Nonextensive Statistical Mechanics

1.4 A Few Words on the Foundations of Statistical Mechanics 15the interactions, or whether the system is on the ordered or on the disordered sideof a continuous phase transition. Generically speaking, the influence of the orderingof t →∞and N →∞limits is typically related to some kind of breakdown ofsymmetry, or of ergodicity, or the alike.The simplest nontrivial dynamical situation is expected to occur for an isolatedmany-body short-range-interacting classical Hamiltonian system (microcanonicalensemble); later on we shall qualify when an interaction is considered short-rangedin the present context. In such a case, the typical microscopic (nonlinear) dynamicsis expected to be strongly chaotic, in the sense that the maximal Lyapunov exponentis positive. Such a system would necessarily be mixing, i.e., it would quickly visitvirtually all the accessible phase-space (more precisely, very close to almost all theaccessible phase-space) for almost any possible initial condition. Furthermore, itwould necessarily be ergodic with respect to some measure in the full phase-space,i.e., time averages and ensemble averages would coincide. In most of the casesthis measure is expected to be uniform in phase-space, i.e., the hypothesis of equalprobabilities would be satisfied.A slightly more complex situation is encountered for those systems which exhibita continuous phase transition. Let us consider the simple case of a ferromagnetwhich is invariant under inversion of the hard axis of magnetization, e.g., the d = 3XY classical nearest-neighbor ferromagnetic model on simple cubic lattice. If thesystem is in its disordered (paramagnetic) phase, the limits t →∞and N →∞commute, and the entire phase-space is expected to be equally well visited. If thesystem is in its ordered (ferromagnetic) phase, the situation is expected to be moresubtle. The lim N→∞ lim t→∞ set of probabilities is, as before, equally distributed allover the entire phase-space for almost any initial condition. But this is not expectedto be so for the lim t→∞ lim N→∞ set of probabilities. The system probably lives, inthis case, only in half of the entire phase-space. Indeed, if the initial condition is suchthat the initial magnetization is positive, even infinitesimally positive (for instance,under the presence of a vanishingly small external magnetic field), then the systemis expected to be ergodic but only in the half phase-space associated with positivemagnetization; the other way around occurs if the initial magnetization is negative.This illustrates, already in this simple example, the importance that the ordering ofthose two limits can have.A considerably more complex situation is expected to occur, if we consider along-range-interacting model, e.g., the same d = 3 XY classical ferromagneticmodel on simple cubic lattice as before, but now with a coupling constant whichdecays with distance as 1/r α , where r is the distance measured in crystal units,and 0 ≤ α ≤ d (the nearest-neighbor model that we just discussed correspondsto α →∞, which is the extreme case of the short-ranged domain α>d). The0 ≤ α/d ≤ 1 model also appears to have a continuous phase transition. In thedisordered phase, the system possibly is ergodic over the entire phase-space. Butin the ordered phase the result can strongly depend on the ordering of the twolimits. The lim N→∞ lim t→∞ set of probabilities corresponds to the system livingin the entire phase-space. In contrast, the lim t→∞ lim N→∞ set of probabilities forthe same (conveniently scaled) total energy might be considerably more complex. Itseems that, for this ordering, phase-space exhibits at least two macroscopic basins