On the equivalence between four versions of thermostatistics based on strongly pseudo-additive entropies

Abstract

The class of SPA entropies, which can be represented as an increasing continuous transformation of Shannon and Rényi entropies, have intensively been studied in previous decades. Although their mathematical structure has thoroughly been explored and established by GSK axioms, the analysis of their thermostatistical properties have mostly been limited to special cases which belong to two parameter Sharma-Mittal entropy class, such as Tsallis, Renyi and Gaussian entropies. In this paper we present a general analysis of SPA entropies thermostatistics by taking into account both linear and escort constraints on internal energy. We develop two types of dualities between the thermostatistics formalisms. By the first one, the formalism of Rényi entropy is transformed in the formalism of SPA entropy under general energy constraint and, by the second one, the generalized thermostatistics which corresponds to the linear constraint is transformed into the one which corresponds to the escort constraint. Thus, we establish the equivalence between four different thermostatistics formalisms based on Rényi and SPA entropies coupled with linear and escort constraints and we provide the transformation formulas. In this way we obtain a general framework which is applicable to the wide class of entropies and constraints previously discussed in the literature. As an example, we rederive maximum entropy distributions for Sharma-Mittal entropy and we establish new relationships between the corresponding thermodynamic potentials. We obtain, as special cases, previously developed expressions for maximum entropy distributions and thermodynamic quantities for Tsallis, Rényi and Gaussian entropies. In addition, the results are applied for derivation of thermostatistical relationships for supra-extensive entropy, which has not previously been considered.

arXiv comment15 pages, 1 figure

References in corpus (23)

• A step beyond Tsallis and Renyi entropies
• Generalized algebra within a nonextensive statistics
• Nonextensive thermodynamic relations
• Generalized entropies and the transformation group of superstatistics
• A critique of non-extensive q-entropy for thermal statistics
• Group entropies, correlation laws and zeta functions
• Zeroth Law compatibility of non-additive thermodynamics
• Maximum Entropy Principle in statistical inference: case for non-Shannonian entropies
• General entropy-like uncertainty relations in finite dimensions
• Connections between Tsallis' formalisms employing the standard linear average energy and ones employing the normalized $q$-average energy
• Formal Groups and $Z$-Entropies
• Generalized Shannon-Khinchin Axioms and Uniqueness Theorem for Pseudo-additive Entropies
• On the thermodynamic stability conditions of Tsallis' entropy
• On q-non-extensive statistics with non-Tsallisian entropy
• On the robust thermodynamical structures against arbitrary entropy form and energy mean value
• A new entropy based on a group-theoretical structure
• Consistency of the structure of Legendre transform in thermodynamics with the Kolmogorov-Nagumo average
• The world according to Renyi: Thermodynamics of multifractal systems
• Properties of the Nonextensive Gaussian entropy
• A unified characterization of generalized information and certainty measures
• Thermal and mechanical equilibrium among weakly interacting systems in generalized thermostatistics framework
• Information geometric duality of $\varphi$-deformed exponential families
• Comments on "On q-non-extensive statistics with non-Tsallisian entropy"

Cited by in corpus (2)

• When Shannon and Khinchin meet Shore and Johnson: equivalence of information theory and statistical inference axiomatics
• Stochastic thermodynamics and fluctuation theorems for non-linear systems