Approach to equilibrium via Tsallis distributions in a realistic ionic-crystal model and in the FPU model

Abstract

In Statistical Mechanics, Tsallis distributions were apparently conceived in connection with systems presenting long-range interactions. In fact, they were observed in numerical computations for models of such a type, as occurring in the approach to equilibrium, i.e., to a Maxwell-Boltzmann distribution. Here we exhibit two apparently new results. The first one is that Tsallis distributions occur also in an ionic-crystal model with long-range Coulomb forces, which is so realistic as to reproduce in an impressively good way the experimental infrared spectra. Thus such distributions may be expected to be actual physical features of crystals. The second result is that Tsallis distributions occur in the standard short-range FPU model too, so that the presence of long-range interactions is not a necessary condition for Tsallis distributions to occur. In fact, this is in agreement with a previous result of the first author in connection with the statistics of return times for the classical FPU model. We thus confirm the thesis advanced by Tsallis himself, that the relevant property for a dynamical system to present Tsallis distributions is that its dynamics should be not fully chaotic, a property which is known to actually pertain to long-range systems. *

Tsallis distributions, their relaxations and the relation $Δt \cdot ΔE \simeq h$, in the dynamical fluctuations of a classical model of a crystal

“…The first phenomenon, exhibited in figure 5, is the existence of a threshold (or a knee, or a crossover) ε 0 , apparently at 100 K, i.e., rather near to the value of 120 K that would be expected from the results on the infrared spectra. 14 One may wonder which role be played by such a threshold. We already recalled how, for the threshold observed in the standard Fermi-Pasta-Ulam, already in the early years 70's Cercignani put forward the suggestive hypothesis that it be the analogue of zero-point energy.…”

“…The first one is that at all times the moduli of the energy-changes seem to follow a well defined distribution, i.e., a Tsallis one (see [13]). Such a fact is not completely unexpected, since Tsallis distributions are known to show up in models involving long-range forces, either Coulomb ones [14] (as the ionic crystal considered here), or gravitational ones (see [15] [16]). As is known, Tsallis distributions are characterized by two parameters: q (the so-called Tsallis entropic index, related to the decay of the tail of the distribution), and β, which determines the specific energy per degree of freedom ε.…”

Reply to Pessoa, P.; Arderucio Costa, B. Comment on “Tsallis, C. Black Hole Entropy: A Closer Look. Entropy 2020, 22, 17”

“…(i) Let us assume that a system can be modelled by a long-ranged-interacting many-body problem (e.g., classical models such as the d -dimensional -XY ferromagnet [ 53 , 54 ], -Heisenberg ferromagnet [ 55 , 56 , 57 , 58 , 59 ], -Fermi–Pasta–Ulam model [ 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 ], -Lennard–Jones gas [ 69 , 70 , 71 , 72 ]), N basically being the number of particles. The notation, including in all of them, comes from the fact that a two-body attractive interaction is assumed in all of them, which asymptotically decays as (with ), r being the distance.…”

Nonextensive statistical mechanics, superstatistics and beyond: theory and applications in astrophysical and other complex systems

“…In particular, Metzler discusses superstatistics and non-Gaussian diffusion [1]; Zhang et al [2] investigate a class of network diffusion equations with large powernonlinearity connected to q-statistics; Carati et al [3] discuss via q-distributions a realistic ionic-crystal model and, within the FPU model, how the system reaches equilibrium; Rozynek and Wilk [4] study the level of nonextensivity of the quarkgluon system described by lattice QCD; Souza et al investigate the ground entangled state of the one-dimensional spin-1/2 Ising ferromagnet at its transverse-field critical point [5]; Suyari et al study the advantages of the q-logarithm representation over the q-exponential one [6]; Korbel et al [7] discuss within the information geometry framework the scaling expansions of non-exponentially growing configuration spaces; Wedemann and Plastino [8] study a nonlinear Fokker Planck equation and its stationary-state solution, proving an H-theorem obeyed by a free-energy functional that involves the generalized entropy S q ; Yoon uses the equivalence of kappa distributions and q-Gaussian distributions to focus on the non-equilibrium statistical mechanical applications to the formation of non-Maxwellian electron distribution in space [9]. Interesting applications of q-statistics for predicting ruptures and earthquakes are then discussed in the paper by Greco et al [10] and in the paper by Skordas et al [11] respectively; Singh and Roy [12] discuss an internetwork synchronisation technique for complex dynamical networks of different kinds.…”