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

Abstract

We report the results of a numerical investigation, performed in the frame of dynamical systems' theory, for a realistic model of a ionic crystal for which, due to the presence of long-range Coulomb interactions, the Gibbs distribution is not well defined. Taking initial data with a Maxwell-Boltzmann distribution for the mode-energies E k , we study the dynamical fluctuations, computing the moduli of the the energychanges |E k (t) − E k (0)|. The main result is that they follow Tsallis distributions, which relax to distributions close to Maxwell-Boltzmann ones; indications are also given that the system remains correlated. The relaxation time τ depends on specific energy ε, and for the curve τ vs, ε one has two results. First, there exists an energy threshold ε 0 , above which the curve has the form τ • ε h , where, unexpectedly, Planck's constant h shows up. In terms of the standard deviation ∆E of a mode-energy (for which one has ∆E = ε), denoting by ∆t the relaxation time τ , the relation reads ∆t • ∆E h, which reminds of the Heisenberg uncertainty relation. Moreover, the threshold corresponds to zero-point energy. Indeed, the quantum value of the latter is hν/2 ( where ν is the characterisic infrared frequency of the system), while we find ε 0 hν/4, so that one only has a discrepancy of a factor 2. So it seems that lack of full chaoticity manifests itself, in Statistical Thermodynamics, through quantum-like phenomena.