Microscopic chaos and Gaussian diffusion processes

Chaos, Solitons & Fractals, 2002

We describe one-dimensional discrete difFusive motion that is driven by a purely deterministic system, the logistic map. We find normal diffusive motion above a critical value k, of the chaotic parameter. At A, i, the diftusion constant D plays the role of an order parameter, with a critical exponent of 1/2. In the presence of external noise, D is expressed in terms of a universal scaling function and the critical exponent associated with the noise is found to be 1. We also consider biased di6'usive motion driven by the logistic map. We find that the di8'usion process thus generated is di6'erent from random biased di6'usion. In contrast to the case of the biased "random walker" where all configurations ("walks") are still possible, here certain configurations are not allowed. However, the statistical properties approach those for random biased di6'usion in the infinite time limit. We define a measure of the e6'ective randomness.

1999

Chaos, 1998

We study the consequences of deterministic chaos for diffusion-controlled reaction. As an example, we analyze a diffusive-reactive deterministic multibaker and a parameter-dependent variation of it. We construct the diffusive and the reactive modes of the models as eigenstates of the Frobenius-Perron operator. The associated eigenvalues provide the dispersion relations of diffusion and reaction and, hence, they determine the reaction rate. For the simplest model we show explicitly that the reaction rate behaves as phenomenologically expected for one-dimensional diffusion-controlled reaction. Under parametric variation, we find that both the diffusion coefficient and the reaction rate have fractal-like dependences on the system parameter.

Physica D: Nonlinear Phenomena, 2001

We examine some new diagnostics for the behavior of a field ρ evolving in an advective-diffusive system. One of these diagnostics is approximately the Fourier second moment (denoted as χ 2 ) and the other is the linear entropy or field intensity S, the latter being significantly easier to compute or measure. We establish that as a result of chaos the increasing structure in ρ is accompanied by χ increasing exponentially rapidly in time at a rate given by ρ-dependent Lyapunov exponents Λ i and dominated by the largest one Λ max . Noise or diffusive coarse-graining of ρ causes χ to decrease as χ 2 ≈ 1 4 Dt, where D is a measure of the diffusion. When both effects are present the competition between the processes leads to metastability for χ followed by a final diffusive tail. The initial stages may be chaotic or diffusive depending upon the value of Λ −1 max 2Dχ 2 (0) but the metastable value of χ 2 is given by χ 2 * = i Λ i /2D irrespective. SinceṠ = −2Dχ 2 , similar analysis applies to S, and in particular there exists a metastable decay rate for S given byṠ * = i Λ i . These arguments are verified for a simple case, the Arnol'd Cat Map with added diffusive noise.

Physica A: Statistical Mechanics and its Applications, 1998

We investigate the in uence of a weak uniform bias on chaotic di usion generated by iterated one-dimensional maps which, in the absence of the bias, lead to subdi usion.

2004

Annales de l'Institut Henri Poincare (B) Probability and Statistics, 1998

In this paper, we are interested in a stochastic di erential equation which is nonlinear in the following sense : both the di usion and the drift coe cients depend locally on the density of the time marginal of the solution. When the law of the initial data has a smooth density with respect to Lebesgue measure, we prove existence and uniqueness for this equation. Under more restrictive assumptions on the density, we approximate the solution by a system of moderately interacting di usion processes and obtain a trajectorial propagation of chaos result. Finally, we study the uctuations associated with the convergence of the empirical measure of the system to the law of the solution of the nonlinear equation. In this situation, the convergence rate is di erent from p n.

Chaos — The Interplay Between Stochastic and Deterministic Behaviour

Deviations from simple Brownian motion have been observed in intermittent chaotic systems; of particular interest has been the case of enhanced diffusion. We review an approach to this anomalous behavior based on L~vy scale-invariant distributions to describe transport in such systems. We introduce the basic ingredients that make the approach useful in describing the non-Brownian behavior and demonstrate the applicability in the cases of the standard map, "egg-crate" potential and a one-dimensional iterated map which shows a combined laminar and dispersive motion.

Physical Review Letters

Consider a chaotic dynamical system generating Brownian motion-like diffusion. Consider a second, non-chaotic system in which all particles localize. Let a particle experience a random combination of both systems by sampling between them in time. What type of diffusion is exhibited by this random dynamical system? We show that the resulting dynamics can generate anomalous diffusion, where in contrast to Brownian normal diffusion the mean square displacement of an ensemble of particles increases nonlinearly in time. Randomly mixing simple deterministic walks on the line we find anomalous dynamics characterised by ageing, weak ergodicity breaking, breaking of self-averaging and infinite invariant densities. This result holds for general types of noise and for perturbing nonlinear dynamics in bifurcation scenarios.

Computational Mathematics and Mathematical Physics, 2010

We consider the problem of classifying the convex pentagons that edge-to-edge tile the plane. We demonstrate that in any of such tiling there exists a pentagon with one of the following degree lists of its vertices: