Bryuno Function and the Standard Map

For the standard map the homotopically non-trivial invariant curves of rotation number ω satisfying the Bryuno condition are shown to be analytic in the perturbative parameter ε, provided |ε| is small enough. The radius of convergence ρ(ω) of the Lindstedt series – sometimes called critical function of the standard map – is studied and the relation with the Bryuno function B(ω) is derived: the quantity |log ρ(ω) + 2 B (ω)| is proved to be bounded uniformly in ω. Received: 8 February 2000/ Accepted: 2 March 2001     

Non-Uniformly Hyperbolic Diffeomorphisms Derived from the Standard Map

We prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non-uniformly hyperbolic.     

Fermi Acceleration in Anti-Integrable Limits of the Standard Map

We consider a dynamical system on the semi-infinite cylinder which models the high energy dynamics of a family of mechanical models. We provide conditions under which we ensure that the set of orbits undergoing Fermi acceleration has measure zero.     

Periodic and Quasi-Periodic Orbits¶for the Standard Map

We consider both periodic and quasi-periodic solutions for the standard map, and we study the corresponding conjugating functions, i.e. the functions conjugating the motions to trivial rotations. We compare the invariant curves with rotation numbers ω satisfying the Bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents ω _N = p _N /q _N . We prove the following results for N→ ∞: (1) for rotation numbers ω _N N we study the radius of convergence of the conjugating functions and we find lower bounds on them, which tend to a limit which is a lower bound on the corresponding quantity for ω; (2) the periodic orbits consist of points which are more and more close to the invariant curve with rotation number ω; (3) such orbits lie on analytical curves which tend uniformly to the invariant curve. Received: 14 December 2001 / Accepted: 16 March 2002?Published online: 2 October 2002     

On the Theory of Stochastic Resonance

The transient stage of a stochastic resonance of considerably long duration for low noise intensity as compared to the potential-barrier height is studied. Phase-amplitude and phase-frequency dependences are found for a weak periodic signal at the output of a bistable system. The signal gain and signal-to-noise ratio, as well as the region of maximum phase mismatch, are determined.     

A Proof of the Exponentially Small Transversality of the Separatrices for the Standard Map

In 1984 V. F. Lazutkin [Laz84, LST89] obtained an asymptotic formula for the separatrix splitting angle for the standard map. The difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter. Lazutkin's proof was based on two conjectures. Probably, the original form of those conjectures was incorrect, but Lazutkin's method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices. The consequent works [Laz91, Laz92, GLS94] and [Gel96] prepared the base to fill all the gaps of the original proof. The present paper contains a complete and self-contained proof of a refined version of the original formula (formula (1.7) of the present paper). In this form the formula was obtained in [GLS94]. The proof is inspired by the ideas of Lazutkin's original paper [Laz84]. Received: 16 February 1998 / Accepted: 20 August 1998     

On the Approximation of the Stochastic Burgers Equation

 We prove mathematical approximation results for the (hyperviscous) Burgers equation driven by additive Gaussian noise. In particular we show that solutions of ``approximating equations' driven by a discretized noise converge towards the solution of the original equation when the discretization parameter gets small. The convergence takes place in the expected value of arbitrary powers of certain norms; i.e., all moments of the difference of the solutions tend to zero in certain function spaces. For the hyperviscous Burgers equation, these results are applied to justify the approximation of certain correlation functions that play a major role in statistical turbulence theory. Received: 10 October 2001 / Accepted: 21 May 2002 Published online: 6 August 2002     

Kinetic Theory of Area-Preserving Maps. Application to the Standard Map in the Diffusive Regime

The evolution of the distribution function of a dynamical system governed by a general two-dimensional area-preserving iterative map is studied by the methods of nonequilibrium statistical mechanics. A closed, non-Markovian master equation determines the angle-averaged distribution function (the density profile). The complementary, angle-dependent part (the fluctuations) is expressed as a non-Markovian functional of the density profile. Whenever there exist two widely separated intrinsic time scales, the master equation can be markovianized, yielding an asymptotic kinetic equation. The general theory is applied to the standard map in the diffusive regime, i.e., for large stochasticity parameter and large scale length. The non-Markovian master equation can be written and solved analytically in this approximation. The two characteristic time scales are exhibited. This permits the thorough study of the evolution of the density profile, its tendency toward the Markovian approximation, and eventually toward a diffusive Gaussian packet. The evolution of the fluctuations is also described in detail. The various relaxation processes are governed asymptotically by a single diffusion coefficient, which is calculated analytically. This model appears as a testing bench for the study of kinetic equations. The various previous approaches to this problem are reviewed and critically discussed.     

On the Theoretical Basics of Charged-Particle-Beam Stochastic Optics

The distribution function describing the process of multiple Coulomb scattering is used to investigate the charged-particle-beam dynamics in scattering media. Formulae are presented that describe the phase portrait of the scattered beam through the incident beam and scattering medium parameters. The notion of a beam matched to the target is introduced. The refraction and reflection phenomena at the tilt interface between the medium and vacuum are studied. A mathematical model of flow tubes is suggested to carry out the appropriate calculations.     

