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.

Universality of kappa distributions

“…( 11) to follow the partitioning Φ A⊕B = Φ A • Φ B . This is physically consistent only with a power law of probabilities [9], hence, we find Φ({p i })…”

“…The existence of particle correlations adds order to the whole system, and, thus, decreases its total disorder, the entropy, according to S A+B = S A + S B − S D . The decrease of entropy S D is called entropy defect [1,[6][7][8][9][10][11][12]. This concept is analogous to the mass defect that arises when nuclear particle systems are assembled: the total mass is less than the sum of the assembled mass by the defect amount M D , because of the mass energy spent in the fields binding the particles together (fig.…”