Nonextensive Statistical Mechanics

54 3 Generalizing What We Learntin a considerable number of examples [128–133, 139–142, 146, 147, 150, 153]. Weshall lengthily come back onto these questions in Chapter 5.If our weakly chaotic system has ν positive q-generalized Lyapunov coefficientsλ q(1) ,λsen q sen (2) ,...,λ q sen (ν) , we expect [172]11 − q ent=This yields, if all the q sen (k) are equal,ν∑k=111 − q (k)sen. (3.62)q ent = 1 − 1 − q sen. (3.63)νIf ν = 1, we recover Eq. (3.60). If q sen = 0, we obtainq ent = 1 − 1 ν . (3.64)3.3 Correlations, Occupancy of Phase-Space, and Extensivityof S q3.3.1 A Remark on the Thermodynamical LimitLet us assume a classical mechanical many-body system characterized by the followingHamiltonian:H = K + V =N∑i=1pi22m + ∑ V (r ij ) , (3.65)i≠ jwhere the two-body potential energy V (r) presents no mathematical difficulties nearthe origin r = 0 (e.g., in the r → 0 limit, either it is repulsive, or, if it is attractive, itis nonsingular or at least integrable), and which behaves at long distances (r →∞)likeV (r) ∼− A (A > 0; α ≥ 0) . (3.66)r αA typical example would be the d = 3 Lennard–Jones gas model, for whichα = 6. Were it not the stong singularity at the origin, another example would havebeen Newtonian d = 3 gravitation, for which α = 1.Let us analyze the characteristic average potential energy U pot per particleU pot (N)N∝−A∫ ∞1dr r d−1 r −α , (3.67)