Thermostatistical aspects of generalized entropies

We investigate the properties concerning a class of generalized entropies given by Sq,r=k{1−[∑ipiq]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 pijA B=piApjB. We show that the transmutation which occurs from Tsallis’ entropy to Rényi’s entropy also occur with Sq,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.