Nonextensive Statistical Mechanics

320 8 Final Comments and Perspectivesit is so in the same sense that the BG theory is convenient. The next point thathas to be addressed in order to satisfactorily handle our initial question follows.Assuming that – because of its convenience and unifying power – we indeed wantto make, whenever possible, a statistical mechanical approach of a given problem,do we need a generalization of the BG theory? The answer is yes. For instance,quasi-stationary and other intermediate states are known to exist for long-rangeinteracting classical Hamiltonian systems whose one-particle velocity distributions(both ensemble-averaged and time-averaged) are not Gaussians. This excludes theexponential form of the BG distribution law for the stationary state. Indeed, themarginal probability for the one-particle velocities derived from an exponentialof the total Hamiltonian necessarily is Gaussian. Therefore, we definitively needsomething more general, if it can be formulated. Nonextensive statistical mechanics(as well as its variations such as the Beck–Cohen superstatistics, and others) appearsto be at the present time a strong operational paradigm. And this is so because ofa variety of reasons which include the following inter-related facts: (i) Many of thefunctions that emerge in long-range interacting systems are known to be preciselyof the q-exponential form; (ii) The entropy S q is consistent with nonergodic (and/orslowly mixing) occupancy of the space; (iii) The entropy S q is, in many nonlineardynamical systems, appropriate when the system is weakly chaotic (vanishingmaximal Lyapunov exponent); (iv) In the presence of long-range interactions, theelements of the system tend to evolve in a rather synchronized manner, which makesvirtually impossible an exponential divergence of nearby trajectories in space: thisprevents the system from quick mixing, and, in some cases, violates ergodicity, oneof the pillars of the BG theory; (v) The central limit theorem, on which the BGtheory is based, has been generalized in the presence of a (apparently large) classof global correlations, and the N →∞basic attractors are q-Gaussians (see, forinstance, [45,46,370,371]); (vi) The block entropy S q of paradigmatic Hamiltoniansystems in quantum entangled collective states is extensive only for a special valueof q which differs from unity (see [201, 202]).(l) Why do we need to use escort distributions and q-expectation values insteadof the ordinary ones?The essential mathematical reason for this can be seen in the set of Eq. (4.81) andthe following ones, and is based on connections that have been shown recently [258].When we are dealing with distributions that decay quickly at infinity (e.g., an exponentialdecay), then their characterization can be done with standard averages (e.g.,first and second moments). This is the typical case within BG statistical mechanics,and such moments precisely are the constraints that are normally imposed for theextremization of the entropy S BG . But if we are dealing with distributions that decayslowly at infinity (e.g., power-law decay), the usual characterization becomes inadmissiblesince all the moments above a given one (which depends on the asymptoticbehavior of the distribution) diverge. The characterization can, however, be donewith mathematically well-defined quantities by using q-expectation values (i.e.,with escort distributions). This is the typical case within nonextensive statisticalmechanics.