Nonextensive Statistical Mechanics

3.6 Nonextensive Statistical Mechanics and Thermodynamics 97(iv) Factorization of the likelihood function (thermodynamically independent systems).This property generalizes [218–220] the celebrated one introduced by Einsteinin 1910 [20] (reversal of Boltzmann formula). The likelihood function satisfiesW q ({p i }) ∝ e S q ({p i })q . (3.217)If A and B are two probabilistically independent systems, it can be immediatelyverified thatwhere we have used e S q (A)+S q (B)+(1−q)S q (A)S q (B)qW q (A + B) = W q (A) W q (B) (∀q) , (3.218)= e S q (A)qe S q (B)q .(v) Onsager reciprocity theorem (microscopic time reversibility). It has beenshown [221–223] that the reciprocal linear coefficients satisfyL jk = L kj (∀q) . (3.219)(vi) Kramers and Kronig relation (causality). Its validity has been proved [222]for all values of q.(vii) Pesin-like identity (relation between sensitivity to the initial conditions andthe entropy production per unit time). It has been conjectured [127] that the q-generalized entropy production per unit time (Kolmogorov-Sinai-like entropy rate)K q and the q-generalized Lyapunov coefficient λ q are related throughK q ={λ q if λ q > 0 ,0 otherwise.(3.220)The actual validity of this relation has been analytically proved and/or numericallyverified for various classes de models [128, 129, 131–133, 139–142, 146, 147, 150,153, 358]. We come back onto this identity later on. Indeed, as we shall see, Eq.(3.220) is in fact one among an infinite countable family of such relations.Properties (i) and (iii–vi) essentially reflect something quite basic. In the theorythat we are presenting here, we have generalized nothing concerning mechanics,either classical, quantum, or whatsoever. What we have generalized is the conceptof information upon mechanics. Consistently, the properties whose essential originlies in mechanics should be expected to be q-invariant, and we verify that indeedthey are.Some physical interpretations of nonextensive statistics are already available inthe literature [327–329]. We come back onto this question later on, in particular inconnection with the Beck–Cohen superstatistics.Let us mention also that various procedures that are currently used in BG statisticalmechanics have been q-generalized. These include the variational method [224–226], the Green-function methods [222, 225, 320, 330–333], the Darwin–Fowler