Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 57Fig. 3.10 The so-called extensive systems (α/d > 1 for the classical ones) typically involve absolutelyconvergent series, whereas the so-called nonextensive systems (0 ≤ α/d < 1 for the classicalones) typically involve divergent series. The marginal systems (α/d = 1 here) typically involveconditionally convergent series, which therefore depend on the boundary conditions, i.e., typicallyon the external shape of the system. Capacitors constitute a notorious example of the α/d = 1 case.The model usually referred to in the literature as the Hamiltonian-Mean-Field (HMF) one lies onthe α = 0axis(∀d > 0). The model usually referred to as the d-dimensional α-XY model [177]lies on the vertical axis at abscissa d (∀α ≥ 0).G(N, T, p, H) = U(N, T, p, H) − TS(N, T, p, H) + pV(N, T, p, H)− HM(N, T, p, H) , (3.70)where (T, p, H) correspond, respectively, to the temperature, pressure, and externalmagnetic field, U is the internal energy, S is the entropy, V is the volume, and Mthe magnetization.If the interactions are short-ranged (i.e., if α/d > 1), we can divide this equationby N and then take the N →∞limit. We obtaing(T, p, H) = u(T, p, H) − Ts(T, p, H) + pv(T, p, H) − Hm(T, p, H) , (3.71)where g(T, p, H) ≡ lim N→∞ G(N, T, p, H)/N, and analogously for the othervariables of the equation.If the interactions are instead long-ranged (i.e., if 0 ≤ α/d ≤ 1), all these quantitiesdiverge, hence thermodynamically speaking they are nonsense. Consequently,the generically correct procedure, i.e., ∀ α/d ≥ 0, must conform to the followinglines:G(N, T, p, H) U(N, T, p, H)lim= limN→∞ NN ⋆N→∞ NN ⋆+ limN→∞− limN→∞p V (N, T, p, H)− limN ⋆ NN→∞T S(N, T, p, H)N ⋆ NH M(N, T, p, H)N ⋆ N(3.72)hence