Deterministic Chaos and Diffusion: From Theory to Experiments

1: Diffusion coefficient D(a) of the map depicted in the lower right corner. Shown are two pieces of the periodically continued map plus a trajectory segment of a diffusing particle. The slope a determining the average step lengths of a walker serves as a control parameter. The dashed lines represent random walk solutions for D(a), the inset depicts a blowup for the initial region of the main graph. 1 Introduction: deterministic chaos and diffusion A century ago Einstein developed a theory of diffusion that is based on the assumption of stochasticity for a Brownian particle. On a microscopic level, however, the particle's dynamics is governed by Newton's deterministic equations of motion, which typically are highly nonlinear. This motivates to replace the hypothesis of stochasticity underlying traditional statistical mechanics by the hypothesis of microscopic deterministic chaos in the equations of motion of a Brownian particle. Such a deterministic chaos approach to transport takes the full memory of a particle into account while in stochastic models dynamical correlations tend to be neglected. One may thus ask whether for a given deterministic dynamical system there exist non-trivial dynamical correlations; if so, whether fundamental principles of transport theory such as Fick's laws are still recovered; or if not, what the precise consequences for diffusive transport are. These questions form the problem of deterministic chaos and diffusion.