Thermodynamic equilibrium and its stability for microcanonical systems described by the Sharma-Taneja-Mittal entropy

Abstract

It is generally assumed that the thermodynamic stability of equilibrium states is reflected by the concavity of entropy. We inquire, in the microcanonical picture, about the validity of this statement for systems described by the two-parametric entropy S(kappa,r) of Sharma, Taneja, and Mittal. We analyze the "composability" rule for two statistically independent systems A and B, described by the entropy S(kappa,r) with the same set of the deformation parameters. It is shown that, in spite of the concavity of the entropy, the "composability" rule modifies the thermodynamic stability conditions of the equilibrium state. Depending on the values assumed by the deformation parameters, when the relation S(kappa,r)(A union B) > S(kappa,r)(A) + S(kappa,r)(B) holds (superadditive systems), the concavity condition does imply thermodynamics stability. Otherwise, when the relation S(kappa,r)(A union B) < S(kappa,r)(A) + S(kappa,r)(B) holds (subadditive systems), the concavity condition does not imply thermodynamical stability of the equilibrium state.

Lie symmetries and related group-invariant solutions of a nonlinear Fokker-Planck equation based on the Sharma-Taneja-Mittal entropy

“…The analysis showed that the generalized Gaussian function, obtained by replacing the standard exponential with its generalized version, is recurrent in the expression of several GIS. In fact, it models the stationary state (29) as well as the traveling wave (36) and, limiting to the q-case, also the self-similar solution (46). In general, the (κ, r)-Gaussian is not a scale invariant solution of the STM-NFPE although it plays a role in the study of the evolution of localized initial states driven by the purely diffusive equation (57) related to the STM-NFPE.…”

“…Trivially, the invariant solution under the action of the generator χ 1 corresponds to the stationary state (29) considered in the previous section. Also trivial is the invariant solution related to the generator χ 2 , which produces a timedependent space translation.…”

“…The thermostatistics theory based on the STM-entropy fulfills the Legendre structure [29]. The main proprieties of a statistical system described by this entropy, in the microcanonical formalism, has been investigated in [30].…”

“…for [29]. The thermostatistics theory based on the STM-entropy fulfills the Legendre structure [29].…”

A mechanism to derive multi-power law functions: An application in the econophysics framework

“…Finally, accounting for the solution (2.5), the entropy (2.1) assumes the form 6) which recovers, in the limit (κ, r) → (0, 0), the Shannon-Boltzmann-Gibbs entropy S = − p(x) ln p(x) dx. This entropic form, introduced previously in literature in [13,14,15], is known as the Sharma-Taneja-Mittal information measure and has been applied recently in the formulation of a possible thermostatistics theory [16,17].…”