arXiv subjects

Giorgio Mantica

Publications and source records attributed to Giorgio Mantica.

24 records · Page 2Linked to original sources

arXiv2009

The global statistics of return times: return time dimensions versus generalized measure dimensions

We investigate the relations holding among generalized dimensions of invariant measures in dynamical systems and similar quantities defined by the scaling of global averages of powers of return times. Because of a heuristic use of Kac theorem, these latter have been used in place of the former in numerical and experimental investigations; to mark this distinction, we call them return time dimensions. We derive a full set of inequalities linking measure and return time dimensions and we comment on their optimality with the aid of two maps due to von Neumann -- Kakutani and to Gaspard -- Wang. We conjecture the behavior of return time dimensions in a typical system. We only assume ergodicity of the dynamical system under investigation.

math.DS↗

arXiv2009

On the statistical distribution of first--return times of balls and cylinders in chaotic systems

We study returns in dynamical systems: when a set of points, initially populating a prescribed region, swarms around phase space according to a deterministic rule of motion, we say that the return of the set occurs at the earliest moment when one of these points comes back to the original region. We describe the statistical distribution of these "first--return times" in various settings: when phase space is composed of sequences of symbols from a finite alphabet (with application for instance to biological problems) and when phase space is a one and a two-dimensional manifold. Specifically, we consider Bernoulli shifts, expanding maps of the interval and linear automorphisms of the two dimensional torus. We derive relations linking these statistics with Renyi entropies and Lyapunov exponents.

nlin.CD↗

arXiv1999

Quantum Algorithmic Integrability: The Metaphor of Polygonal Billiards

An elementary application of Algorithmic Complexity Theory to the polygonal approximations of curved billiards-integrable and chaotic-unveils the equivalence of this problem to the procedure of quantization of classical systems: the scaling relations for the average complexity of symbolic trajectories are formally the same as those governing the semi-classical limit of quantum systems. Two cases-the circle, and the stadium-are examined in detail, and are presented as paradigms.

chao-dyn↗

arXiv1996

Quantum Intermittency in Almost-Periodic Lattice Systems Derived from their Spectral Properties

Hamiltonian tridiagonal matrices characterized by multi-fractal spectral measures in the family of Iterated Function Systems can be constructed by a recursive technique here described. We prove that these Hamiltonians are almost-periodic. They are suited to describe quantum lattice systems with nearest neighbours coupling, as well as chains of linear classical oscillators, and electrical transmission lines. We investigate numerically and theoretically the time dynamics of the systems so constructed. We derive a relation linking the long-time, power-law behaviour of the moments of the position operator, expressed by a scaling function $β$ of the moment order $α$, and spectral multi-fractal dimensions, D_q, via $β(α) = D_{1-α}$. We show cases in which this relation is exact, and cases where it is only approximate, unveiling the reasons for the discrepancies.

cond-mat.mes-hall↗

arXiv1996

Electron Wave Filters from Inverse Scattering Theory

Semiconductor heterostructures with prescribed energy dependence of the transmittance can be designed by combining: {\em a)} Padé approximant reconstruction of the S-matrix; {\em b)} inverse scattering theory for Schrodinger's equation; {\em c)} a unitary transformation which takes into account the variable mass effects. The resultant continuous concentration profile can be digitized into an easily realizable rectangular-wells structure. For illustration, we give the specifications of a 2 narrow band-pass 12 layer $Al_cGa_{1-c}As$ filter with the high energy peak more than {\em twice narrower} than the other.

cond-mat↗

arXiv1996

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, $P_n(z)=ϕ(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $ϕ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like $|\hbox{\rm Im}\,z|$. We present the low and high disorder asymptotic behaviors. Then we particularize to the large $n$ limit of the average density of complex roots of monic algebraic polynomials of the form $P_n(z) = z^n +\sum_{k=1}^{n}c_kz^{n-k}$ with real independent, identically distributed Gaussian coefficients having zero mean and dispersion $δ= \frac 1{\sqrt{nλ}}$. The average density tends to a simple, {\em universal} function of $ξ={2n}{\log |z|}$ and $λ$ in the domain $ξ\coth \fracξ{2}\ll n|\sin \arg (z)|$ where nearly all the roots are located for large $n$.

chao-dyn↗