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

## TL;DR

This paper investigates the ergodicity and relaxation times in a realistic ionic-crystal model with long-range interactions, revealing nonergodic behavior at low temperatures and discussing implications for a generalized statistical mechanics framework.

## Contribution

It demonstrates nonergodicity in a realistic ionic-crystal model at low temperatures and explores the emergence of q-statistics, challenging traditional Gibbs-based thermodynamics.

## Key findings

- Nonergodicity occurs at low temperatures in the model.
- Gibbs prescriptions are not dynamically justified at low temperatures.
- Preliminary evidence of q-statistics in the model.

## 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.