716 research outputs found
Transport Properties of the Lorentz Gas in Terms of Periodic Orbits
We establish a formula relating global diffusion in a space periodic
dynamical system to cycles in the elementary cell which tiles the space under
translations.Comment: 8 pages, Postscript, A
Long wave expansions for water waves over random topography
In this paper, we study the motion of the free surface of a body of fluid
over a variable bottom, in a long wave asymptotic regime. We assume that the
bottom of the fluid region can be described by a stationary random process
whose variations take place on short length scales and which
are decorrelated on the length scale of the long waves. This is a question of
homogenization theory in the scaling regime for the Boussinesq and KdV
equations. The analysis is performed from the point of view of perturbation
theory for Hamiltonian PDEs with a small parameter, in the context of which we
perform a careful analysis of the distributional convergence of stationary
mixing random processes. We show in particular that the problem does not fully
homogenize, and that the random effects are as important as dispersive and
nonlinear phenomena in the scaling regime that is studied. Our principal result
is the derivation of effective equations for surface water waves in the long
wave small amplitude regime, and a consistency analysis of these equations,
which are not necessarily Hamiltonian PDEs. In this analysis we compute the
effects of random modulation of solutions, and give an explicit expression for
the scattered component of the solution due to waves interacting with the
random bottom. We show that the resulting influence of the random topography is
expressed in terms of a canonical process, which is equivalent to a white noise
through Donsker's invariance principle, with one free parameter being the
variance of the random process . This work is a reappraisal of the paper
by Rosales & Papanicolaou \cite{RP83} and its extension to general stationary
mixing processes
Experimental study of out of equilibrium fluctuations in a colloidal suspension of Laponite using optical traps
This work is devoted to the study of displacement fluctuations of
micron-sized particles in an aging colloidal glass. We address the issue of the
validity of the fluctuation dissipation theorem (FDT) and the time evolution of
viscoelastic properties during aging of aqueous suspensions of a clay (Laponite
RG) in a colloidal glass phase. Given the conflicting results reported in the
literature for different experimental techniques, our goal is to check and
reconcile them using \emph{simultaneously} passive and active microrheology
techniques. For this purpose we measure the thermal fluctuations of micro-sized
brownian particles immersed in the colloidal glass and trapped by optical
tweezers. We find that both microrheology techniques lead to compatible results
even at low frequencies and no violation of FDT is observed. Several
interesting features concerning the statistical properties and the long time
correlations of the particles are observed during the transition
A renormalization approach to irrational rotations
We introduce a renormalization procedure which allows us to study in a
unified and concise way different properties of the irrational rotations on the
unit circle , \alpha \in \R\setminus \Q. In
particular we obtain sharp results for the diffusion of the walk on
generated by the location of points of the sequence on a
binary partition of the unit interval. Finally we give some applications of our
method.Comment: 27 page
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