Nonextensive Statistical Mechanics

76 3 Generalizing What We Learntprobabilities are redistributed into a small number of all the other possible states,in such a way that norm is preserved. Notice in both Tables 3.8 and 3.9 that only a“left” strip of width d + 1 has nonvanishing probabilities. All the other probabilitiesare strictly zero. To complete the description of these models we need to indicatethe values of the nonvanishing probabilities.The first model (Table 3.8), hereafter referred to as the restricted uniform one,has, for a fixed value of N, all nonvanishing r Nn equal. This is to saywithr (d)N,n = 1/2N (if N ≤ d) ,r (d)N,n = 1W ef f (N, d)(if N > d and n ≤ d) , (3.133)r (d)N,n= 0 (if N > d and n > d) ,W ef f (N, d) =d∑n=0N!(N − n)! n! , (3.134)where eff stands for effective. For example, W eff (N, 0) = 1, W eff (N, 1) = N + 1,W eff (N, 2) = 1 2 N(N + 1) + 1, W eff (N, 3) = 1 6 N(N 2 + 5) + 1, and so on. For fixedd and N →∞, we have thatThe entropy is given byW eff (N, d) ∼ N dd! ∝ N d . (3.135)S q (N) = ln q W eff (N, d) . (3.136)Therefore, by using Eq. (3.120), we obtain that S q (N) is extensive if and only ifq = 1 − 1 d . (3.137)If q > 1 − 1 d (q < 1 − 1 d ) we have that lim N→∞ S q (N)/N vanishes (diverges).But this limit converges to a finite value for the special value of q. More precisely,S 1−1/d (N)limN→∞ N= d . (3.138)(d !)1/dLet us address now the second model (Table 3.9). The probabilities are given by