arXiv subjects

Andrea Carati

Publications and source records attributed to Andrea Carati.

At least 19 recordsLinked to original sources

arXiv2026

Cooling rate and glassy behavior in the Fermi--Pasta--Ulam system

In this work, we numerically studied the cooling process of a Fermi--Pasta--Ulam system, which occurs when the FPU system is placed in contact with a gas whose temperature $T$ is reduced with a certain cooling rate $\xi$. It was found that the existence of a weak stochastic threshold has a significant impact on the cooling process, because below such a stochastic threshold the specific FPU energy is larger than its temperature, i.e., the FPU system falls out of equilibrium. The difference remains finite in the limit $T \to 0$, so that the FPU system maintains a residual amount of energy $E_0$ at vanishing temperature. Our numerical simulations reveal that this energy exhibits a power--law dependence on both the system size and the cooling rate, scaling approximately as $E_{0} \sim (\xi N)^{2/3}$.

cond-mat.stat-mech↗

arXiv2026

Spectra as a classical phenomenon, and the Einstein classical program

According to Born (\emph{Atomic Physics, page 103}), spectra are \emph{``quantum phenomena, which from a classical standpoint are perfectly unintelligible''}. However we illustrate results on classical calculations of infrared spectra of ionic crystals (actually LiF) which show that the situation is much more complex. Indeed it turns out that: 1) At room temperature and at higher ones (up to 1060 K) the classical computations reproduce the experimental data, even better than the \emph{presently available} quantum ones do; 2) At lower temperatures (even at 7.5 K), the classical computations reproduce pretty well the data, if one accepts the idea advanced in 1916 by Nernst (the inventor of the third principle) that zero-point energy has room in classical physics too. It is eventually pointed out that the mentioned results might be regarded as a first step towards an implementation of the Einstein Classical Program, which aims at deducing quantum physics (admittedly the correct theory) from a realistic theory. In fact, we are considering the Einstein classical program in the extreme version in which the realistic theory is just (\emph{essentially, see below}) classical electrodynamics of matter in bulk, involving phase space orbits, solutions of Newton equations. An Appendix is devoted to illustrate the Nernst approach, which concerns also the relation between equipartition and Planck's law.

cond-mat.stat-mech↗

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 dynamical approach to the $α$-$β$ displacive transition of quartz

General features of the $α-β$ transition of quartz are investigated. Molecular dynamics methods are mainly used, an analytic treatment being deferred to a work in preparation. A basic preliminary observation is that the transition involves only a subsystem of four normal modes on which the remaining ones just act as a reservoir. The dynamics of the relevant subsystem turns out to be Hamiltonian, being governed by an effective potential that depends on the specific energy of the total system. The effective potential is actually calculated through time averages. It describes the transition as a pitchfork bifurcation, and also explains the phenomenon of the soft mode, since it exhibits a frequency that vanishes at the transition. The critical exponent too is estimated.

cond-mat.stat-mech↗

arXiv2022

Numerical study of the Transverse Diffusion coefficient for a one component model of plasma

In this paper we discuss the results of some Molecular Dynamics simulations of a magnetized One Component Plasma, targeted to estimate the diffusion coefficient $D_{\perp}$ in the plane orthogonal to the magnetic field lines. We find that there exists a threshold with respect to the magnetic field strength $|\vec B|$: for weak magnetic field the diffusion coefficients scales as $1/|\vec B|^2$, while a slower decay appears at high field strength. The relation of this transition with the different mixing properties of the microscopic dynamics is investigated by looking at the behavior of the velocity auto correlation.

physics.plasm-ph↗

arXiv2020

Tsallis distributions, their relaxations and the relation $Δt \cdot ΔE \simeq h$, in the dynamical fluctuations of a classical model of a crystal

