arXiv subjects

Alberto Maiocchi

Publications and source records attributed to Alberto Maiocchi.

At least 19 recordsLinked to original sources

arXiv2026

A survey on rigorous results for the dynamics of periodic FPU chains

In this paper we review some analytic results on the dynamics of the FPU system. In the first part of the paper, having in mind that the FPU Hamiltonian and the Toda Hamiltonian are close each other, we present some results on the action angle variables of the Toda system and deduce some stability properties for the dynamics of the FPU system. We first focus on the case of finitely many particles and then we study the limit $N\to\infty$. We present also some results on the continous limit of the Toda chain showing that it is well described by a couple of KdV equations. Then we study directly the dynamics of the function interpolating the FPU system and show that the dynamics is Hamiltonian and that the Hamiltonian is very close to a function of the first three Hamiltonians of the KdV hierarchy. In the second part of the paper we present some results valid in the thermodynamic limit, according to which the time autocorrelation functions of some suitably constructed observables decay slowly implying lower bounds on the thermalization times of the system.

math-ph↗

arXiv2023

A refinement of Heath-Brown's theorem on quadratic forms

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 assumed to be $C_0^\infty$-smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not vanish at the singularity and have an explicit decay at infinity. The paper uses only elementary results from the number theory and is available to readers without a number-theoretical background.

math.NT↗

arXiv2021

The large-period limit for equations of discrete turbulence

We consider the damped/driven cubic NLS equation on the torus of a large period $L$ with a small nonlinearity of size $λ$, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first $λ\to 0$ and then $L\to \infty$. The first limit, called the limit of discrete turbulence, is known to exist, and in this work we study the second limit $L\to\infty$ for solutions to the equations of discrete turbulence. Namely, we decompose the solutions to formal series in amplitude and study the second order truncation of this series. We prove that the energy spectrum of the truncated solutions becomes close to solutions of a damped/driven nonlinear wave kinetic equation. Kinetic nonlinearity of the latter is similar to that which usually appears in works on wave turbulence, but is different from it (in particular, it is non-autonomous). Apart from tools from analysis and stochastic analysis, our work uses two powerful results from the number theory.

math.AP↗

arXiv2021

Some remarks on Heath-Brown's theorem on quadratic forms

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 weight function to be finitely smooth and not vanish near the singularity, and we give also an explicit dependence on the weight function.

math.NT↗

arXiv2018

A large probability averaging Theorem for the defocousing NLS

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 for times of order $β^{2+ς}$, $β$ being the inverse of the temperature and $ς$ a positive number (we prove $ς= 1/10$). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.

math-ph↗

arXiv2018

Classical infrared spectra of ionic crystals and their relevance for statistical mechanics

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 between the problem of spectra in ionic crystal models on the one hand, and that of energy equipartition in the Fermi--Pasta--Ulam model, on the other. Indeed at low temperatures the computations of the spectra show that the dynamics of the considered system is not completely chaotic, so that the use of the Boltzmann--Gibbs statistics is put in question, as in the Fermi--Pasta--Ulam case. Here, however, at variance with the equipartition problem, a first positive indication is given on the modifications that should be introduced in a classical statistical treatment: the new results at low temperatures show that it is indeed the Clausius identification of temperature that has to be modified. In fact, at very low temperatures the theoretical spectra fail to reproduce the experimental ones, if the temperature is taken as proportional to mean kinetic energy, but agreement is recovered through the only expedient of introducing a suitable temperature rescaling. Analogous results are also found in connection with thermal expansion.

cond-mat.stat-mech↗

arXiv2016

Infrared optical properties of $α$ quartz by molecular dynamics simulations

This paper is concerned with theoretical estimates of the refractive--index curves for quartz, obtained by the Kubo formulæ 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 nonlinearities in situations where they are very large, as at the $α$--$β$ structural phase transition. We show that on the one hand they don't play an essential role in connection with the form of the spectra in the infrared. On the other hand they play an essential role in introducing a chaoticity which involves a definite normal mode. This might explain why that mode is Raman active in the $α$ phase, but not in the $β$ phase. The second objective concerns whether it is possible in a microscopic model to obtain normal mode frequencies, or peak frequencies in the optical spectra, that are in good agreement with the experimental data for quartz. Notwithstanding a lot of effort, we were unable to find results agreeing better than about 6%, as apparently also occurs in the whole available literature. We interpret this fact as indicating that some essential qualitative feature is lacking in all models which consider, as the present one, only short--range repulsive potentials and unretarded long--range electric forces.

