2,471 research outputs found
Corrections to Einstein's relation for Brownian motion in a tilted periodic potential
In this paper we revisit the problem of Brownian motion in a tilted periodic
potential. We use homogenization theory to derive general formulas for the
effective velocity and the effective diffusion tensor that are valid for
arbitrary tilts. Furthermore, we obtain power series expansions for the
velocity and the diffusion coefficient as functions of the external forcing.
Thus, we provide systematic corrections to Einstein's formula and to linear
response theory. Our theoretical results are supported by extensive numerical
simulations. For our numerical experiments we use a novel spectral numerical
method that leads to a very efficient and accurate calculation of the effective
velocity and the effective diffusion tensor.Comment: 29 pages, 7 figures, submitted to the Journal of Statistical Physic
Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations
In this paper we study the diffusion approximation of a swarming model given
by a system of interacting Langevin equations with nonlinear friction. The
diffusion approximation requires the calculation of the drift and diffusion
coefficients that are given as averages of solutions to appropriate Poisson
equations. We present a new numerical method for computing these coefficients
that is based on the calculation of the eigenvalues and eigenfunctions of a
Schr\"odinger operator. These theoretical results are supported by numerical
simulations showcasing the efficiency of the method
Anomalous diffusion and response in branched systems: a simple analysis
We revisit the diffusion properties and the mean drift induced by an external
field of a random walk process in a class of branched structures, as the comb
lattice and the linear chains of plaquettes. A simple treatment based on
scaling arguments is able to predict the correct anomalous regime for different
topologies. In addition, we show that even in the presence of anomalous
diffusion, Einstein's relation still holds, implying a proportionality between
the mean square displacement of the unperturbed systems and the drift induced
by an external forcing.Comment: revtex.4-1, 16 pages, 7 figure
Periodic Homogenization for Inertial Particles
We study the problem of homogenization for inertial particles moving in a
periodic velocity field, and subject to molecular diffusion. We show that,
under appropriate assumptions on the velocity field, the large scale, long time
behavior of the inertial particles is governed by an effective diffusion
equation for the position variable alone. To achieve this we use a formal
multiple scale expansion in the scale parameter. This expansion relies on the
hypo-ellipticity of the underlying diffusion. An expression for the diffusivity
tensor is found and various of its properties studied. In particular, an
expansion in terms of the non-dimensional particle relaxation time (the
Stokes number) is shown to co-incide with the known result for passive
(non-inertial) tracers in the singular limit . This requires the
solution of a singular perturbation problem, achieved by means of a formal
multiple scales expansion in Incompressible and potential fields are
studied, as well as fields which are neither, and theoretical findings are
supported by numerical simulations.Comment: 31 pages, 7 figures, accepted for publication in Physica D. Typos
corrected. One reference adde
Phase Diagram for Turbulent Transport: Sampling Drift, Eddy Diffusivity and Variational Principles
We study the long-time, large scale transport in a three-parameter family of
isotropic, incompressible velocity fields with power-law spectra. Scaling law
for transport is characterized by the scaling exponent and the Hurst
exponent , as functions of the parameters. The parameter space is divided
into regimes of scaling laws of different {\em functional forms} of the scaling
exponent and the Hurst exponent. We present the full three-dimensional phase
diagram.
The limiting process is one of three kinds: Brownian motion (),
persistent fractional Brownian motions () and regular (or smooth)
motion (H=1).
We discover that a critical wave number divides the infrared cutoffs into
three categories, critical, subcritical and supercritical; they give rise to
different scaling laws and phase diagrams. We introduce the notions of sampling
drift and eddy diffusivity, and formulate variational principles to estimate
the eddy diffusivity. We show that fractional Brownian motions result from a
dominant sampling drift
Langevin dynamics with space-time periodic nonequilibrium forcing
We present results on the ballistic and diffusive behavior of the Langevin
dynamics in a periodic potential that is driven away from equilibrium by a
space-time periodic driving force, extending some of the results obtained by
Collet and Martinez. In the hyperbolic scaling, a nontrivial average velocity
can be observed even if the external forcing vanishes in average. More
surprisingly, an average velocity in the direction opposite to the forcing may
develop at the linear response level -- a phenomenon called negative mobility.
The diffusive limit of the non-equilibrium Langevin dynamics is also studied
using the general methodology of central limit theorems for additive
functionals of Markov processes. To apply this methodology, which is based on
the study of appropriate Poisson equations, we extend recent results on
pointwise estimates of the resolvent of the generator associated with the
Langevin dynamics. Our theoretical results are illustrated by numerical
simulations of a two-dimensional system
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