5,278 research outputs found
Transitions from deterministic to stochastic diffusion
We examine characteristic properties of deterministic and stochastic
diffusion in low-dimensional chaotic dynamical systems. As an example, we
consider a periodic array of scatterers defined by a simple chaotic map on the
line. Adding different types of time-dependent noise to this model we compute
the diffusion coefficient from simulations. We find that there is a crossover
from deterministic to stochastic diffusion under variation of the perturbation
strength related to different asymptotic laws for the diffusion coefficient.
Typical signatures of this scenario are suppression and enhancement of normal
diffusion. Our results are explained by a simple theoretical approximation.Comment: 6 pages (revtex) with 3 figures (postscript
Linear and fractal diffusion coefficients in a family of one dimensional chaotic maps
We analyse deterministic diffusion in a simple, one-dimensional setting
consisting of a family of four parameter dependent, chaotic maps defined over
the real line. When iterated under these maps, a probability density function
spreads out and one can define a diffusion coefficient. We look at how the
diffusion coefficient varies across the family of maps and under parameter
variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated
in terms of generalised Takagi functions, we derive exact, fully analytical
expressions for the diffusion coefficients. Typically, for simple maps these
quantities are fractal functions of control parameters. However, our family of
four maps exhibits both fractal and linear behavior. We explain these different
structures by looking at the topology of the Markov partitions and the ergodic
properties of the maps.Comment: 21 pages, 19 figure
Capturing correlations in chaotic diffusion by approximation methods
We investigate three different methods for systematically approximating the
diffusion coefficient of a deterministic random walk on the line which contains
dynamical correlations that change irregularly under parameter variation.
Capturing these correlations by incorporating higher order terms, all schemes
converge to the analytically exact result. Two of these methods are based on
expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method
approximates Markov partitions and transition matrices by using the escape rate
theory of chaotic diffusion. We check the practicability of the different
methods by working them out analytically and numerically for a simple
one-dimensional map, study their convergence and critically discuss their
usefulness in identifying a possible fractal instability of parameter-dependent
diffusion, in case of dynamics where exact results for the diffusion
coefficient are not available.Comment: 11 pages, 5 figure
Density-dependent diffusion in the periodic Lorentz gas
We study the deterministic diffusion coefficient of the two-dimensional
periodic Lorentz gas as a function of the density of scatterers. Results
obtained from computer simulations are compared to the analytical approximation
of Machta and Zwanzig [Phys.Rev.Lett. 50, 1959 (1983)] showing that their
argument is only correct in the limit of high densities. We discuss how the
Machta-Zwanzig argument, which is based on treating diffusion as a Markovian
hopping process on a lattice, can be corrected systematically by including
microscopic correlations. We furthermore show that, on a fine scale, the
diffusion coefficient is a non-trivial function of the density. We finally
argue that, on a coarse scale and for lower densities, the diffusion
coefficient exhibits a Boltzmann-like behavior, whereas for very high densities
it crosses over to a regime which can be understood qualitatively by the
Machta-Zwanzig approximation.Comment: 9 pages (revtex) with 9 figures (postscript
Negative and Nonlinear Response in an Exactly Solved Dynamical Model of Particle Transport
We consider a simple model of particle transport on the line defined by a
dynamical map F satisfying F(x+1) = 1 + F(x) for all x in R and F(x) = ax + b
for |x| < 0.5. Its two parameters a (`slope') and b (`bias') are respectively
symmetric and antisymmetric under reflection x -> R(x) = -x. Restricting
ourselves to the chaotic regime |a| > 1 and therein mainly to the part a>1 we
study not only the `diffusion coefficient' D(a,b), but also the `current'
J(a,b). An important tool for such a study are the exact expressions for J and
D as obtained recently by one of the authors. These expressions allow for a
quite efficient numerical implementation, which is important, because the
functions encountered typically have a fractal character. The main results are
presented in several plots of these functions J(a,b) and D(a,b) and in an
over-all `chart' displaying, in the parameter plane, all possibly relevant
information on the system including, e.g., the dynamical phase diagram as well
as invariants such as the values of topological invariants (kneading numbers)
which, according to the formulas, determine the singularity structure of J and
D. Our most significant findings are: 1) `Nonlinear Response': The parameter
dependence of these transport properties is, throughout the `ergodic' part of
the parameter plane (i.e. outside the infinitely many Arnol'd tongues)
fractally nonlinear. 2) `Negative Response': Inside certain regions with an
apparently fractal boundary the current J and the bias b have opposite signs.Comment: corrected typos and minor reformulations; 28 pages (revtex) with 7
figures (postscript); accepted for publication in JS
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