Classical microscopic theory of dispersion, emission and absorption of light in dielectrics

This paper is a continuation of a recent one in which, apparently for the first time, the existence of polaritons in ionic crystals was proven in a microscopic electrodynamic theory. This was obtained through an explicit computation of the dispersion curves. Here the main further contribution consists in studying electric susceptibility, from which the spectrum can be inferred. We show how susceptibility is obtained by the Green--Kubo methods of Hamiltonian statistical mechanics, and give for it a concrete expression in terms of time--correlation functions. As in the previous paper, here too we work in a completely classical framework, in which the electrodynamic forces acting on the charges are all taken into account, both the retarded forces and the radiation reaction ones. So, in order to apply the methods of statistical mechanics, the system has to be previously reduced to a Hamiltonian one. This is made possible in virtue of two global properties of classical electrodynamics, namely, the Wheeler--Feynman identity and the Ewald resummation properties, the proofs of which were already given for ordered system. The second contribution consists in formulating the theory in a completely general way, so that in principle it applies also to disordered systems such as glasses, or liquids or gases, provided the two general properties mentioned above continue to hold. A first step in this direction is made here by providing a completely general proof of the Wheeler--Feynman identity, which is shown to be the counterpart of a general causality property of classical electrodynamics. Finally it is shown how a line spectrum can appear at all in classical systems, as a counterpart of suitable stability properties of the motions, with a broadening due to a coexistence of chaoticity

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

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

Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem

AbstractWe consider an n-degrees of freedom Hamiltonian system near an elliptic equilibrium point. The system is put in normal form (up to an arbitrary order and with respect to some resonance module) and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form. A bound to the rate of diffusion is thus found, and by optimizing the order of normalization exponential estimates of Nekhoroshev's type are obtained. This provides explicit estimates for the stability properties of the elliptic point, and leads in some cases to “effective stability,” i.e., stability up to finite but long times. An application to the stability of the triangular libration points in the spatial restricted three body is also given

The Fermi-Pasta-Ulam system as a model for glasses

Abstract We show that the standard Fermi-Pasta-Ulam system, with a suitable choice for the inter particle potential, constitutes a model for glasses, and indeed an extremely simple and manageable one. Indeed, it allows one to describe the landscape of the minima of the potential energy and to deal concretely with any one of them, determining the spectrum of frequencies and the normal modes. A relevant role is played by the harmonic energy E relative to a given minimum, i.e., the expansion of the Hamiltonian about the minimum up to second order. Indeed we find that there exists an energy threshold in E such that below it the harmonic energy E appears to be an approximate integral of motion for the whole observation time. Consequently, the system remains trapped near the minimum, in what may be called a vitreous or glassy state. Instead, for larger values of E the system rather quickly relaxes to a final equilibrium state. Moreover we find that the vitreous states present peculiar statistical behaviors, still involving the harmonic energy E. Indeed, the vitreous states are described by a Gibbs distribution with an effective Hamiltonian close to E and with a suitable effective inverse temperature. The final equilibrium state presents instead statistical properties which are in very good agreement with the Gibbs distribution relative to the full Hamiltonian of the system