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Deterministic Diffusion
One-Dimensional Chaotic Dynamical Systems

R. Klages, Ph.D. thesis, TU Berlin

A theoretical analysis of simple one-dimensional models for deterministic diffusion has been performed. These models consist of arrays of identical scatterers with a moving particle. The microscopic chaotic scattering process of the particle can be changed continuously by switching a single control parameter. This induces a parameter dependence for the macroscopic diffusion coefficient of the system.

For the calculation of the diffusion coefficient, new analytical and numerical methods have been developed. They are based on the theory of chaotic dynamical systems and on the theory of transport of statistical mechanics. The computed parameter-dependent difffusion coefficient shows a surprisingly complex fractal structure, which is obtained for the first time in a dynamical system.

A detailed analysis of the process of deterministic diffusion, which leads to this fractal diffusion coefficient, shows that the system has macroscopic dynamical properties analogous to the ones of a simple statistical diffusion process. On a fine scale, however, structures appear which are inherent to the deterministic microscopic properties of the scatterers.

To explain the structure of the fractal diffusion coefficient, qualitative and quantitative methods are developed. These methods relate the sequence of oscillations in the strength of the parameter-dependent diffusion coefficient to the microscopic coupling of the single scatterers, which changes by varying the control parameter. By employing a newly-defined class of fractal functions, a systematic analytical and numerical approximation procedure is introduced, which provides a better understanding of certain details of the parameter-dependent diffusion coefficient. Simple random walk models are applied to the process of deterministic diffusion. This leads to the prediction of universal laws for the parameter-dependent diffusion coefficient on a large scale. Moreover, there is evidence for the existence of a dynamical phase transition. In analogy to related phenomena recently discovered in dynamical systems, this transition can be understood as a crisis in deterministic diffusion.

It is supposed that fractal diffusion coefficients, and characteristic properties of their underlying diffusion processes, are obtained for a variety of deterministic dynamical systems. To a certain extent, such systems could already be realized experimentally. Thus, the results presented in this work should not only give more insight into fundamental questions concerning the microscopic origin of macroscopic transport, but they could also become important for special technological applications as, e.g., in the field of semi-conductor devices.

This thesis has been published (in English) by:

Wissenschaft & Technik Verlag (Berlin), April 1996; 137 pages, 28 figures;

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Rainer Klages