We report the results of a numerical investigation, performed in the frame of dynamical systems' theory, for a realistic model of a ionic crystal for which, due to the presence of long--range Coulomb interactions, the Gibbs distribution is not well defined. Taking initial data with a Maxwell-Boltzmann distribution for the mode-energies $E_k$, we study the dynamical fluctuations, computing the moduli of the the energy-changes $|E_k(t)-E_k(0)|$. The main result is that they follow Tsallis distributions, which relax to distributions close to Maxwell-Boltzmann ones; indications are also given that the system remains correlated. The relaxation time $τ$ depends on specific energy $\varepsilon$, and for the curve $τ$ vs, $\varepsilon$ one has two results. First, there exists an energy threshold $\varepsilon_0$, above which the curve has the form $$ τ\cdot \varepsilon \simeq h\ , $$ where, unexpectedly, Planck's constant $h$ shows up. In terms of the standard deviation $ΔE$ of a mode-energy (for which one has $ΔE=\varepsilon$), denoting by $Δt$ the relaxation time $τ$, the relation reads $Δt \cdot ΔE \simeq h$, which reminds of the Heisenberg uncertainty relation. Moreover, the threshold corresponds to zero-point energy. Indeed, the quantum value of the latter is $hν/2$ ( where $ν$ is the characterisic infrared frequency of the system), while we find $\varepsilon \simeq hν/4$, so that one only has a discrepancy of a factor 2. So it seems that lack of full chaoticity manifests itself, in Statistical Thermodynamics, through quantum-like phenomena.

cond-mat.stat-mech↗

arXiv2019

Chemical bond by Newtonian trajectories in the $H_2^+$ ion

According to the correspondence principle, classical mechanics and quantum mechanics agree in the semiclassical limit, although presently it has become more and more clear how intriguing would be to try to fix a boundary between them. Here we give a significant example in which the agreement concerns Newtonian trajectories of an electron with initial data corresponding to a quantum ground state. The example is the simplest case in which a chemical bond occurs, i.e. the $H_2^+$ ion. By molecular dynamics simulations for the full system (two protons and one electron) we show that there exist initial data producing an ``effective potential'' among the protons, which superposes in a surprisingly good way the quantum one computed in the Born-Oppenheimer approximation (Fig~1). Preliminarily, following the perturbation procedure first exhibited by Born and Heisenberg in the year 1924, we recall why an effective potential should exist in a classical frame, and also describe the numerical procedure employed in computing it.

physics.chem-ph↗

arXiv2019

Approach to equilibrium via Tsallis distributions in a realistic ionic--crystal model and in the FPU model

In Statistical Mechanics, Tsallis distributions were apparently conceived in connection with systems presenting long--range interactions. In fact, they were observed in numerical computations for models of such a type, as occurring in the approach to equilibrium, i.e., to a Maxwell--Boltzmann distribution. Here we exhibit two apparently new results. The first one is that Tsallis distributions occur also 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. Thus such distributions may be expected to be actual physical features of crystals. The second result is that Tsallis distributions occur in the standard short--range FPU model too, so that the presence of long--range interactions is not a necessary condition for Tsallis distributions to occur. In fact, this is in agreement with a previous result of the first author in connection with the statistics of return times for the classical FPU model. We thus confirm the thesis advanced by Tsallis himself, 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 to long--range systems.

cond-mat.stat-mech↗

arXiv2019

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.

cond-mat.stat-mech↗

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↗

arXiv2017

Chopping time of the FPU $α$-model

We study, both numerically and analytically, the time needed to observe the breaking of an FPU $α$-chain in two or more pieces, starting from an unbroken configuration at a given temperature. It is found that such a "chopping" time is given by a formula that, at low temperatures, is of the Arrhenius-Kramers form, so that the chain does not break up on an observable time-scale. The result explains why the study of the FPU problem is meaningful also in the ill-posed case of the $α$-model

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↗

arXiv2016

Numerical study of the Transverse Diffusion coefficient for a one component model of a plasma

We report the results of MD numerical simulations for a one component model of a plasma in the weakly coupled regime, at different values of temperature $T$ and applied magnetic field $\vec B$, in which the diffusion coefficient $D_{\perp}$ transverse to the field is estimated. We find that there exists a threshold in temperature, at which an inversion occurs, namely, for $T$ above the threshold the diffusion coefficient $D_{\perp}$ starts decreasing as $T$ increases. This is at variance with the behavior predicted by the Bohm law $D_{\perp}\sim T/B$, which actually holds below the threshold. In addition we find that, for temperatures above such a threshold, another transition occurs, now with respect to the values of the magnetic field: for weak magnetic fields the diffusion coefficients scales as $1/B^2$, in agreement with the predictions of the standard kinetics theory, while it apparently saturates when the field strength is sufficiently increased.

physics.plasm-ph↗

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↗

arXiv2014

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.

cond-mat.stat-mech↗

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↗