Nonextensive Statistical Mechanics

8.1 Falsifiable Predictions and Conjectures, and Their Verifications 307S q (t) would increase linearly with t only for q = q sen , and that the slope (entropyproduction per unit time) would satisfy K qsen = λ qsen , thus q-generalizing the q = 1Pesin-like identity K 1 = λ 1 . This scenario was verified in various systems since1997, and analytically proved since 2002: see [128–133, 139–142, 146, 147, 150,153, 172, 358], among others.(e) Scaling with N ∗ for long-range-interacting systemsAn important class of two-body potentials V (r) ind dimensions consists in beingsmooth or integrable at short distances, and satisfying V (r) ∼ − A (A >r α0; α ≥ 0) at long distances. If the system is classical, such potentials are said shortrange-interactingif α/d > 1, and long-range-interacting if 0 ≤ α/d ≤ 1 (see, forinstance, Eq. (3.69)). The usual thermodynamical recipes address the short-rangecases. Special scaling must be used in the long-range cases.Since the successful verification done in 1995 [869] for ferro-fluids, it becamenatural to conjecture that, in order to have finite equations of states in the N →∞limit, it was necessary to divide by N ∗ (defined in Eq. (3.69)) quantities such astemperature, pressure, external magnetic field, chemical potential, etc., by N quantitiessuch as volume, magnetization, entropy, number of particles, etc., and by NN ∗quantities such as the internal energy and all the thermodynamical potentials. Theseprescriptions were verified since 1996 in many kinds of systems, such as Lennard–Jones-like fluids [870, 874], magnets [174, 175, 177, 871, 872, 875, 877], anomalousdiffusion [873], and percolation [878, 879].(f) Vanishing Lyapunov spectrum for classical long-range-interacting many-bodyHamiltonian systemsIt was first realized in 1977 [127] that the q-exponential functions emerge whenthe maximal Lyapunov exponent vanishes (see point (d) here above). It then becamenatural to conjecture that, in any anomalous stationary (or quasi-stationary) state,the Lyapunov spectrum should exhibit a generic tendency to approach zero at theN →∞limit for classical long-range-interacting Hamiltonian systems (whereas itis of course expected to be positive for short-range-interacting Hamiltonians). Thiswas indeed verified, first in 1998 [177] for the α-XY ferromagnet (see Figs. 5.47and 5.48), and since then in many other systems [178, 376–378] (see Figs. 5.49,5.50, 5.51, and 5.52). In all these cases it was numerically verified that, in theN >> 1 limit, the Lyapunov spectrum vanishes for 0 ≤ α/d ≤ 1, and is nonzero forα/d > 1.(g) Nonuniform convergence for long-range Hamiltonians associated with a divergentlim N→∞ t crossover (N)It was conjectured in 1999 (see Fig. 4 in [63]) that classical long-range-interactingmany-body systems could evolve, before attaining thermal equilibrium, through one(or more) nonequilibrium long-standing states. The departure from the longstandingstates towards equilibration would occur (slowly, as indicated in Fig. 4, alongsomething such as a logarithmic scale for time) at t crossover (N). Furthermore, it wasconjectured that lim N→∞ t crossover (N) =∞. Since 1999, the entire scenario wasverified many times: see [45, 46, 373, 376–379, 820, 838–842], among others.