Nonextensive Statistical Mechanics

Chapter 6Generalizing Nonextensive Statistical MechanicsAqui... onde a terra se acaba e o mar começa...Luís Vaz de CamõesCantoOitavo–LUSÍADASWe have schematically represented in Fig. 6.1 the various thermostatisticaltheories that are in principle possible. The present chapter is dedicated to a briefexploration of the non q-describable region.6.1 Crossover StatisticsEquations (5.49) (paradigmatic for BG statistics) and (5.54) (paradigmatic fornonextensive statistics) can be unified in the following one [282]:dydx =−a 1 y − (a q − a 1 ) y q . (6.1)We recover the BG equation for q = 1(∀ a q )orfora q = a 1 (∀q). We recoverthe nonextensive equation for a 1 = 0. The instances of Eq. (6.1) for which q isa natural number are particular cases of the Bernoulli differential equations [382].The solution of Eq. (6.1) is given by1y = [ a 1 −qa 1+ a qa 1e ] (q−1) a 1 x 1q−1(x ≥ 0) . (6.2)It can be straightforwardly verified that it contains, as particular instances, thesolutions of Eqs. (5.49) and (5.54). We can also verify that⎧1 − a q x if 0 ≤ x