Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 55where we have integrated from a typical distance (taken equal to unity) on. This isthe typical energy one would calculate within a BG approach. We see immediatelythat this integral converges for α/d > 1 (hereafter referred to as short-range interactionsfor classical systems) but diverges for 0 ≤ α/d ≤ 1 (hereafter referredto as long-range interactions). This already indicates that something anomalousmight happen. 3 By the way, it is historically fascinating the fact that Gibbs himselfwas aware of the possibility of such difficulty! (see, in Section 1.2, Gibbs’ remarksconcerning long-range interactions).On a vein slightly differing from the standard BG recipe, which would demandintegration up to infinity in Eq. (3.67), let us assume that the N-particle system isroughly homogeneously distributed within a limited sphere. Then Eq. (3.67) has tobe replaced by the following one:U pot (N)N∫ N 1/d∝−A dr r d−1 r −α =− A1d N ∗ , (3.68)withN ⋆ ≡ N 1−α/d − 11 − α/d⎧1if α/d > 1;α/d − 1⎪⎨= ln α/d N ∼ ln N if α/d = 1;(3.69)N 1−α/d⎪⎩1 − α/dif 0 1, and diverges like N 1−α/d /(1 − α/d) if0 ≤ α/d < 1 (it divergeslogarithmically if α/d = 1). 4 In other words, the energy is extensive for short-3 This is essentially the very same reason for which virtually all statistical mechanics textbooksdiscuss paradigmatic systems like a particle in a square well, the harmonic oscillator, the rigidrotator, and a spin 1/2 in the presence of an external magnetic field, but not the Hydrogen atom!All these simple systems, including of course the Hydrogen atom, are discussed in the quantummechanics textbooks. But, in what concerns statistical mechanics, the Hydrogen atom constitutesan illustrious absence. Amazingly enough, and in spite of the existence of an almost centennialrelated literature [160–171], this highly important system passes without comments in almost allthe textbooks on thermal statistics. The – understandable but not justifiable – reason of course isthat, since the system involves the long-range Coulombian attraction between electron and proton,the energy spectrum exhibits an accumulation point at the ionization energy (frequently taken tobe zero), which makes the BG partition function to diverge.4 These results turn out afterwards to be consistent with those discussed in relation to Eq. (1.67)of [340], in the frame of how strongly can N random variables be correlated, and be still applicableto the standard Central Limit Theorem, in the sense of the corresponding attractor be a Gaussiandistribution.