Alberto Mario Maiocchi

• PhD
• Researcher at Università degli Studi di Milano-Bicocca

We study the exchange of energy between the modes of the optical branch and those of the acoustic one in a diatomic chain of particles, with masses $m_1$ and $m_2$. We prove that, at small temperature and provided $m_1\gg m_2$, for the majority of the initial data the energy of each branch is approximately constant for times of order $\beta^{S/2}$,...

We study the exchange of energy between the modes of the optical branch and those of the acoustic one in a diatomic chain of particles, with masses m1 and m2. We prove that, at small temperature and provided m1≫m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \u...

We consider the damped/driven cubic NLS equation on the torus of a large period L with a small nonlinearity of size \(\lambda \), a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first \(\lambda \rightarrow 0\) and then \(L\rightarrow \infty \). The first limit, called the limit of discrete...

В своей статье 1996 г. о квадратичных формах Хис-Браун разработал версию кругового метода для подсчета числа точек пересечения неограниченной квадрики с решеткой короткого периода, когда каждой точке придан вес, и аппроксимировал эту величину интегралом от весовой функции по некоторой мере на квадрике. При этом весовая функция предполагается $C_0^\...

In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of small period, when each point is assigned a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function i...

In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function is ass...

We consider the damped/driven cubic NLS equation on the torus of a large period $L$ with a small nonlinearity of size $\lambda$, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first $\lambda\to 0$ and then $L\to \infty$. The first limit, called the limit of discrete turbulence, is known to...

In his paper from 1996 on quadratic forms Heath-Brown developed a version of circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight. The weight function is assumed to be $C_0^\infty$-smooth and to vanish near the singularity of the quadric. In out work we allow the w...

We consider the nonlinear Schrödinger equation on the one dimensional torus, with a defocusing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measures with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant fo...

We 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 t...

We consider the nonlinear Schroedinger equation on the one dimensional torus, with a defocousing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measure with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant f...

It was recently shown that the experimental infrared spectra of ionic crystals at room temperature are very well reproduced by classical realistic models, and here new results are reported on the temperature dependence of the spectra, for the LiF crystal. The principal aim of the present work is however to highlight the deep analogy existing betwee...

It was recently shown that the experimental infrared spectra of ionic crystals at room temperature are very well reproduced by classical realistic models, and here new results are reported on the temperature dependence of the spectra, for the LiF crystal. The principal aim of the present work is however to highlight the deep analogy existing betwee...

We consider the free linear Schrödinger equation on a torus 𝕋 d , perturbed by a Hamiltonian nonlinearity, driven by a random force and subject to a linear damping: Here u = u ( t, x ), x ∈ 𝕋 d , 0 < ν ≪ 1, q* ∈ ℕ, f is a positive continuous function, ρ is a positive parameter and β𝒌 ( t ) are standard independent complex Wiener processes. We are...

A review is given of the studies aimed at extending to the thermodynamic limit stability results of Nekhoroshev type for nearly integrable Hamiltonian systems. The physical relevance of such an extension, i. e., of proving the persistence of regular (or ordered) motions in that limit, is also discussed. This is made in connection both with the old...

This paper is concerned with theoretical estimates of the refractive– index curves for quartz, obtained by the Kubo formulae in the classical approximation, through MD simulations for the motions of the ions. Two objectives are considered. The first one is to understand the role of non-linearities in situations where they are very large, as at the...

We propose a new phenomenological law for the shape of the spectral lines in the infrared, which accounts for the exponential decay of the extinction coefficient in the high-frequency region, observed in many spectra. We apply this law to the measured infrared spectra of LiF, NaCl, and , finding good agreement over a wide range of frequencies.

We show that the standard Fermi--Pasta--Ulam system, with a suitable choice for the interparticle 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 o...

We suggest a new derivation of a wave kinetic equation for the spectrum of the weakly nonlinear Schrödinger equation with stochastic forcing. The kinetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of th...

The theoretical dispersion curves $n(\omega)$ (refractive index versus 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 a...

In this chapter we present a general method of constructing the effective equation which describes the behaviour of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the re...

