Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 67W effA+B ≤ W A+B . (3.112)If A and B are independent, then the equality holds. The opposite is not true:correlation might exist between A and B and, nevertheless, the equality be satisfied.It is not, however, this kind of (weak) correlation that we are interested here. Ourfocus is on a special type of (strong) correlation, which necessarily decreases thenumber of joint states whose probability differs from zero. More specifically, wefocus on a correlation such thatW effA+B = W A ⊗ q W B = (W 1−qA+ W 1−qB− 1) 1/(1−q) (q ≤ 1) . (3.113)We can verify that W effA+B /W A+B generically decreases from unity to zero whenq decreases from unity to −∞.Let us generalize the above to N subsystems A 1 , A 2 ,...,A N (they typically arethe elements of the system) whose numbers of states (with nonzero probabilities)are, respectively, W A1 , W A2 ,...,W AN . We then haveandW A1 +A 2 +...+A N=N∏W Ar , (3.114)r=1W ef fA 1 +A 2 +...+A N= W A1 ⊗ q W A2 ⊗ q ...⊗ q W AN =[( N ∑r=1) ]W Ar − (N − 1) . (3.115)It will generically be W effA 1 +A 2 +...+A N≤ W A1 +A 2 +...+A Nfor q ≤ 1, the equality genericallyholds for and only for q = 1.A frequent and important case is that in which the N subsystems are all equal(hence W Ar = W A1 ≡ W 1 , ∀r). In such a case, we haveW eff (N) = [NW 1−q1− (N − 1)] 1/(1−q) ≤ W N 1 (q ≤ 1) , (3.116)where the notation W eff (N) is self-explanatory. This equality immediately yieldsthe following very suggestive result:ln q [W eff (N)] = N ln q W 1 . (3.117)If q = 1, W eff (N) = W (N) = W1 N , and this result recovers the well-knownadditivity of S BG , i.e., S BG (N) = NS BG (1) for the case of equal probabilities. Indeed,in the q = 1 case, the hypothesis of simultaneously having equal probabilitiesin each of the N equal subsystems as well as in the total system is admissible: theprobability of each state of any single subsystem is 1/W 1 , and the probability ofeach state of the entire system is 1/W = 1/W1 N .