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.