Nonextensive Statistical Mechanics

36 2 Learning with Boltzmann–Gibbs Statistical MechanicsWe can prove also thatthat the Helmholtz free energy is given by1T = S BGU , (2.66)F BG ≡ U − TS BG =− 1 β ln Z BG , (2.67)and that the internal energy is given byU =− β ln Z BG . (2.68)In the limit T →∞we recover the microcanonical ensemble.2.4.3 OthersThe system may be exchanging with the thermostat not only energy, so that thetemperature is that of the thermostat, but also particles, so that also the chemicalpotential is fixed by the reservoir. This physical situation corresponds to the socalledgrand-canonical ensemble. This and other similar physical situations can betreated along the same path, as shown by Gibbs. We shall not review here thesetypes of systems, which are described in detail in [35], for instance.Another important physical case, which we do not review here either, is when theparticles cannot be considered as distinguishable. Such is the case of bosons (leadingto Bose–Einstein statistics), fermions (leading to Fermi–Dirac statistics), and the socalledgentilions (leading to Gentile statistics, also called parastatistics [101–103],which unifies those of Bose–Einstein and Fermi–Dirac).All these various physical systems, and even others, constitute what is currentlyreferred to as BG statistical mechanics, essentially because at its basis we find, inone way or another, the entropic functional S BG . It is this entire theoretical bodythat in principle we intend to generalize in the rest of the book, through the generalizationof S BG itself.