Av. Milovanov et Lm. Zelenyi, Functional background of the Tsallis entropy: "coarse-grained" systems and"kappa" distribution functions, NONL PR GEO, 7(3-4), 2000, pp. 211-221
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.