On the Stochastic Dynamics of Disordered Spin Models

In this article we discuss several aspects of the stochastic dynamics of spin models. The paper has two independent parts. Firstly, we explore a few properties of the multi-point correlations and responses of generic systems evolving in equilibrium with a thermal bath. We propose a fluctuation principle that allows us to derive fluctuation–dissipation relations for many-time correlations and linear responses. We also speculate on how these features will be modified in systems evolving slowly out of equilibrium, such as finite-dimensional or dilute spin-glasses. Secondly, we present a formalism that allows one to derive a series of approximated equations that determine the dynamics of disordered spin models on random (hyper) graphs.     

On the Maximal Scarring for Quantum Cat Map Eigenstates

We consider the quantized hyperbolic automorphisms on the 2-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the semiclassical measure corresponding to a sequence of normalized eigenstates has a pure point component (phenomenon of strong scarring), then the weight of this component cannot be larger than the weight of the Lebesgue component, and therefore admits the sharp upper bound 1/2.     

Stochastic electrodynamics. I. On the stochastic zero-point field

This is the first in a series of papers that present a new classical statistical treatment of the system of a charged harmonic oscillator (HO) immersed in an omnipresent stochastic zero-point (ZP) electromagnetic radiation field. This paper establishes the Gaussian statistical properties of this ZP field using Bourret's postulate that all statistical moments of the stochastic field plane waves at a given space-time point should agree with their corresponding quantized field vacuum expectations. This postulate is more than adequate to derive the Planck spectrum classically via Boyer's and Theimer's methods, but it requires that the stochastic amplitude of each linearly polarized plane wave in the field contain two independent Gaussian random variables, not just a random phase as has sometimes been assumed. In the succeeding papers in the series, the total motion of a charged HO is described by a fully renormalized dipole-approximation Abraham-Lorentz equation. This leads without further approximation to the following major results concerning this stochastic electrodynamics (SED) of the HO: i) The ensemble-average Liouville equation for the oscillator-ZP field system in the presence of an arbitrary applied classical radiation field is exactly equivalent to the usual time-dependent Schrödinger equation supplemented by an explicit radiation reaction vector potential similar to that of the Crisp-Jaynes-Stroud theory; ii) this SED Schrödinger equation for the HO is incomplete, insmuch as there exists a companion equation that restricts initial conditions such that the corresponding Wigner phase-space distribution is always positive; iii) the wave function of the SED Schrödinger equation has thea priori significance of position probability amplitude; iv) first-order transition rates predicted for the HO by this theory agree with those predicted by quantum electrodynamics for resonance absorption and spontaneous emission, which occurs with no triggering necessary; and v) if SED is taken seriously, then the concepts of quantized energies and photons must be abandoned.     

On the eigenperiods in the Tyrrhenian Sea level oscillations

Summary In this paper the eigenperiods of the Tyrrhenian Sea are examined by a hydrodinamical model in two dimensions. The knowledge of these eigenperiods is important to evaluate the seiches that often appear, in some point or circumstance, stronger than the tides. This investigation is performed by energizing the basin through the Sardinia channel, in the same way as it was already done for the tides. This excitation simulates a large set of waves inducing oscillations inside the basin. The periods of the waves are narrow and their amplitude equal. An analogous method of investigation is that of the fluorescence. In addition, the results are compared with those previously obtained leaving the basin swinging, after having increased the sea level by a uniform displacement. The response of the bidimensional model agrees with that of a monodimensional one as far as possible; thus the last one can give useful information, also for a sea that does not have a channel shape, like the Tyrrhenian Sea. Nevertheless, this basin has other openings, besides the Sardinia channel, it seems that they do not influence the eigenperiod behaviour. This has been confirmed by performing the same computations over a sea 500 m less deep: the uniform lowering of the sea surface is equivalent to close the four minor openings. The results carried out in this case agree with the previous one. Among all the periods pointed out by our investigation, one is always exhibited in every point and by each system. The period is that of 5.70 h and it can be considered a fundamental mode of the Tyrrhenian Sea.     

On the eigenperiods in the Tyrrhenian Sea level oscillations

The online version of the original article can be found at     

On the Higgs mass generation mechanism in the Standard Model

The mass-generation mechanism is the most urgent problem of modern particle physics. The discovery and study of the Higgs boson with the Large Hadron Collider at CERN are the highest priority steps to solve the problem. In this paper, the Standard Model Higgs mechanism of elementary particle mass generation is reviewed with pedagogical details. The discussion of the Higgs quadric self-coupling λ parameter and the bounds to the Higgs boson mass are presented. In particular, the unitarity, triviality, and stability constraints on the Higgs boson mass are discussed. The generation of a finite value for the λ parameter due to quantum corrections via effective potential is illustrated. Some simple predictions for the top quark and the Higgs boson masses are given when both the top Yukawa coupling and the Higgs self-coupling λ are equal to 1. The text was submitted by the authors in English.