cond-mat.mtrl-sci↗

arXiv2015

Time-averaging for weakly nonlinear CGL equations with arbitrary potentials

Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-Δu+V(x)u)=εμΔu+ε\mathcal{P}( u),\quad x\in {R^d}\,, \quad(*) $$ under the periodic boundary conditions, where $μ\geqslant0$ and $\mathcal{P}$ is a smooth function. Let $\{ζ_1(x),ζ_2(x),\dots\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-Δ+V(x)$. For a complex function $u(x)$, write it as $u(x)=\sum_{k\geqslant1}v_kζ_k(x)$ and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{ε=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if equation $(*)$ with a sufficiently smooth real potential $V(x)$ is well posed on time-intervals $t\lesssim ε^{-1}$, then for any its solution $u^ε(t,x)$, the limiting behavior of the curve $I(u^ε(t,\cdot))$ on time intervals of order $ε^{-1}$, as $ε\to0$, can be uniquely characterized by a solution of a certain well-posed effective equation: $$ u_t=εμ\triangle u+εF(u), $$ where $F(u)$ is a resonant averaging of the nonlinearity $\mathcal{P}(u)$. We also prove a similar results for the stochastically perturbed equation, when a white in time and smooth in $x$ random force of order $\sqrtε$ is added to the right-hand side of the equation. The approach of this work is rather general. In particular, it applies to equations in bounded domains in $R^d$ under Dirichlet boundary conditions.

math.AP↗

arXiv2015

A replacement of the Lorentz law for the shape of the spectral lines in the infrared region

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 MgF$_2$, finding a good agreement, over a wide range of frequencies.

physics.optics↗

arXiv2015

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

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 of frequencies and the normal modes. A relevant role is played by the harmonic energy $\mathcal E$ 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 $\mathcal E$ such that below it the harmonic energy $\mathcal E$ 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 $\mathcal E$ 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 $\mathcal E$. Indeed, the vitreous states are described by a Gibbs distribution with an effective Hamiltonian close to $\mathcal E$ 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.

cond-mat.stat-mech↗

arXiv2015

Agreement of classical Kubo theory with the infrared dispersion curves $n(ω)$ of ionic crystals

The theoretical dispersion curves $n(ω)$ (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 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.

cond-mat.mtrl-sci↗

arXiv2015

The effective equation method

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 resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behaviour of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three-- and four--wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. In the case of the NLS equation we use next some heuristic approximation from the arsenal of wave turbulence to show that under the iterated limit "the volume goes to infinity", taken after the limit "the amplitude of oscillations goes to zero", the energy spectrum of solutions for the effective equation is described by a Zakharov-type kinetic equation. Evoking the Zakharov ansatz we show that stationary in time and homogeneous in space solutions for the latter equation have a power law form. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanology.

math-ph↗

arXiv2014

The limit of small Rossby numbers for randomly forced quasi-geostrophic equation on $β$-plane

We consider the 2d quasigeostrophic equation on the $β$-plane for the stream function $ψ$, with dissipation and a random force: $$ (*)\qquad (-Δ+K)ψ_t - ρJ(ψ, Δψ) -βψ_x= \langle \text{random force}\rangle -κΔ^2ψ+Δψ, $$ where $ψ=ψ(t,x,y), \ x\in\mathbb{R}/2πL\mathbb{Z}, \ y\in \mathbb{R}/2π\mathbb{Z}$. 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 of its complex Fourier coefficients) converges, as $β\to\infty$, to the law of an action-vector for solution of an auxiliary effective equation, and the stationary distribution of the action-vector for solutions of $(*)$ converges to that of the effective equation. Moreover, this convergence is uniform in $κ\in(0,1]$. The effective equation is an infinite system of stochastic equations which splits into invariant subsystems of complex dimension $\le3$; each of these subsystems is an integrable hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits $\lim_{L=ρ\to\infty} \lim_{β\to\infty}$ and $\lim_{κ\to 0} \lim_{β\to\infty}$ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies.

