1,198 research outputs found
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 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 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 , and a continuous spectrum on the real line . 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
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