Nonextensive Statistical Mechanics

7.1 Physics 2417.1.8 Astrophysics7.1.8.1 Self-Gravitating SystemsA vast literature explores the possible connections of q-statistics with self-gravitatingsystems and related astrophysical phenomena. The first such connection wasestablished in 1993 by Plastino and Plastino [217]. 2 It provided a possible wayout for an old gravitational difficulty, namely the impossibility of existence of aself-gravitating system such that its total mass, total energy, and total entropy areall three simultaneously finite. Within a Vlasov–Poisson polytropic description ofa Newtonian self-gravitating system (i.e., D = 3), a connection was put forwardbetween the polytropic index n and the entropic index q, namely (see [481,482] andreferences therein)11 − q = n − 1 2 . (7.6)The limit n →∞(hence q = 1) recovers the isothermal sphere case (responsiblefor the paradox mentioned previously); n = 5 (hence q c = 7/9, where the subindexc stands for critical) corresponds to the so-called Schuster sphere; for n < 5 (henceq < q c = 7/9), simultaneous finiteness of mass, energy, and entropy naturallyemerges. Equation (7.6) can be generalized to the D-dimensional Vlasov–Poissonproblem, and the following result is obtained [481]11 − q = n − D − 2 . (7.7)2The critical case corresponds to the Schuster D-dimensional sphere, for whichReplacing this expression into Eq. (7.7), we obtainq c (D) =n = D + 2D − 2 . (7.8)8 − (D − 2) 28 − (D − 2) 2 + 2(D − 2) . (7.9)We see that q c decreases below unity when D increases above D = 2. Thefact that the limiting case q c = 1 occurs at D = 2 is quite natural. Indeed, theD-dimensional gravitational potential energy decays, for D > 2, as −1/r D−2 withdistance r. Consequently, the dimension below which BG statistical mechanics canbe legitimately used is precisely D = 2.2 This contribution constitutes in fact a historical landmark in nonextensive statistical mechanics.Indeed, it was the very first connection of the present theory with any concrete physical system.