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.

Consider an FPU chain composed of N ≫ 1 particles, and endow the phase space with the Gibbs measure corresponding to a small temperature β −1 . Given a fixed K < N , we construct K packets of normal modes whose energies are adiabatic invariants (i.e., are approximately constant for times of order β 1−a , a > 0) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order β. The restrictions on the shape of the packets are very mild. All estimates are uniform in the number N of particles and thus hold in the thermodynamic limit N → ∞, β > 0.

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It is known that a plasma in a magnetic field, conceived microscopically as a system of point charges, can exist in a magnetized state, and thus remain confined, inasmuch as it is in an ordered state of motion, with the charged particles performing gyrational motions transverse to the field. Here we give an estimate of a threshold, beyond which transverse motion become chaotic, the electrons being unable to perform even one gyration, so that a breakdown should occur, with complete loss of confinement. The estimate is obtained by the methods of perturbation theory, taking as perturbing force acting on each electron that due to the so-called microfield, i.e., the electric field produced by all the other charges. We first obtain a general relation for the threshold, which involves the fluctuations of the microfield. Then, taking for such fluctuations the fomula given by Iglesias, Lebowitz and MacGowan for the model of a one component plasma with neutralizing background, we obtain a definite formula for the threshold, which corresponds to a density limit increasing as the square of the imposed magnetic field. Such a theoretical density limit is found to fit pretty well the empirical data for collapses of fusion machines.

The theoretical dispersion curves n(ω) (refractive index vs. frequency) of ionic crystals in the infrared domain are expressed, within the Green-Kubo theory, in terms of a time correlation function involving the motion of the ions only. The aim of this paper is to investigate how well the experimental data are reproduced by a classical approximation of the theory, in which the time correlation functions are expressed in terms of the ions orbits. We report the results of molecular-dynamics (MD) simulations for the ions motions of a LiF lattice of 4096 ions at room temperature. The theoretical curves thus obtained are in surprisingly good agreement with the experimental data, essentially over the whole infrared domain. This shows that at room temperature the motion of the ions develops essentially in a classical regime.