Equivalence between four versions of thermostatistics based on strongly pseudoadditive entropies

Abstract

The class of strongly pseudo-additive 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 generalized Shannon-Khinchin 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 the strongly pseudo-additive 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.

New probability distribution describing emergence in state space

“…Here we will concentrate on the statistics of a minimal model such of cases, namely the pairing model introduced in [2]. The paring model has been studied in the context of relating the N dependence of W (N ) to generalised entropies in a number of publications, see [3,4,5,6,7,8,9,10,11,12]. One may think of the components of the paring model as coins.…”

A study of Rényi entropy based on the information geometry formalism

“…For this reason, both the entropies S R α (p) or S T α (p) give the same actual form for the optimized distribution family under the same constraints (i.e. equations (1.2) versus (1.8)), although redefine the Lagrange multipliers [35]. In an information geometry language, this means that they share the same statistical manifold but with a different definition of the vector parameter θ so that they recover different geometric structures.…”