Legendre structure of the thermostatistics theory based on the Sharma–Taneja–Mittal entropy

The statistical proprieties of complex systems can differ deeply for those of classical systems governed by Boltzmann–Gibbs entropy. In particular, the probability distribution function observed in several complex systems shows a power-law behavior in the tail which disagrees with the standard exponential behavior showed by Gibbs distribution. Recently, a two-parameter deformed family of entropies, previously introduced by Sharma, Taneja and Mittal (STM), has been reconsidered in the statistical mechanics framework. Any entropy belonging to this family admits a probability distribution function with an asymptotic power-law behavior. In the present work we investigate the Legendre structure of the thermostatistics theory based on this family of entropies. We introduce some generalized thermodynamical potentials, study their relationships with the entropy and discuss their main proprieties. Specialization of the results to some one-parameter entropies belonging to the STM family are presented.