Tsallis distributions, their relaxations and the relation $\Delta t \cdot \Delta E \simeq h$, 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 energy-changes $|E_k(t)-E_k(0)|$. 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 $\tau$ depends on specific energy $\varepsilon$, and for the curve $\tau$ vs, $\varepsilon$ one has two results. First, there exists an energy threshold $\varepsilon_0$, above which the curve has the form $$ \tau \cdot \varepsilon \simeq h\ , $$ where, unexpectedly, Planck's constant $h$ shows up. In terms of the standard deviation $\Delta E$ of a mode-energy (for which one has $\Delta E=\varepsilon$), denoting by $\Delta t$ the relaxation time $\tau$, the relation reads $\Delta t \cdot \Delta E \simeq h$, which reminds of the Heisenberg uncertainty relation. Moreover, the threshold corresponds to zero-point energy. Indeed, the quantum value of the latter is $h\nu/2$ ( where $\nu$ is the characterisic infrared frequency of the system), while we find $\varepsilon \simeq h\nu/4$, 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.