Relaxation times and ergodic properties in a realistic ionic-crystal model, and the modern form of the FPU problem

Abstract

It is well known that Gibbs' statistical mechanics is not justified for systems presenting long-range interactions, such as plasmas or galaxies. In a previous work we considered a realistic FPU-like model of an ionic crystal (and thus with long-range interactions), and showed that it reproduces the experimental infrared spectra from 1000 K down to 7 K, provided one abandons the Gibbs identification of temperature in terms of specific kinetic energy, at low temperatures. Here we investigate such a model in connection with its ergodicity properties. The conclusion we reach is that at low temperatures ergodicity does not occur, and thus the Gibbs prescriptions are not dynamically justified, up to geological time scales. We finally give a preliminary result indicating how the so-called "nonclassical" q-statistics show up in the realistic ionic-crystal model. How to formulate a consistent statistical mechanics, with the corresponding suitable identification of temperature in such nonergodicity conditions, remains an open problem, which apparently constitutes the modern form of the FPU problem.

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.…”

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

“…The contribution of the electrons, which don't show up in the model but are known to produce polarization forces on the ions, is taken into account in a phenomenological way by introducing a shortrange potential V phen acting among the ions, with suitable "effective" charges substituted for the real ones. In our first paper [7] the phenomenological potential was just that originally proposed by Born, namely, V phen (r) = C/r 6 , whereas a more complex potential, depending on the pair of ions, was used in the subsequent papers [8,9]. In the end, the Hamiltonian reads…”

“…The first result is that Tsallis distributions converging to a Maxwell-Boltzmann one are met also for the normal-mode energies of a realistic 3-d FPU-like model. We are referring to an ionic crystal model (actually, a LiF model) with Coulomb long-range interactions (see [7,8,9]), which has such a realistic character as to reproduce in an impressively good way (and indeed within a classical frame) the experimental infrared spectra. An agreement between experimental data and theory over 9 orders of magnitude for the infrared spectra of LiF, is exhibited in the first two figures of the paper [8].…”

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

“…The unexpected empirical result of the work [28] was that an agreement between experimental data and numerical computations at low temperatures can be restored, provided one abandons the familiar identification of temperature as proportional to mean kinetic energy. Indeed the experimental data at a temperature of 85 K are reproduced if one takes a specific energy ε = 180 K, and analogously for the data at 7.5 K if one takes ε = 125 K. One should notice that a procedure of such a type for constructing the relation between temperature and mechanical energy has a noble precursor in statistical mechanics since, in the case of dilute gases, Clausius obtained such a relation by comparing the theoretical expression of the product pV in terms of kinetic energy, with the empirical expression in terms of temperature.…”