is an intensive variable of state, i.e. the entropic index $q$ is a universal constant, the requirements of the equilibrium thermodynamics are violated.Comment: 7 page

Foundation of Equilibrium Statistical Mechanics Based on Generalized Entropy

“…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 z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17].…”

“…In modern physics, there exist alternative theories for the equilibrium statistical mechanics [1,2] based on the generalized statistical entropy [3][4][5][6][7][8][9][10][11][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.…”

Statistical Distribution of a Completely Open System Based on Incomplete Shannon Entropy

“…(4)(5) and the corresponding thermodynamic formulas in Eqs. (6)(7)(8)(9)(10)(11).When the deformation parameter q 1,both results reduce to the statistical distribution deduced by T. L. Hill by the method of ensemble transformation in small systems [11]. However, our approach belongs to the family of non-extensive statistical mechanics, and is suitable for systems with any size.…”

“…Wang also discussed some physical systems on the basis of incomplete entropy [3][4][5] and incomplete Shannon entropy [6,7]. Others made a contribution to the field [8][9][10].…”

Nonextensive statistical effects in the quark-gluon plasma formation at relativistic heavy-ion collisions energies

“…The behavior of hot, dense nuclear matter is usually described in terms of the so-called relativistic mean field models, wherein thermodynamical quantities are usually obtained through the common Boltzmann-Gibbs statistics. But, in recent years, there have been growing theoretical and experimental indications that, under extreme conditions and during the possible phase transition to quark-gluon matter reachable in high energy heavy ion collisions, strong dynamical correlations, long-range color interactions and microscopic memory effects can take place [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In this context the non-extensive statistical mechanics proposed by Tsallis [1][2][3] can be used to describe and investigate such physical phenomena.…”