Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 59a simple hypercubic lattice with unit crystalline parameter we have r ij = 1, 2, 3,...if d = 1, r ij = 1, √ 2, 2,...if d = 2, r ij = 1, √ 2, √ 3, 2,...if d = 3, and so on.For such a case, we have thatN ⋆ ≡N∑i=2r −α1i, (3.75)which has in fact the same asymptotic behaviors as indicated in Eq. (3.69).In other words, here again α/d > 1 corresponds to short-range interactions, and0 ≤ α/d ≤ 1 corresponds to long-range ones.For example, the α/d = 0 particular case corresponds to the usual mean fieldapproach. Indeed, in this case we have N ∗ = N − 1 ∼ N, which is equivalent tothe usual rescaling of the microscopic coupling constant through division by N (seealso [177]). In fact, to accommodate with the common use of dividing by N (insteadof N − 1) for the α/d = 0 case, it is sometimes practical to use, as done in Fig. 3.9,Ñ ≡ N ∗ + 1 = N 1−α/d − (α/d)1 − α/d. (3.76)For short-range interactions, N ∗ → constant, consequently we recover the usualextensivity of Gibbs, Helmholtz, and internal thermodynamical energies, entropy,volume, and magnetization, as well as the intensivity of temperature, pressure, andmagnetic field. But for long-range interactions, N ∗ diverges with N, therefore thesituation is quite more subtle. Indeed, in order to have nontrivial equations of stateswe must express the nonextensive Gibbs, Helmholtz, and internal thermodynamicalenergies, as well as the extensive entropy, volume and magnetization in termsof the rescaled variables (T ∗ , p ∗ , H ∗ ). In general, i.e., ∀ (α/d), we see that thevariables that are intensive when the interactions are short-ranged remain a singleclass (although scaling with N ∗ ) in the presence of long-ranged interactions. But,in what concerns the variables that are extensive when the interactions are shortranged,the situation is more complex. Indeed, they split into two classes. One ofthem contains all types of thermodynamical energies (G, F, U), which scale withNN ∗ . The other one contains all those variables (S, V, M) that appear in pairs inthe thermodynamical energies. These variables remain extensive, in the sense thatthey scale with N. 5By no means this implies that thermodynamical equilibrium between two systemsoccurs in general when they share the same values of say (T ∗ , p ∗ , H ∗ ). Itonly means that, in order to have finite mathematical functions for their equations ofstates, the variables (T ∗ , p ∗ , H ∗ ) must be used. Although this has to be verified,thermodynamical equilibrium might still be directly related to sharing the usualvariables (T, p, H).5 Consequently, for 0 ≤ α/d < 1, we expect U(N, T ) ∼ N 2−α/d u(T/N 1−α/d ), S(N, T ) ∼Ns(T/N 1−α/d ), the specific heat C(N, T ) ∼ Nc(T/N 1−α/d ), etc.