math-ph↗

arXiv2014

Some analytic results on the FPU paradox

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 its explanation, we concentrate on more recent rigorous results which are based on the use of (i) canonical perturbation theory and KdV equation, (ii) Toda lattice, (iii) a new approach based on the construction of functions which are adiabatic invariants with large probability in the Gibbs measure.

math-ph↗

arXiv2014

Resonant averaging for small solutions of stochastic NLS equations

We consider the free linear Schrödinger equation on a torus $\mathbb T^d$, perturbed by a hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$ u_t -iΔu +iνρ|u|^{2q_*}u = - νf(-Δ) u + \sqrtν\,\frac{d}{d t}\sum_{k\in \mathbb Z^d} b_lβ^k(t)e^{ik\cdot x} \ . $$ Here $u=u(t,x),\ x\in\mathbb T^d$, $0<ν\ll 1$, $q_*\in\mathbb N$, $f$ is a positive continuous function, $ρ$ is a positive parameter and $β^k(t)$ are standard independent complex Wiener processes. We are interested in limiting, as $ν\to0$, behaviour of distributions of solutions for this equation and of its stationary measure. Writing the equation in the slow time $τ=νt$, we prove that the limiting behaviour of the both is described by the effective equation $$ u_τ+ f(-Δ) u = -iF(u)+\frac{d}{dτ}\sum b_kβ^k(τ)e^{ik\cdot x} \, $$ where the nonlinearity $F(u)$ is made out of the resonant terms of the monomial $ |u|^{2q_*}u$. We explain the relevance of this result for the problem of weak turbulence

math-ph↗

arXiv2013

Derivation of the Kolmogorov-Zakharov equation from the resonant-averaged stochastic NLS equation

We suggest a new derivation of a kinetic equation of Kolmogorov-Zakharov (KZ) type for the spectrum of the weakly nonlinear Schrödinger 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 of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analyzed with full rigour in arXiv:1311.6793, and show how the second limit leads to a kinetic equation for the spectrum, if some further hypotheses (commonly employed in the weak turbulence theory) are accepted. Finally we show how to derive from these equations the KZ spectra.

math-ph↗

arXiv2013

Resonant averaging for weakly nonlinear stochastic Schrödinger equations

We consider the free linear Schroedinger equation on a torus $\mathbb T^d$, perturbed by a Hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$u_t -iΔu +iνρ|u|^{2q_*}u = - νf(-Δ) u + \sqrtν\,\frac{d}{d t}\sum_{k\in \mathbb Z^d} b_kβ^k(t)e^{ik\cdot x} \ . $$ Here $u=u(t,x),\ x\in\mathbb T^d$, $0<ν\ll1$, $q_*\in\mathbb N\cup\{0\}$, $f$ is a positive continuous function, $ρ$ is a positive parameter and $β^k(t)$ are standard independent complex Wiener processes. We are interested in limiting, as $ν\to0$, behaviour of solutions for this equation and of its stationary measure. Writing the equation in the slow time $τ=νt$, we prove that the limiting behaviour of the both is described by the effective equation $$ u_τ+ f(-Δ) u = -iF(u)+\frac{d}{dτ}\sum b_kβ^k(τ)e^{ik\cdot x} \, $$ where the nonlinearity $F(u)$ is made out of the resonant terms of the monomial $ |u|^{2q_*}u$. We explain the relevance of this result for the problem of weak turbulence.

math-ph↗

arXiv2013

An averaging theorem for FPU in the thermodynamic limit

Consider an FPU chain composed of $N\gg 1$ particles, and endow the phase space with the Gibbs measure corresponding to a small temperature $β^{-1}$. Given a fixed $K 0$) 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 $N\to\infty$, $β>0$.

math-ph↗