Universality of attractors at weak dissipation and particles distribution in turbulence

TL;DR

This paper investigates the universal properties of attractors in weakly dissipative flows and predicts particle distribution patterns in turbulence, highlighting the role of pressure fluctuations.

Contribution

It introduces a universal representation of strange attractors in weakly dissipative systems and derives their fractal spectrum, connecting theory with turbulence particle distribution.

Findings

Space-averaged properties follow a universal log-normal distribution.

Derived spectrum of fractal dimensions for attractors.

Provides testable predictions for particle distribution in turbulence.

Abstract

We study stationary solutions to the continuity equation for weakly compressible flows. These describe non-equilibrium steady states of weakly dissipative dynamical systems. Compressibility is a singular perturbation that changes the steady state density from a constant "microcanonical" distribution into a singular multifractal measure supported on the "strange attractor". We introduce a representation of the latter and show that the space-averaged properties are described universally by a log-normal distribution determined by a single structure function. The spectrum of fractal dimensions is derived. Application to the problem of distribution of particles in turbulence gives testable predictions for real turbulence and stresses the role of pressure fluctuations.