Fractal catastrophes

• 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.

Ämnesord

NATURVETENSKAP  -- Fysik (hsv//swe)

NATURAL SCIENCES  -- Physical Sciences (hsv//eng)

Nyckelord

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

Publikations- och innehållstyp

ref (ämneskategori)

art (ämneskategori)