Thermostatistical aspects of generalized entropies

Abstract

We investigate the properties concerning a class of generalized entropies given by S q, r = k{1−[∑ i p i q ] r }/[ r( q−1)] which include Tsallis’ entropy ( r=1), the usual Boltzmann–Gibbs entropy ( q=1), Rényi’s entropy ( r=0) and normalized Tsallis’ entropy ( r=−1). In order to obtain the generalized thermodynamic relations we use the laws of thermodynamics and considering the hypothesis that the joint probability of two independent systems is given by p ij A∪ B = p i A p j B . We show that the transmutation which occurs from Tsallis’ entropy to Rényi’s entropy also occur with S q, r . In this scenario, we also analyze the generalized variance, covariance and correlation coefficient of a non-interacting system by using extended optimal Lagrange multiplier approach. We show that the correlation coefficient tends to zero in the thermodynamic limit. However, Rényi’s entropy related to this non-interacting system presents a certain degree of non-extensivity.