Nonextensive Statistical Mechanics

322 8 Final Comments and PerspectivesFurther analytical expressions for q in a variety of other physical systems arepresented in [232]. See also [A.B. Adib, A.A. Moreira, J.S. Andrade Jr. and M.P.Almeida, Tsallis thermostatistics for finite systems: A Hamiltonian approach,PhysicaA 322, 276 (2003)] for connections between q and finite-sized systems.As we readily verify, in some cases q characterizes universality classes (ofnonadditivity), in total analogy with those of standard critical phenomena. Relations(5.58), (3.145), and (5.11) constitute such examples. In other cases, analogouslyto the two-dimensional short-range-interacting isotropic XY ferromagneticmodel and to the Baxter line of the square-lattice Ashkin–Teller ferromagnet[233] (whose critical exponents depend on the temperature and on the detailsof the Hamiltonian), q depends on model details. Relations (7.36), (4.107),and (4.109) constitute such examples. A case which is believed to be of the universalityclass type is that of classical long-range Hamiltonian systems. The indexq is expected to depend only on α (which characterizes the range of theforces) and on d (spatial dimension of the system), possibly even only on α/d.However, this remains an open problem at the time when this book is beingwritten.(n) Why are there so many different values of q for the same system?The basic function ubiquitously emerging in the BG theory is a very universalone, namely the exponential one. It is present in the sensitivity to the initial conditions,in the relaxation of many physical quantities, in the distribution of energystates at thermal equilibrium (in particular, in the distribution of velocities), in thesolution of the linear Fokker–Planck equation in the absence of external forces (andeven for linear external forces), in the attractor in the sense of the Central Limit Theorem(CLT). In all these cases, the only quantity which is not universal is the scaleof the independent variable. Of course, functions different from the exponential alsoappear in BG statistical mechanics, but at the crucial and generic points we find itagain and again.For many complex systems (the realm of nonextensive statistical mechanics), thisfunction is generalized into a less universal one, namely the q-exponential function(a power-law, in the asymptotic region). It is this one which ubiquitously emergesnow at the same crucial and generic points. The q-exponential function depends notonly on the scale, but also on the exponent (i.e., on the value of q) of the powerlaw.Therefore, for a given system, different physical quantities are associated withdifferent values of q. The indices q are expected to appear in the theory in infinitenumber. However, only a few of them should be necessary to characterize the mostimportant features of the system. And several of these few are expected to be interrelatedin such a way that only very few would be independent. A paradigmatic casehas been analytically shown to occur in the context of the q-generalization of theCLT: see Eq. (4.91) and Fig. 4.20. Once the values of α and q ≡ q α,0 are fixed, theentire family of infinite countable indices q is uniquely determined. Analogously,it is expected that, for classical d-dimensional long-range-interacting many-bodyHamiltonians, all relevant values of q would be fixed once the exponent α (whichfixes how quickly the force decays with distance, independently from the intensityof the force as long as it is nonzero) is fixed.