Decorrelating a compressible turbulent flow: an experiment

Floating particles that are initially distributed uniformly on the surface of a turbulent fluid, subsequently coagulate, until finally a steady state is reached. This being so, they manifestly form a compressible system. In this experiment, the information dimension D_1, and the Lyapunov exponents of the coagulated floaters, are measured. The trajectories and the velocity fields of the particles are captured in a sequence of rapidly acquired images. Then the velocity sequence is randomly shuffled in time to generate new trajectories. This analysis mimics the Kraichnan ensemble and yields properties of a velocity correlation function that is delta-correlated in time (but not in space). The measurements are compared with theoretical expectations and with simulations of Boffetta et al., that closely mimic the laboratory experiment reported here.Comment: 6 pages, 5 figure

The Fractality of the Hydrodynamic Modes of Diffusion

Transport by normal diffusion can be decomposed into the so-called hydrodynamic modes which relax exponentially toward the equilibrium state. In chaotic systems with two degrees of freedom, the fine scale structure of these hydrodynamic modes is singular and fractal. We characterize them by their Hausdorff dimension which is given in terms of Ruelle's topological pressure. For long-wavelength modes, we derive a striking relation between the Hausdorff dimension, the diffusion coefficient, and the positive Lyapunov exponent of the system. This relation is tested numerically on two chaotic systems exhibiting diffusion, both periodic Lorentz gases, one with hard repulsive forces, the other with attractive, Yukawa forces. The agreement of the data with the theory is excellent

Measurement of entropy production rate in compressible turbulence

The rate of change of entropy $\dot S$ is measured for a system of particles floating on the surface of a fluid maintained in a turbulent steady state. The resulting coagulation of the floaters allows one to relate $\dot S$ to the velocity divergence and to the Lyapunov exponents characterizing the behavior of this system. The quantities measured from experiments and simulations are found to agree well with the theoretical predictions.Comment: 7 Pages, 4 figures, 1 tabl

The gauging of two-dimensional bosonic sigma models on world-sheets with defects

We extend our analysis of the gauging of rigid symmetries in bosonic two-dimensional sigma models with Wess-Zumino terms in the action to the case of world-sheets with defects. A structure that permits a non-anomalous coupling of such sigma models to world-sheet gauge fields of arbitrary topology is analysed, together with obstructions to its existence, and the classification of its inequivalent choices.Comment: 94 pages, 1 figur

Understanding deterministic diffusion by correlated random walks

Low-dimensional periodic arrays of scatterers with a moving point particle are ideal models for studying deterministic diffusion. For such systems the diffusion coefficient is typically an irregular function under variation of a control parameter. Here we propose a systematic scheme of how to approximate deterministic diffusion coefficients of this kind in terms of correlated random walks. We apply this approach to two simple examples which are a one-dimensional map on the line and the periodic Lorentz gas. Starting from suitable Green-Kubo formulas we evaluate hierarchies of approximations for their parameter-dependent diffusion coefficients. These approximations converge exactly yielding a straightforward interpretation of the structure of these irregular diffusion coeficients in terms of dynamical correlations.Comment: 13 pages (revtex) with 5 figures (postscript

Spectral analysis and an area-preserving extension of a piecewise linear intermittent map

We investigate spectral properties of a 1-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius--Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius--Perron operator has two simple real eigenvalues 1 and $\lambda_d \in (-1,0)$, and a continuous spectrum on the real line $[0,1]$. From these spectral properties, we also found that this system exhibits power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.Comment: 12 pages, 3 figure

Drift of particles in self-similar systems and its Liouvillian interpretation

We study the dynamics of classical particles in different classes of spatially extended self-similar systems, consisting of (i) a self-similar Lorentz billiard channel, (ii) a self-similar graph, and (iii) a master equation. In all three systems the particles typically drift at constant velocity and spread ballistically. These transport properties are analyzed in terms of the spectral properties of the operator evolving the probability densities. For systems (i) and (ii), we explain the drift from the properties of the Pollicott-Ruelle resonance spectrum and corresponding eigenvectorsComment: To appear in Phys. Rev.

Steady-state conduction in self-similar billiards

The self-similar Lorentz billiard channel is a spatially extended deterministic dynamical system which consists of an infinite one-dimensional sequence of cells whose sizes increase monotonically according to their indices. This special geometry induces a nonequilibrium stationary state with particles flowing steadily from the small to the large scales. The corresponding invariant measure has fractal properties reflected by the phase-space contraction rate of the dynamics restricted to a single cell with appropriate boundary conditions. In the near-equilibrium limit, we find numerical agreement between this quantity and the entropy production rate as specified by thermodynamics

Time-dependent mode structure for Lyapunov vectors as a collective movement in quasi-one-dimensional systems

Time dependent mode structure for the Lyapunov vectors associated with the stepwise structure of the Lyapunov spectra and its relation to the momentum auto-correlation function are discussed in quasi-one-dimensional many-hard-disk systems. We demonstrate mode structures (Lyapunov modes) for all components of the Lyapunov vectors, which include the longitudinal and transverse components of their spatial and momentum parts, and their phase relations are specified. These mode structures are suggested from the form of the Lyapunov vectors corresponding to the zero-Lyapunov exponents. Spatial node structures of these modes are explained by the reflection properties of the hard-walls used in the models. Our main interest is the time-oscillating behavior of Lyapunov modes. It is shown that the largest time-oscillating period of the Lyapunov modes is twice as long as the time-oscillating period of the longitudinal momentum auto-correlation function. This relation is satisfied irrespective of the particle number and boundary conditions. A simple explanation for this relation is given based on the form of the Lyapunov vector.Comment: 39 pages, 21 figures, Manuscript including the figures of better quality is available from http://www.phys.unsw.edu.au/~gary/Research.htm

Random walk approach to the d-dimensional disordered Lorentz gas

A correlated random walk approach to diffusion is applied to the disordered nonoverlapping Lorentz gas. By invoking the Lu-Torquato theory for chord-length distributions in random media [J. Chem. Phys. 98, 6472 (1993)], an analytic expression for the diffusion constant in arbitrary number of dimensions d is obtained. The result corresponds to an Enskog-like correction to the Boltzmann prediction, being exact in the dilute limit, and better or nearly exact in comparison to renormalized kinetic theory predictions for all allowed densities in d=2,3. Extensive numerical simulations were also performed to elucidate the role of the approximations involved.Comment: 5 pages, 5 figure