Fractal catastrophes

TL;DR

This paper develops a quantitative theory for how fractal attractors and caustics influence spatial clustering and Lyapunov exponents of particles in random force fields, with applications to turbulence and wave propagation.

Contribution

It introduces a method to analyze the effect of projection on the distribution of finite-time Lyapunov exponents for fractal attractors, revealing universal contributions of fractal catastrophes.

Findings

Caustics from fractal attractors significantly affect spatial Lyapunov exponent distributions.

The study explains the breakdown of a fluctuation relation upon projection.

Results have implications for turbulence particle dynamics and wave propagation in media.

Abstract

We analyse the spatial inhomogeneities ('spatial clustering') in the distribution of particles accelerated by a force that changes randomly in space and time. To quantify spatial clustering, the phase-space dynamics of the particles must be projected to configuration space. Folds of a smooth phase-space manifold give rise to catastrophes ('caustics') in this projection. When the inertial particle dynamics is damped by friction, however, the phase-space manifold converges towards a fractal attractor. It is believed that caustics increase spatial clustering also in this case, but a quantitative theory is missing. We solve this problem by determining how projection affects the distribution of finite-time Lyapunov exponents. Applying our method in one spatial dimension we find that caustics arising from the projection of a dynamical fractal attractor ('fractal catastrophes') make a distinct and universal contribution to the distribution of spatial finite-time Lyapunov exponents. Our results explain a projection formula for the spatial fractal correlation dimension, and how a fluctuation relation for the distribution of finite-time Lyapunov exponents for white-in-time Gaussian force fields breaks upon projection. We explore the implications of our results for heavy particles in turbulence, and for wave propagation in random media.