Recent Results on the Periodic Lorentz Gas

The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption -- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases -- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) vol. 185 (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.Comment: 62 pages. Course at the conference "Topics in PDEs and applications 2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem 4.6 corrected in the new versio

Relative entropy via non-sequential recursive pair substitutions

The entropy of an ergodic source is the limit of properly rescaled 1-block entropies of sources obtained applying successive non-sequential recursive pairs substitutions (see P. Grassberger 2002 ArXiv:physics/0207023 and D. Benedetto, E. Caglioti and D. Gabrielli 2006 Jour. Stat. Mech. Theo. Exp. 09 doi:10.1088/1742.-5468/2006/09/P09011). In this paper we prove that the cross entropy and the Kullback-Leibler divergence can be obtained in a similar way.Comment: 13 pages , 2 figure

Self-Structuring of Granular Media under Internal Avalanches

We study the phenomenon of internal avalanching within the context of recently proposed ``Tetris'' lattice models for granular media. We define a recycling dynamics under which the system reaches a steady state which is self-structured, i.e. it shows a complex interplay between textured internal structures and critical avalanche behavior. Furthermore we develop a general mean-field theory for this class of systems and discuss possible scenarios for the breakdown of universality.Comment: 4 pages RevTex, 3 eps figures, revised version to appear in Phys. Rev. Let

Coarsening and Slow-Dynamics in Granular Compaction

We address the problem of the microscopic reorganization of a granular medium under a compaction process in the framework of Tetris-like models. We point out the existence of regions of spatial organization which we call domains, and study their time evolution. It turns out that after an initial transient, most of the activity of the system is concentrated on the boundaries between domains. One can then describe the compaction phenomenon as a coarsening process for the domains, and a progressive reduction of domain boundaries. We discuss the link between the coarsening process and the slow dynamics in the framework of a model of active walkers on active substrates.Comment: Revtex 4 pages, 4 figures, in press in PRL. More info http://axtnt3.phys.uniroma1.it/Tetri

Random Assignment Problems on 2d Manifolds

We consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold Ω of unit area. It is known that the average cost scales as EΩ(N) ∼ 1 / 2 πln N with a correction that is at most of order lnNlnlnN. In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first Ω -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on Ω. We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics

Free-volume kinetic models of granular matter

We show that the main dynamical features of granular media can be understood by means of simple models of fragile-glass forming liquid provided that gravity alone is taken into account. In such lattice-gas models of cohesionless and frictionless particles, the compaction and segregation phenomena appear as purely non-equilibrium effects unrelated to the Boltzmann-Gibbs measure which in this case is trivial. They provide a natural framework in which slow relaxation phenomena in granular and glassy systems can be explained in terms of a common microscopic mechanism given by a free-volume kinetic constraint.Comment: 4 pages, 6 figure

Algebraic Correlation Function and Anomalous Diffusion in the HMF model

In the quasi-stationary states of the Hamiltonian Mean-Field model, we numerically compute correlation functions of momenta and diffusion of angles with homogeneous initial conditions. This is an example, in a N-body Hamiltonian system, of anomalous transport properties characterized by non exponential relaxations and long-range temporal correlations. Kinetic theory predicts a striking transition between weak anomalous diffusion and strong anomalous diffusion. The numerical results are in excellent agreement with the quantitative predictions of the anomalous transport exponents. Noteworthy, also at statistical equilibrium, the system exhibits long-range temporal correlations: the correlation function is inversely proportional to time with a logarithmic correction instead of the usually expected exponential decay, leading to weak anomalous transport properties

Liquid hot isostatic pressing of QE22A magnesium alloy: a preliminary test

A preliminary experimental comparison of the behaviour of aluminium and magnesium alloys subjected to Liquid Hot Isostatic Pressing (LHIP) is proposed. The two metals melt at approximately the same temperature.However, as a consequence of a larger deformability of magnesium at elevated temperatures, the choice of LHIP parameters – and especially the temperature at which the pressure is applied – in the present exploratory case was constrained to values far smaller than those one would like to select in order to improve the ultimate tensile stress and the elongation to fracture