Suppression and enhancement of diffusion in disordered dynamical systems

Abstract

The impact of quenched disorder on deterministic diffusion in chaotic dynamical systems is studied. As a simple example, we consider piecewise linear maps on the line. In computer simulations we find a complex scenario of multiple suppression and enhancement of normal diffusion, under variation of the perturbation strength. These results are explained by a theoretical approximation, showing that the oscillations emerge as a direct consequence of the unperturbed diffusion coefficient, which is known to be a fractal function of a control parameter.

Deterministic diffusion in flower-shaped billiards

“…The joint efforts compiled in Refs. [8,9,12] may therefore be considered as first steps towards answering the conjecture of Refs. [5,6], which suggested a possible universality of fractal diffusion coefficients in low-dimensional fully chaotic dynamical systems exhibiting some spatial periodicity, for the case of chaotic Hamiltonian dynamical systems such as particle billiards.…”

“…[12] appears to be most suitable for understanding the irregular behavior of the parameter dependent diffusion coefficient, because it conveniently transforms the diffusive dynamics into a sum over the velocity correlation function, whose specific parameter dependence can in turn be analyzed step by step. In particular, this approach yields an exact convergence to the parameter dependent diffusion coefficient as obtained from simulations.…”

“…These deficiencies were essentially resolved in Ref. [12] by the derivation of a Green-Kubo formula which employs the symbolic dynamics on the hexagonal lattice of traps introduced in section V. The result reads…”

“…The diffusion coefficient of this standard periodic Lorentz gas has been studied in various ways both analytically and numerically, where recent work focused particularly onto its density dependence (see Refs. [8,12] and further references therein). However, the density of this model cannot be varied much because of the condition of finite horizon, which prohibits collision-free ballistic motion and thus guarantees the existence of normal diffusion [10].…”

“…However, whether the numerically detected irregularities in the diffusion coefficient were of a fractal nature remains an open question. More recently, a third approximation scheme was proposed by deriving a Green-Kubo formula that exactly generalizes the Machta-Zwanzig approximation [12]. Applying all these methods led to the conclusion that including long-term correlations, or memory effects, was indispensable to reproduce the precise functional form of the parameter dependent diffusion coefficient for the standard periodic Lorentz gas.…”