Functional background of the Tsallis entropy: "coarse-grained" systems and"kappa" distribution functions

The concept of the generalized entropy is analyzed, with the particular att ention to the definition postulated by Tsallis [J. Stat. Phys. 52, 479 (198 8)]. We show that the Tsallis entropy can be rigorously obtained as the sol ution of a nonlinear functional equation; this equation represents the entr opy of a complex system via the partial entropies of the subsystems involve d, and includes two principal parts. The first part is linear (additive) an d leads to the conventional, Boltzmann, definition of entropy as the logari thm of the statistical weight of the system. The second part is muitiplicat ive and contains all sorts of multilinear products of the partial entropies ; inclusion of the multiplicative terms is shown to reproduce the generaliz ed entropy exactly in the Tsallis sense. We speculate that the physical bac kground for considering the multiplicative terms is the role of the long-ra nge correlations supporting the "macroscopic" ordering phenomena (e.g., for mation of the "coarse-grained" correlated patterns). We prove that the cano nical distribution corresponding to the Tsallis definition of entropy, coin cides with the so-called "kappa" distribution which appears in many physica l realizations. This has led us to associate the origin of the "kappa" dist ributions with the "macroscopic" ordering ("coarse-graining") of the system . Our results indicate that an application of the formalism based on the Ts allis notion of entropy might actually have sense only for the systems whos e statistical weights, Omega, are relatively small. (For the "coarse-graine d" systems, the weight Omega could be interpreted as the number of the "gra ins".) For large Omega (i.e., Omega --> infinity), the standard statistical mechanical formalism is advocated, which implies the conventional, Boltzma nn definition of entropy as In Omega.