A. Carati

Nonuniqueness properties of the physical solutions of the Lorentz-Dirac equation

The solutions of the Lorentz-Dirac equation are investigated, for the problem of a one-dimensional scattering of a charged particle by a potential barrier, and a phenomenon is found having some similarity to the quantum weak-reflection effect. Namely, there exists an energy strip, slightly above the maximum of the barrier, such that for any given initial energy in the strip there is a certain number of physical (or nonrunaway) solutions of two types, i.e. Those of mechanical type, transmitted beyond the barrier, and those of nonmechanical type, reflected by the barrier. From the mathematical point of view, the existence of this phenomenon is related to the nonuniqueness of the physical solutions of the Lorentz-Dirac equation for given initial data of position and velocity. This in turn is strictly related to a property recently pointed out, namely the asymptotic character of the relevant series expansions occurring for that equation. Correspondingly, the width of the energy strip where the phenomenon occurs is found to decrease exponentially fast, as the small parameter entering the problem tends to zero.

The Fermi—Pasta—Ulam Problem and the Metastability Perspective

A review is given of the works on the FPU problem that were particularly relevant in connection with the metastability perspective, proposed in the year 1982. The idea is that there exists a specific energy threshold above which the time-averages of the relevant quantities quickly agree with the predictions of classical equilibrium statistical mechanics, whereas below it there exist two time scales. First there is a quick formation of a packet of low-frequency modes which do share the energy, and this produces a metastable state that lasts for a long time; then the system attains the final equilibrium state. There are strong indications that the specific energy threshold does not vanish in the limit of infinitely many particles. The review is given for the case of a one-dimensional FPU chain.

The Fermi-Pasta-Ulam problem as a challenge for the foundations of physics

The Fermi-Pasta-Ulam (FPU) problem is discussed in connection with its physical relevance, and it is shown how apparently there exist only two possibilities: either the FPU problem is just a curiosity, or it has a fundamental role for the foundations of physics, casting a new light on the relations between classical and quantum mechanics. To this end, a short review is given of the main conceptual proposals that have been advanced. Particular emphasis is given to the perspective of a metaequilibrium scenario, which appears to be the only possible one for the FPU paradox to survive in the physically relevant case of infinitely many particles.

Exponentially Long Stability Times for a Nonlinear Lattice in the Thermodynamic Limit

In this paper, we construct an adiabatic invariant for a large 1–d lattice of particles, which is the so called Klein Gordon lattice. The time evolution of such a quantity is bounded by a stretched exponential as the perturbation parameters tend to zero. At variance with the results available in the literature, our result holds uniformly in the thermodynamic limit. The proof consists of two steps: first, one uses techniques of Hamiltonian perturbation theory to construct a formal adiabatic invariant; second, one uses probabilistic methods to show that, with large probability, the adiabatic invariant is approximately constant. As a corollary, we can give a bound from below to the relaxation time for the considered system, through estimates on the autocorrelation of the adiabatic invariant.

Fermi-Pasta-Ulam phenomenon for generic initial data

The well-known Fermi-Pasta-Ulam (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 Monte Carlo 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 actually occurs, i.e., whether the autocorrelations vanish as time tends to infinity.

Thermodynamics and time averages

For a dynamical system far from equilibrium, one has to deal with empirical probabilities defined through time averages, and the main problem then is 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 to a functional expression for the entropy which is in general different from that of Boltzmann–Gibbs. In particular, the Gibbs entropy is recovered for empirical probabilities of the Poisson type, while the Tsallis entropies are recovered for a deformation of the Poisson distribution.

An averaging theorem for FPU in the thermodynamic limit

Consider an FPU chain composed of

Boundary effects on the dynamics of chains of coupled oscillators

N\gg 1