Nonextensive Statistical Mechanics

56 3 Generalizing What We Learntrange interactions (α/d > 1), and nonextensive for long-range interactions (0 ≤α/d ≤ 1). Satisfactorily enough, Eqs. (3.69) recover the characterization with Eq.(3.67) in the limit N →∞, but they have the great advantage of providing, forfinite N, afinite value. This fact will be now shown to enable to properly scale themacroscopic quantities in the thermodynamic limit (N → ∞), for all values ofα/d ≥ 0 (See Figs. 3.9 and 3.10).A totally similar situation occurs if we have, playing the role of Hamiltonian 3.65,say N coupled rotators localized on a lattice. We further detail this case later on.We are now prepared to address the thermodynamical consequences of the microscopicinteractions being short- or long-ranged ( [173], and references within [174]).To present a slightly more general illustration, we shall assume from now on that ourhomogeneous and isotropic classical fluid is made by magnetic particles. Its Gibbsfree energy is then given by87(a)N ∼ N = 1Fig. 3.9 The rescalingfunction Ñ(N,α/d) ≡N ∗ (N,α/d) + 1vs.α/d fortypical values of N (a), andvs. N for typical values ofα/d (b). For fixed α/d ≥ 0,Ñ monotonically increaseswith N increasing from 1 to∞;forfixedN > 1, Ñmonotonically decreases forα/d increasing from 0 to ∞.Ñ(N, 0) = N, thusrecovering the Mean FieldApproximation usualrescaling; lim N→∞ Ñdiverges for 0 ≤ α/d ≤ 1,thus separating the extensivefrom the nonextensive region;Ñ(∞,α/d) =(α/d)/[(α/d) − 1] ifα/d > 1; lim α/d→∞ Ñ = 1,thus recovering precisely thetraditional intensive andextensive thermodynamicalquantities; Ñ(N, 1) = ln N(from [176]).5N ∼4326543210N = 5N→∞0 1 2 3 α /d 4(b)α /d = 0α /d = 1α /d = 21α/d→∞1 2 3 4 N 5