Consider weakly nonlinear complex Ginzburg–Landau (CGL) equation of the form: $$\displaystyle{ u_{t} + i(-\bigtriangleup u + V (x)u) =\varepsilon \mu \varDelta u +\varepsilon \mathcal{P}(\nabla u,u),\quad x \in \mathbb{R}^{d}\,, }$$ (*) under the periodic boundary conditions, where μ ≥ 0 and \(\mathcal{P}\) is a smooth function. Let \(\{\zeta _{1}(...

We consider the 2d quasigeostrophic equation on the ß-plane for the stream function ψ, with dissipation and a random force: (∗)(−Δ+K)ψ−ρJ(ψ,Δψ)−βψx= −κΔ2ψ+Δψ where ψ=ψ(t,x,y), x∈R/2πLZ, y∈R/2πZ. For typical values of the horizontal period L we prove that the law of the action-vector of a solution for (∗) (formed by the halves of the squared norms o...

We present some analytic results aiming at explaining the lack of thermalization observed by Fermi Pasta and Ulam in their celebrated numerical experiment. In particular we focus on results which persist as the number $N$ of particles tends to infinity. After recalling the FPU experiment and some classical heuristic ideas that have been used for it...

The present paper is a numerical counterpart to the theoretical work [Carati et al., Chaos 22, 033124 (2012)]. We are concerned with the transition from order to chaos in a one-component plasma (a system of point electrons with mutual Coulomb interactions, in a uniform neutralizing background), the plasma being immersed in a uniform stationary magn...

We consider the free linear Schr\"odinger equation on a torus $\mathbb T^d$, perturbed by a hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$ u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u + \sqrt\nu\,\frac{d}{d t}\sum_{k\in \mathbb Z^d} b_l\beta^k(t)e^{ik\cdot x} \ . $$ Here $u=u(t,x),\ x\in\mathbb T...

We suggest a new derivation of a kinetic equation of Kolmogorov-Zakharov (KZ) type for the spectrum of the weakly nonlinear Schr\"odinger equation with stochastic forcing. The kynetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size...

We consider the free linear Schroedinger equation on a torus $\T^d$, perturbed by a Hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u + \sqrt\nu\,\frac{d}{d t}\sum_{\bk\in \Z^d} b_\bk\bb^\bk(t)e^{i\bk\cdot x} \ . $$ Here $u=u(t,x),\ x\in\T^d$, $0<\nu\ll1$,...

Consider an FPU chain composed of $N\gg 1$ particles, and endow the phase space with the Gibbs measure corresponding to a small temperature $\beta^{-1}$. 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 $\beta^{1-a}$, $a>0$) for initial data in...

We show how statistical thermodynamics can be formulated in situations in which thermodynamics applies, while equilibrium statistical mechanics does not. A typical case is, in the words of Landau and Lifshitz, that of partial (or incomplete) equilibrium. One has a system of interest in equilibrium with the environment, and measures one of its quant...

We report the results of numerical simulations for a model of a one component plasma (a system of N point electrons with mutual Coulomb interactions) in a uniform stationary magnetic field. We take N up to 512, with periodic boundary conditions, and macroscopic parameters corresponding to the weak coupling regime, with a coupling parameter \Gamma=1...

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 tr...

We present here a general iterative formula which gives a (formal) series expansion for the time autocorrelation of smooth dynamical variables, for all Hamiltonian systems endowed with an invariant measure. We add some criteria, theoretical in nature, which enable one to decide whether the decay of the correlations is exponentially fast or not. One...

A crucial problem concerning a large variety of fusion devices is that the confinement due to an external magnetic field is lost above a critical density, while a widely accepted first principles explanation of such a fact is apparently lacking. In the present paper, making use of standard methods of statistical mechanics in the Debye--H\"uckel app...

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...

Usually, the relaxation times of a gas are estimated in the frame of the Boltzmann equation. In this paper, instead, we deal with the relaxation problem in the frame of the dynamical theory of Hamiltonian systems, in which the definition itself of a relaxation time is an open question. We introduce a lower bound for the relaxation time, and give a...