Nonextensive Statistical Mechanics

8.2 Frequently Asked Questions 315one, but involving three instead of two systems, is expected to be able to illustratethe possible validity of the zeroth principle for anomalous systems such as the HMFone.(f) Does the quasi-stationary state in long-range-interacting Hamiltonians reallyexist?A re-analysis was done by Zanette and Montemurro [819] of the moleculardynamics approach and results presented in [373] for the infinitely-long-rangeinteracting planar rotators already discussed here. They especially focus on the timedependence of the temperature T (t) defined as the mean kinetic energy per particle.For total energy slightly below the second-order critical point and a non-zeromeasureclass of initial conditions, a long-standing nonequilibrium state emergesbefore the system achieves the terminal BG thermal equilibrium. When T vs.s log tis plotted, an inflection point exists. If we call t crossover the value of t at whichthe inflection point is located, it has been repeatedly verified numerically by variousauthors, including Zanette and Montemurro [819], that lim N→∞ t crossover (N)diverges. Therefore, if the system is very large (in the limit N →∞, mathematicallyspeaking) it remains virtually for ever in the anomalous state, currently called quasistationarystate or metastable state. Zanette and Montemurro point out (correctly)that, if a linear scale is used for t, the inflection point disappears. 2 For increasinglylarge N, T (t) remains constant, and different from the BG value, within a quite smallerror bar. This effect appears in an even more pronounced way because of a slightminimum that T (t) presents just before going up to the BG value. This intriguingminimum had already been observed in [373], and has been detected with higherprecision in [819]. Further details are presented in [820]. It remains nevertheless afact that the nature of this quasi-stationary state is quite unusual (with aging andother indications of glassy-like dynamics [821–824]), and surely deserves furtherstudies.(g) Are the q-exponential distributions compatible with the central limit theoremswhich only allow, in the thermodynamic limit, for Gaussian and Lévy distributions?This interesting issue has been raised in several occasions by several people.For example, soon after their previous critique, Zanette and Montemurro advanceda second one [825] objecting the validity of nonextensive statistical mechanics forthermodynamical systems. This line of critique addresses the possibility of the ubiquityof the q-exponential form as a stable law in nature. The argument essentiallygoes that only Gaussians and Lévy distributions would be admissible, because of the2 From this, these authors conclude that this well-known metastable state is but a kind of mathematicalartifact, and no physically relevant quasi-stationarity exists. Such an argument is mathematicallysimilar to stating that the high-to-low energies crossing occurring, at a given temperature,in Fermi–Dirac statistics would have no physical meaning! Indeed, if instead of using the linearscale for the energies we were to use a faster scale (e.g., an exponential scale), the well-knowninflection point would disappear. Nevertheless, there is no point to conclude from this that thetextbook crossing in Fermi–Dirac statistics is but a mathematical artifact. In fact, any inflectionpoint on any curve will disappear by sufficiently “accelerating” the abscissa. The crossover willobviously remain.