Extensive entropy: the case of zero entropy defect

Abstract

This paper shows that the Rényi and Boltzmann-Gibbs (BG) extensive entropies share the same functional relationship with the nonextensive entropy associated with kappa distributions, which coincides with the well-known Havrda/Charvát/Daróczy/Tsallis (HCDT) entropy. We find that while the relationship between kappa/HCDT and Rényi entropies is merely a mathematical identity between their entropic statistical definitions, the relationship between kappa/HCDT and BG entropies is based on their thermodynamic connection. The latter connects the entropy between a system characterized by correlations among their all constituents (kappa/HCDT entropy) and the entropy of the same system but with no correlations among their constituents (BG entropy). The origin of this relationship, and its connection with thermodynamics, is examined using the concept of entropy defect, that is, the decrease in a system’s entropy caused by the presence of long-range correlations among its constituents; in the limiting case of zero correlations, the entropy defect vanishes and the entropy becomes extensive and expressed by the BG formulation.