Nonextensive Statistical Mechanics

310 8 Final Comments and Perspectivesinitial conditions) is of the q-exponential form, in contrast with extensive systems,which are therefore those whose stationary-state (thermal equilibrium) distribution(or similar properties) is of the usual BG exponential form. So, in extensive (BG)statistical mechanics, both the total energy and the total entropy are additive andextensive, whereas, in nonextensive statistical mechanics, the total energy is nonextensivebut the total entropy is nonadditive and extensive! Regretfully it remainstrue that there was an inadvertence when the book [69] was named “NonextensiveEntropy” instead of “Nonadditive Entropy”!(c) How come ordinary differential equations play an important role in nonextensivestatistical mechanics?Some remarks related to ordinary differential equations might surprise somereaders, hence deserve a clarification. Indeed, in virtually all the textbooks of statisticalmechanics, functions such as the energy distribution at thermal equilibrium arediscussed using a variational principle, namely referring to the entropy functional,and not using ordinary differential equations and their solutions. In our opinion, itis so not because of some basic (and unknown) principle of exclusivity, but ratherbecause the first-principle dynamical origin of the BG factor still remains, mathematicallyspeaking, at the status of a dogma [34]. Indeed, as already mentioned, tothe best of our knowledge, no theorem yet exists which establishes the necessaryand sufficient first-principles conditions for being valid the use of the celebratedBG factor. Moreover, one must not forget that it was precisely through a differentialequation that Planck heuristically found, as described in his famous October 1900paper [312, 831], 1 the black-body radiation law. It was only in his equally famousDecember 1900 paper that he made the junction with the – at the time, quite controversial– Boltzmann factor by assuming the – at the time, totally bizarre – hypothesisof discretized energies.A further point which deserves clarification is why have we also interpreted thelinear ordinary differential equation in Section 5.5 as providing the typical time evolutionof both the sensitivity to the initial conditions and the relaxation of relevantquantities. Although the bridging was initiated by Krylov [832], the situation still isfar from completely clear on mathematical grounds. However, intuitively speaking,it seems quite natural to think that the sensitivity to the initial conditions is preciselywhat makes the system to relax to equilibrium, and therefore opens the door for the1 The celebrated equation in Planck’s 19 October 1900 paper is −( 2 S) −1 = αU + βU 2 (whereU 2α and β are constants), the heuristic interpolation between a term proportional to U and one proportionalto U 2 . By replacing in this equation the thermodynamic relation SU = T −1 , one obtainsU(1/T ) =−αU−βU 2 , which is precisely the q = 2 particular case of the differential equation (6.1)!From the solution of this equation (see Eq. (6.2)), Planck readily arrived to his famous black-bodyradiation law u(ν, T ) = (aν 3 /c 3 )/(e bν/T − 1). Two months later, in his 14 December 1900 paper,by incorporating a discretized energy within Boltzmann’s thermostatistical theory, he obtained theform which is used nowadays, namely u(ν, T ) = (8πν 2 /c 3 )hν/(e hν/kT −1) (where b was replacedby h/k). The constant k (introduced and named Boltzmann constant by Planck) was the ratiobetween the gas constant R and the Avogadro number N ; the constant h was obtained by fittingthe black-body experimental data available at the time.