Chopping Time of the FPU alpha-Model

We study, both numerically and analytically, the time needed to observe the breaking of an FPU \u3b1-chain in two or more pieces, starting from an unbroken configuration at a given temperature. It is found that such a \u201cchopping\u201d time is given by a formula that, at low temperatures, is of the Arrhenius-Kramers form, so that the chain does not break up on an observable time-scale. The result explains why the study of the FPU problem is meaningful also in the ill-posed case of the \u3b1-model

On the definition of temperature using time--averages

This paper is a natural continuation of a previous one by the author, which was concerned with the foundations of statistical thermodynamics far from equilibrium. One of the problems left open in that paper was the correct definition of temperature. In the literature, temperature is in general defined through the mean kinetic energy of the particles of a given system. In this paper, instead, temperature is defined "a la Caratheodory", the system being coupled to a heat bath, and temperature being singled out as the ``right'' integrating factor of the exchanged heat. As a byproduct, the ``right'' expression for the entropy is also obtained. In particular, in the case of a q-distributions the entropy turns out to be that of Tsallis, which we however show to be additive, at variance with what is usually maintained

On the definition of temperature in FPU systems

It is usually assumed, in classical statistical mechanics, that the temperature should coincide, apart from a suitable constant factor, with the mean kinetic energy of the particles. We show that this is not the case for \FPU systems, in conditions in which energy equipartition between the modes is not attained. We find that the temperature should be rather identified with the mean value of the energy of the low frequency modes.Comment: 12 pages, 4 Figure

FPU phenomenon for generic initial data

The well known FPU phenomenon (lack of attainment of equipartition of the mode--energies at low energies, for some exceptional initial data) suggests that the FPU model does not have the mixing property at low energies. We give numerical indications that this is actually the case. This we show by computing orbits for sets of initial data of full measure, sampled out from the microcanonical ensemble by standard Montecarlo techniques. Mixing is tested by looking at the decay of the autocorrelations of the mode--energies, and it is found that the high--frequency modes have autocorrelations that tend instead to positive values. Indications are given that such a nonmixing property survives in the thermodynamic limit. It is left as an open problem whether mixing obtains within time--scales much longer than the presently available ones

Thermodynamics and time-average

For a dynamical system far from equilibrium, one has to deal with empirical probabilities defined through time-averages, and the main problem is then how to formulate an appropriate statistical thermodynamics. The common answer is that the standard functional expression of Boltzmann-Gibbs for the entropy should be used, the empirical probabilities being substituted for the Gibbs measure. Other functional expressions have been suggested, but apparently with no clear mechanical foundation. Here it is shown how a natural extension of the original procedure employed by Gibbs and Khinchin in defining entropy, with the only proviso of using the empirical probabilities, leads for the entropy to a functional expression which is in general different from that of Boltzmann--Gibbs. In particular, the Gibbs entropy is recovered for empirical probabilities of Poisson type, while the Tsallis entropies are recovered for a deformation of the Poisson distribution.Comment: 8 pages, LaTex source. Corrected some misprint

Replacement of the Lorentz law for the shape of the spectral lines in the infrared region

We propose a new phenomenological law for the shape of the spectral lines in the infrared, which accounts for the exponential decay of the extinction coefficient in the high-frequency region, observed in many spectra. We apply this law to the measured infrared spectra of LiF, NaCl, and MgF2, finding good agreement over a wide range of frequencies