121 research outputs found
Transition to stochasticity in a one-dimensional model of a radiant cavity
No full textWe make a numerical study of the solutions of the equations of motion for the electromagnetic field in a one-dimensional model of a radiant cavity. Our main results are as follows: (1) There exist Stochasticity thresholds such that below them one has ordered motions without energy exchanges, while chaotic motions with intense energy exchanges occur above them; (2) above thresholds there is a trend toward equipartition of energy (in time average) among the normal modes of the field, but this occurs in the sense of Boltzmann and Jeans, namely, with the higher frequencies requiring longer and longer times in order to be involved in the energy sharing
Kolmogorov entropy and numerical experiments
No full textNumerical investigations of dynamical systems allow one to give estimates of the rate of divergence of nearby trajectories, by means of a quantity which is usually assumed to be related to the Kolmogorov (or metric) entropy. In this paper it is shown first, on the basis of mathematical results of Oseledec and Piesin, how such a relation can be made precise. Then, as an example, a numerical study of the Kolmogorov entropy for the Hénon-Heiles model is reported
Boltzmann's ultraviolet cutoff and Nekhoroshev's theorem on Arnold diffusion
No full textThe greatest difficulty with classical statistical mechanics may be the fact that some degrees of freedom do not seem to attain the energy expected from the equipartition principle. This is typical of the vibrational modes in molecules and of the high frequencies in the electromagnetic radiation (problem of the ultraviolet cutoff). Boltzmann1 had the intuition that an explanation might be provided if each degree of freedom were to have a characteristic relaxation time until equilibrium, and that such a time should greatly increase with frequency; he spoke in terms of relaxation times of days or centuries. This possibility was seriously considered by Rayleigh and Jeans2, but they could not produce a clear classical mechanism to explain the freezing of the high frequency degrees of freedom; so, the equilibrium concept of the ultraviolet cutoff provided by quantum mechanics was accepted, and Boltzmann's hypothesis was forgotten. Here we point out that a general framework for understanding the ultraviolet cutoff in Boltzmann's dynamical sense in a classical context seems to be provided by Nekhoroshev's3 theorem on Arnold diffusion
Approach to equilibrium via Tsallis distributions in a realistic ionic-crystal model and in the FPU model
No full textWe report results of dynamical simulations exhibiting the occurrence of Tsallis distributions, and their eventual approach to Maxwell–Boltzmann distributions, for the normal-mode energies of FPU-like systems. The first result is that Tsallis distributions occur in an ionic crystal model with long-range Coulomb forces, which is so realistic as to reproduce in an impressively good way the experimental infrared spectra. So, such distributions may be expected to represent actual physical features of crystals. The second result is that Tsallis distributions for the normal mode energies occur in the classical FPU model too. This is in agreement with previous results obtained in the latter model, namely: by Antonopoulos, Bountis and Basios for the distributions of local observables (particles’ properties as energies or momenta), and by the first of the present authors for the statistical properties of return times. All such results thus confirm the thesis advanced by Tsallis himself, i.e., that the relevant property for a dynamical system to present Tsallis distributions is that its dynamics should be not fully chaotic, a property which is known to actually pertain, in particular, to systems with long-range interactions
Relaxation times and ergodic properties in a realistic ionic-crystal model, and the modern form of the FPU problem
No full textIt 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 ergodic 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
A Dynamical Approach to the – Displacive Transition of Quartz
No full textThe problem of displacive phase transitions (by which crystals pass on heating from a less symmetric to a more symmetric form) is investigated through numerical integration of the Newton equations of motion for a realistic model, in the paradigmatic case of quartz. Usually such transitions are discussed in terms of the positions of the atoms, while the role of normal modes is emphasized here. The key preliminary property established, in agreement with the indications given by Landau in his thermodynamic-like approach, is that four well definite modes are sufficient to describe the transition, the remaining modes just acting as a noise. The main result is then that such four modes constitute a closed Hamiltonian subsystem presenting an effective potential parametrically dependent on specific energy. The effective potential is actually computed, through (appropriately defined) time-averages of the accelerations of the relevant modes, and is found to describe, as energy is varied, a pitchfork bifurcation, once more confirming in dynamical terms the Landau result. The effective potential also allows one to advance a possible explanation of the "soft mode" phenomenon, namely the occuring, in the Raman spectrum, of a peak whose frequency depends on temperature and vanishes at the transition
The FPU Problem as a Statistical-mechanical Counterpart of the KAM Problem, and Its Relevance for the Foundations of Physics
No full textWe give a review of the Fermi – Pasta – Ulam (FPU) problem from the perspective of its possible impact on the foundations of physics, concerning the relations between classical and quantum mechanics. In the first part we point out that the problem should be looked upon in a wide sense (whether the equilibrium Gibbs measure is attained) rather than in the original restricted sense (whether energy equipartition is attained). The second part is devoted to some very recent results of ours for an FPU-like model of an ionic crystal, which has such a realistic character as to reproduce in an impressively good way the experimental infrared spectra. Since the existence of sharp spectral lines is usually considered to be a characteristic quantum phenomenon, even unconceivable in a classical frame, this fact seems to support a thesis suggested by the original FPU result. Namely, that the relations between classical and quantum mechanics are much subtler than usually believed, and should perhaps be reconsidered under some new light
Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory
No full textSince several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper
Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application
No full textThe present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit method for computing all Lyapunov Characteristic Exponents of a dynamical system. After the general theory on such exponents developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B)
Apparent fractal dimensions in conservative dynamical systems
No full textFor conservative dynamical systems, the invariant sets which are in a sense the analog of the strange attractors of dissipative systems, namely the closures of homoclinic orbits of hyperbolic points, are known to have in general integral dimensions. We show however, working numerically on a particular model, that actual numerical estimates for perturbations of an integrable system will necessarily exhibit an apparent fractal dimension, which will be the effective one to all practical purpose. A simple scheme of interpretation is also given
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