246 research outputs found
Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes
Consider a linear Boltzmann equation posed on the Euclidian plane with a
periodic system of circular holes and for particles moving at speed 1. Assuming
that the holes are absorbing -- i.e. that particles falling in a hole remain
trapped there forever, we discuss the homogenization limit of that equation in
the case where the reciprocal number of holes per unit surface and the length
of the circumference of each hole are asymptotically equivalent small
quantities. We show that the mass loss rate due to particles falling into the
holes is governed by a renewal equation that involves the distribution of
free-path lengths for the periodic Lorentz gas. In particular, it is proved
that the total mass of the particle system at time t decays exponentially fast
as t tends to infinity. This is at variance with the collisionless case
discussed in [Caglioti, E., Golse, F., Commun. Math. Phys. 236 (2003), pp.
199--221], where the total mass decays as Const./t as the time variable t tends
to infinity.Comment: 29 pages, 1 figure, submitted; figure 1 corrected in new versio
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Complex systems approach to language games
The mechanisms leading language conventions to be socially accepted and adopted by a group are object of an intense debate. The issue can be of course addressed by different points of view, and recently also complex system science has started to contribute, mainly by means of computer simulations and analytical approaches. In this paper we study a very simple multi-agent model of convention spreading and investigate some of the crucial aspects of its dynamics, resorting, whenever possible, to quantitative analytic methods. In particular, the model is able to account for the emergence of global consensus out of local (pairwise) interactions. In this regard, a key question concerns the role of the size of the population. We investigate in detail how the cognitive efforts of the agents in terms of memory and the convergence time scale with the number of agents. We also point out the existence of an hidden timescale ruling a fundamental aspect of the dynamics, and we discuss the nature of the convergence process
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
Response properties in a model for granular matter
We investigate the response properties of granular media in the framework of
the so-called {\em Random Tetris Model}. We monitor, for different driving
procedures, several quantities: the evolution of the density and of the density
profiles, the ageing properties through the two-times correlation functions and
the two-times mean-square distance between the potential energies, the response
function defined in terms of the difference in the potential energies of two
replica driven in two slightly different ways. We focus in particular on the
role played by the spatial inhomogeneities (structures) spontaneously emerging
during the compaction process, the history of the sample and the driving
procedure. It turns out that none of these ingredients can be neglected for the
correct interpretation of the experimental or numerical data. We discuss the
problem of the optimization of the compaction process and we comment on the
validity of our results for the description of granular materials in a
thermodynamic framework.Comment: 22 pages, 35 eps files (21 figures
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
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Self-organizing communication in language games
From the point of view of semiotic dynamics language is an evolving complex dynamical system. In this perspective, unrevealing the mechanisms that allow for the birth of shared conventions is a major issue. Here we describe a very simple model in which agents negotiate conventions and reach a global agreement without any intervention from the outside. In particular we focus on the possibility of predicting on which of the several competing conventions the agreement is reached. We find from simulations that early created conventions are favored in the competition process and this advantage can be quantified. Beyond the specific results presented here, we think that this paper provides an example of a new way of investigating language features where simple models allow for the investigation of precise problems and, possibly, for analytical approaches
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
Spherical averages in the space of marked lattices
A marked lattice is a -dimensional Euclidean lattice, where each lattice
point is assigned a mark via a given random field on . We prove
that, if the field is strongly mixing with a faster-than-logarithmic rate, then
for every given lattice and almost every marking, large spheres become
equidistributed in the space of marked lattices. A key aspect of our study is
that the space of marked lattices is not a homogeneous space, but rather a
non-trivial fiber bundle over such a space. As an application, we prove that
the free path length in a crystal with random defects has a limiting
distribution in the Boltzmann-Grad limit
Inviscid dynamical structures near Couette flow
Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow
v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation
at v_0 decays in time. At the nonlinear level, such inviscid damping has not
been proved. First, we show that in any (vorticity) H^{s}(s<(3/2)) neighborhood
of Couette flow, there exist non-parallel steady flows with arbitrary minimal
horizontal period. This implies that nonlinear inviscid damping is not true in
any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow and for any
horizontal period. Indeed, the long time behavior in such neighborhoods are
very rich, including nontrivial steady flows, stable and unstable manifolds of
nearby unstable shears. Second, in the (vorticity) H^{s}(s>(3/2)) neighborhood
of Couette, we show that there exist no non-parallel steadily travelling flows
v(x-ct,y), and no unstable shears. This suggests that the long time dynamics in
H^{s}(s>(3/2)) neighborhoods of Couette might be much simpler. Such contrasting
dynamics in H^{s} spaces with the critical power s=(3/2) is a truly nonlinear
phenomena, since the linear inviscid damping near Couette is true for any
initial vorticity in L^2
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