On the distribution of free-path lengths for the periodic Lorentz gas III

In a flat 2-torus with a disk of diameter $r$ removed, let $\Phi_r(t)$ be the distribution of free-path lengths (the probability that a segment of length larger than $t$ with uniformly distributed origin and direction does not meet the disk). We prove that $\Phi_r(t/r)$ behaves like $\frac{2}{\pi^2 t}$ for each $t>2$ and in the limit as $r\to 0^+$, in some appropriate sense. We then discuss the implications of this result in the context of kinetic theory.Comment: 26 pages, 5 figures, to be published in Commun. Math. Phy

Euclidean random matching in 2D for non-constant densities

We consider the 2-dimensional random matching problem in $\mathbb{R}^2.$ In a challenging paper, Caracciolo et. al. arXiv:1402.6993 on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent $2,$ between two samples of $N$ uniformly distributed points in the unit square is $\log N/2\pi N$ plus corrections, while the expected value of the square of the Wasserstein distance between one sample of $N$ uniformly distributed points and the uniform measure on the square is $\log N/4\pi N$. These conjectures has been proved by Ambrosio et al. arXiv:1611.04960. Here we consider the case in which the points are sampled from a non uniform density. For first we give formal arguments leading to the conjecture that if the density is regular and positive in a regular, bounded and connected domain $\Lambda$ in the plane, then the leading term of the expected values of the Wasserstein distances are exactly the same as in the case of uniform density, but for the multiplicative factor equal to the measure of $\Lambda$. We do not prove these results but, in the case in which the domain is a square, we prove estimates from above that coincides with the conjectured result.Comment: 14 pages, 3 figure

The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions

The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius $r$ centered at the integer points, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this motion on time intervals of order $1/r$ and in the limit as $r\to 0^+$, in the case of two space dimensions

Language Trees and Zipping

In this letter we present a very general method to extract information from a generic string of characters, e.g. a text, a DNA sequence or a time series. Based on data-compression techniques, its key point is the computation of a suitable measure of the remoteness of two bodies of knowledge. We present the implementation of the method to linguistic motivated problems, featuring highly accurate results for language recognition, authorship attribution and language classification.Comment: 5 pages, RevTeX4, 1 eps figure. In press in Phys. Rev. Lett. (January 2002

On the complete phase synchronization for the Kuramoto model in the mean-field limit

We study the Kuramoto model for coupled oscillators. For the case of identical natural frequencies, we give a new proof of the complete frequency synchronization for all initial data; extending this result to the continuous version of the model, we manage to prove the complete phase synchronization for any non-atomic measure-valued initial datum. We also discuss the relation between the boundedness of the entropy and the convergence to an incoherent state, for the case of non identical natural frequencies

Measuring complexity with zippers

Physics concepts have often been borrowed and independently developed by other fields of science. In this perspective a significant example is that of entropy in Information Theory. The aim of this paper is to provide a short and pedagogical introduction to the use of data compression techniques for the estimate of entropy and other relevant quantities in Information Theory and Algorithmic Information Theory. We consider in particular the LZ77 algorithm as case study and discuss how a zipper can be used for information extraction.Comment: 10 pages, 3 figure

Dephasing of Kuramoto oscillators in kinetic regime towards a fixed asymptotically free state

We study the kinetic Kuramoto model for coupled oscillators. We prove that for any regular asymptotically free state, if the interaction is small enough, it exists a solution which is asymptotically close to it. For this class of solution the order parameter vanishes to zero, showing a behavior similar to the phenomenon of Landau damping in plasma physics. We obtain an exponential decay of the order parameter in the case on analytical regularity of the asymptotic state, and a polynomial decay in the case of Sobolev regularity

On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas

43 pages with 3 figures; some typos corrected and references updated; final draft to appear in J. Stat. Phys.,International audienceThe two-dimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle and the obstacles, this dynamical system is studied in the Boltzmann-Grad limit, assuming that the obstacle radius r and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle's distribution function is slowly varying in the space variable. In this limit, the periodic Lorentz gas cannot be described by a linear Boltzmann equation (see [F. Golse, Ann. Fac. Sci. Toulouse 17 (2008), 735--749]), but involves an integro-differential equation conjectured in [E. Caglioti, F. Golse, C.R. Acad. Sci. Ser. I Math. 346 (2008) 477--482] and proved in [J. Marklof, A. Stroembergsson, preprint arXiv:0801.0612], set on a phase-space larger than the usual single-particle phase-space. The main purpose of the present paper is to study the dynamical properties of this integro-differential equation: identifying its equilibrium states, proving a H Theorem and discussing the speed of approach to equilibrium in the long time limit. In the first part of the paper, we derive the explicit formula for a transition probability appearing in that equation following the method sketched in [E. Caglioti, F. Golse, loc. cit.]