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It is well known that Gibbs' statistical mechanics is not justified forsystems presenting long-range interactions, such as plasmas or galaxies. In aprevious work we considered a realistic FPU-like model of an ionic crystal (andthus with long-range interactions), and showed that it reproduces theexperimental infrared spectra from 1000 K down to 7 K, provided one abandonsthe Gibbs identification of temperature in terms of specific kinetic energy, atlow temperatures. Here we investigate such a model in connection with itsergodicity properties. The conclusion we reach is that at low temperaturesergodicity does not occur, and thus the Gibbs prescriptions are not dynamicallyjustified, up to geological time scales. We finally give a preliminary resultindicating how the so-called `nonclassical' q-statistics show up in therealistic ionic-crystal model. How to formulate a consistent statisticalmechanics, with the corresponding suitable identification of temperature insuch nonergodicity conditions, remains an open problem, which apparentlyconstitutes the modern form of the FPU problem.

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