Nonextensive Statistical Mechanics

90 3 Generalizing What We Learntwhere we have used condition (3.181) to eliminate the Lagrange parameter α. Thisdistribution can be straightforwardly rewritten aswithp opt (x) =β (2)′q≡e −β(2)′ q x 2q∫ ∞, (3.185)−∞ dx′ e −β(2)′ x ′2qβ q(2)1 + (1 − q) β q (2) X q(2) . (3.186)We thus see that, in the same way Gaussians are deeply connected to S BG ,thepresent distributions, frequently referred to as q-Gaussians, are connected to the S qentropy.3.5.3 OthersA quite general situation would be to impose, in addition to∫dx p(x) = 1 , (3.187)the constraint∫〈 f (x)〉 q ≡dx f(x) P(x) = F q , (3.188)where f (x) is some known function and F q a known number. We obtainp opt (x) =e −β q ( f (x)−F q )q∫dx′ e −β q ( f (x ′ )−F q. (3.189))qAs for the BG case, it is clear that, by appropriately choosing f (x), we can forcep opt (x) to be virtually any distribution we wish. For example, by choosing f (x) =|x| γ (γ ∈ R), we obtain a generic stretched q-exponential p opt (x) ∝ e −β|x|γq .3.6 Nonextensive Statistical Mechanics and ThermodynamicsWe arrive now to the central goal of the present introduction to nonextensive statisticalmechanics. This theory was first introduced in 1988 [39] as a possible generalizationof Boltzmann–Gibbs statistical mechanics. The idea first emerged in mymind in 1985 during a meeting in Mexico City. The inspiration was related to the