On the Kaniadakis Distributions Applied in Statistical Physics and Natural Sciences.

Constitutive relations are fundamental and essential to characterize physical systems. By utilizing the κ-deformed functions, some constitutive relations are generalized. We here show some applications of the Kaniadakis distributions, based on the inverse hyperbolic sine function, to some topics belonging to the realm of statistical physics and natural science.

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics.

As known, a method to introduce non-conventional statistics may be realized by modifying the number of possible combinations to put particles in a collection of single-particle states. In this paper, we assume that the weight factor of the possible configurations of a system of interacting particles can be obtained by generalizing opportunely the combinatorics, according to a certain analytical function f{π}(n) of the actual number of particles present in every energy level. Following this approach, the configurational Boltzmann entropy is revisited in a very general manner starting from a continuous deformation of the multinomial coefficients depending on a set of deformation parameters {π}. It is shown that, when f{π}(n) is related to the solutions of a simple linear difference-differential equation, the emerging entropy is a scaled version, in the occupational number representation, of the entropy of degree (κ,r) known, in the framework of the information theory, as Sharma-Taneja-Mittal entropic form.

Equivalence between four versions of thermostatistics based on strongly pseudoadditive entropies.

The class of strongly pseudoadditive (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 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 pseudoadditive 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 supraextensive entropy, which has not previously been considered.

Cholesteric-nematic transition induced by a magnetic field in the strong-anchoring model.

We investigate the cholesteric-nematic transition induced by an external bulk field in a sample of finite thickness ℓ. The analysis is performed by considering a tilted magnetic field with respect to the easy direction imposed by rigid boundary conditions inducing planar orientation. In the case of parallel orientation between the magnetic field and of the easy direction, in the limit of ℓ→∞ we reobtain the results of de Gennes where the effective pitch of the cholesteric is a continuous function of the magnetic field diverging at the critical field related to the cholesteric-nematic transition. For finite ℓ we obtain a cascade of transitions, where the bulk expels a half-pitch at a time to avoid divergences in the elastic energy, in a similar manner as solids expel defects in the presence of strong deformation. In the case of oblique orientation between the magnetic field and the easy direction, only the completely untwisted state depends on the tilt angle. Therefore, only the cholesteric-nematic transition depends on the tilt angle while all the other magnetic transition values are unchanged.