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