Validity and failure of the Boltzmann weight

Abstract

The dynamics and thermostatistics of a classical inertial XY model, characterized by long-range interactions, are investigated on d-dimensional lattices (d = 1, 2, and 3), through molecular dynamics. The interactions between rotators decay with the distance r ij like 1/r α ij (α ≥ 0), where α → ∞ and α = 0 respectively correspond to the nearest-neighbor and infinite-range interactions. We verify that the momenta probability distributions are Maxwellians in the short-range regime, whereas q-Gaussians emerge in the long-range regime. Moreover, in this latter regime, the individual energy probability distributions are characterized by long tails, corresponding to q-exponential functions. The present investigation strongly indicates that, in the long-range regime, central properties fall out of the scope of Boltzmann-Gibbs statistical mechanics, depending on d and α through the ratio α/d. * Electronic address: cirto@cbpf.br † Electronic address: antonio.rodriguezm@upm.es ‡ Electronic address: fdnobre@cbpf.br § Electronic address: tsallis@cbpf.br

Scaling properties of d -dimensional complex networks

“…The third and last regime, from α A /d ≃ 5 on, is Boltzmannian-like and is characterized by short-range interactions. The existence of the intermediate regime has also been observed in classical many-body Hamiltonians, namely the α-generalized XY [16,17] and Heisenberg [18] rotator models as well as the Fermi-Pasta-Ulam model [19,20,[29][30][31]. The present work neatly illustrates a fact which is not always obvious to the community working with complex networks, more precisely those exhibiting asymptotic scale-free behavior.…”

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

“…Again, all data collapse, now in the energy range (see details in Figure 3 ); since different dimensions present distinct densities of states, the data does not collapse near the origin. One should call attention to the fact that the value of Figure 3 coincides with the recent result for the -XY model, obtained also with [ 34 ]. Moreover, as found in the study of the -XY model [ 34 ], a curious discrepancy in the values of the index q occurred in the estimates of Figure 1 and Figure 3 , i.e., .…”

“…A curious behavior occurs also for the ensemble-averaged momenta probability distributions. The same departure from the BG predictions is observed for the time-averaged energy distributions; in this case, instead of the usual BG exponential, q -exponential probability distributions are obtained, with [ 34 ]. Similar universality features have been recently obtained in quite different contexts, such as a generalized Fermi-Pasta-Ulam model with long-range interactions [ 35 , 36 , 37 , 38 , 39 ], complex networks with preferential attachment growth [ 40 , 41 ], and a system of particles under overdamped motion interacting repulsively with power-law interactions [ 42 ].…”

“…We have illustrated the one-particle momenta and energy distributions in the long-range region with one example, namely for , and the results have shown a clear collapse of data and were well fitted with a q -Gaussian and a q -exponential respectively. As found in the recent study of the -XY model [ 34 ], an interesting discrepancy in the values of the index q occurred, leading to . Whether this may be a real effect, or a consequence of finite-size effects, remains a point for further investigation.In addition, we have analyzed similar distributions in the short-range region, namely .…”

“…The -XY model has also been largely studied [ 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 ] and, in the d -dimensional case, it was shown [ 29 ] that in its high-energy regime, the maximum Lyapunov exponent scales with the number of rotators as , where the exponent depends on and d through the ratio , as in the case of other quantities, with in the long-range regime ( ) and (thus yielding positive Lyapunov exponents) in the short-range regime ( ). In addition, q -Gaussians have been obtained for the time-averaged momenta probability distributions not only before, but even after the crossover to the very-large-time QSS, with a value of the entropic index depending again on the ratio [ 34 ]. The usual Maxwellian distributions are recovered as increases above unity.…”