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Fractal catastrophes

Journal article
Authors Jan Meibohm
Kristian Gustavsson
J. Bec
Bernhard Mehlig
Published in New Journal of Physics
Volume 22
Issue 1
ISSN 1367-2630
Publication year 2020
Published at Department of Physics (GU)
Language en
Keywords fractals, statistical mechanics, large deviation theory, spatial, clustering, catastrophe theory, non-equilibrium physics, inertial particles, preferential concentration, generalized dimensions, large deviations, heavy-particles, turbulent, statistics, stability, exponents, number, Physics
Subject categories Physical Sciences


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 (FTLEs). 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 FTLEs. Our results explain a projection formula for the spatial fractal correlation dimension, and how a fluctuation relation for the distribution of FTLEs 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.

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