Fractality of the Hydrodynamic Modes of Diffusion

Abstract

Transport by normal diffusion can be decomposed into hydrodynamic modes which relax exponentially toward the equilibrium state. In chaotic systems with 2 degrees of freedom, the fine scale structures of these modes are singular and fractal, characterized by a Hausdorff dimension given in terms of Ruelle's topological pressure. For long-wavelength modes, we relate the Hausdorff dimension to the diffusion coefficient and the Lyapunov exponent. This relationship is tested numerically on two Lorentz gases, one with hard repulsive forces, the other with attractive, Yukawa forces. The agreement with theory is excellent.

Log-periodic drift oscillations in self-similar billiards

“…In a statistically stationary state, this model represents a mechanical system at equilibrium, whose distribution, measured along the boundary of the scattering discs, is uniform in the position and normal velocity angles. Meanwhile the process of relaxation to equilibrium itself is characterized by deterministic hydrodynamic modes of diffusion with fractal properties [7]. One can also induce a non-equilibrium stationary state on a cylindrical version of this billiard by coupling it to stochastic reservoirs of point-particles at its boundaries [8]; these are the so-called flux boundary conditions.…”

Fractal dimension of transport coefficients in a deterministic dynamical system

“…Similar behaviour was found for the diffusion coefficient of a particle thrown onto a periodically corrugated floor and subject to various types of external force, including Hamiltonian [4] and dissipative [5,6] systems. In chaotic systems with two degrees of freedom the modes governing the relaxation to the thermodynamic equilibrium form a fractal structure with a nontrivial fractal dimension which can be related to the Lapunov exponent and the diffusion coefficient of the system [7,8,9]. Another example is provided by a deterministic dynamical system with dynamics defined by iteration of a one-dimensional map: the mean velocity and the drift coefficient in this system were shown to be irregular, nowhere differentiable functions of the system control parameters [10,11,12,13,14].…”

Lifetimes of noisy repellors

“…The simulations by Paar and Pavin [8] also indicate strong variations in escape rate near short periodic orbits which are connected to higher orders in perturbation theory and the complicated spatial structures that are characteristic for next to leading eigenvectors [7]. …”

“…Since the kernel K is not self adjoint left-and right-eigenvectors are different. Simple examples show that the left eigenvectors develop fractal features [7]. With noise the finest scale structures are washed out, but higher levels of the fractal hierarchy survive.…”

“…The next to leading eigenvalue is λ 1 = −0.6. Its left eigenvector shows the fractal structures one expects for such maps [7]. …”