Nonextensive Statistical Mechanics

186 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics5.4.1 Metastability, Nonergodicity, and Distribution of VelocitiesThe model is analytically solvable in the BG canonical ensemble (equilibrium witha thermostat at temperature T ). The molecular dynamics approach coincides with itif the initial conditions for the velocities are described by a Gaussian. But, if we usea water-bag, a longstanding metastable or quasi-stationary state (QSS), appears atvalues of u below 0.75 and not too small (typically between 0.5 and 0.75). A valueat which the effect is numerically very noticeable is u = 0.69, hence many studiesare done precisely at this value. See Figs. 5.36 and 5.37. In Fig. 5.38 we can see theinfluence on T QSS of the initial value of m.On the thermal equilibrium plateau one expects, for all α/d, the velocity distributionto be, for N →∞, one and the same Maxwellian distribution for bothensemble-average and time-average. This is of course consistent with the BG resultfor the canonical ensemble, based on the hypothesis of ergodicity. The situation iscompletely different on the QSS plateau 6 emerging for long-range interactions (i.e.,0 ≤ α/d ≤ 1). Indeed, the ensemble- and time-averages do not coincide [45, 46],thus exhibiting nonergodicity (which, as we shall see, is consistent with the fact that,along this longstanding metastable state, the entire Lyapunov spectrum collapsesonto zero when N →∞). The situation is illustrated in Figs. 5.39, 5.40, 5.41, 5.42,5.43, 5.44, 5.45, and 5.46.5.4.2 Lyapunov SpectrumA set of 2dN Lyapunov exponents is associated with the d-dimensional Hamiltonian(5.44), half of them positive and half of them negative (coupled two by twoin absolute value) since the system is symplectic. We focus on the maximal value˜λ maxN; if this value vanishes, the entire spectrum vanishes. This property is extremelyrelevant for the foundations of statistical mechanics. Indeed, if ˜λ maxN> 0, the systemwill be mixing and ergodic, which is the basis of BG statistical mechanics. If ˜λ maxNvanishes, there is no such guarantee. This is the realm of nonextensive statisticalmechanics, as we have already verified for paradigmatic dissipative and conservativelow-dimensional maps. The scenario for the d = 1 α-XY model is described inFigs. 5.47 and 5.48. The corresponding scenarios for d = 2, 3 have been discussed6 The lifetime τ QSS of this QSS plateau has been conjectured (see Fig. 4 in [63], wherelim N→∞ lim t→∞ is expected to yield the standard BG canonical thermal equilibrium andlim t→∞ lim N→∞ is expected to yield the nonextensive statistical mechanics results) to diverge,for fixed α ≤ d if N →∞. Also, for the d = 1 model, it has been suggested [374] that, for fixedN, τ QSS decreases exponentially with α increasing above zero. All these results are consistentwith τ QSS ∝ (N ⋆ ) a with N ⋆ given in Eq. (3.69) and a > 0. Indeed, such scaling yields, for0 ≤ α/d < 1, τ QSS (α/d, N) ∝ N a[1−(α/d)] (N →∞), which implies τ QSS (0, N) ∝ N a ,andexponentially decreasing with α/d for fixed N. All authors do not always use the same definitionfor τ QSS . The definition used in [373] implies a = 1; the definitions used by other authors implya > 1.