872 research outputs found
Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases
The kinetic theory of gases provides methods for calculating Lyapunov
exponents and other quantities, such as Kolmogorov-Sinai entropies, that
characterize the chaotic behavior of hard-ball gases. Here we illustrate the
use of these methods for calculating the Kolmogorov-Sinai entropy, and the
largest positive Lyapunov exponent, for dilute hard-ball gases in equilibrium.
The calculation of the largest Lyapunov exponent makes interesting connections
with the theory of propagation of hydrodynamic fronts. Calculations are also
presented for the Lyapunov spectrum of dilute, random Lorentz gases in two and
three dimensions, which are considerably simpler than the corresponding
calculations for hard-ball gases. The article concludes with a brief discussion
of some interesting open problems.Comment: 41 pages (REVTEX); 7 figs., 4 of which are included in LaTeX source.
(Fig.7 doesn't print well on some printers) This revised paper will appear in
"Hard Ball Systems and the Lorentz Gas", D. Szasz ed., Encyclopaedia of
Mathematical Sciences, Springe
Dynamical estimates of chaotic systems from Poincar\'e recurrences
We show that the probability distribution function that best fits the
distribution of return times between two consecutive visits of a chaotic
trajectory to finite size regions in phase space deviates from the exponential
statistics by a small power-law term, a term that represents the deterministic
manifestation of the dynamics, which can be easily experimentally detected and
theoretically estimated. We also provide simpler and faster ways to calculate
the positive Lyapunov exponents and the short-term correlation function by
either realizing observations of higher probable returns or by calculating the
eigenvalues of only one very especial unstable periodic orbit of low-period.
Finally, we discuss how our approaches can be used to treat data coming from
complex systems.Comment: subm. for publication. Accepted fpr publication in Chao
Chaos properties and localization in Lorentz lattice gases
The thermodynamic formalism of Ruelle, Sinai, and Bowen, in which chaotic
properties of dynamical systems are expressed in terms of a free energy-type
function - called the topological pressure - is applied to a Lorentz Lattice
Gas, as typical for diffusive systems with static disorder. In the limit of
large system sizes, the mechanism and effects of localization on large clusters
of scatterers in the calculation of the topological pressure are elucidated and
supported by strong numerical evidence. Moreover it clarifies and illustrates a
previous theoretical analysis [Appert et al. J. Stat. Phys. 87,
chao-dyn/9607019] of this localization phenomenon.Comment: 32 pages, 19 Postscript figures, submitted to PR
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving maps on a d-dimensional torus. The quantity we use to measure
the irreversibility of the dynamics is the dissipation time. We focus on the
asymptotic behaviour of this time in the limit of small noise. We derive
universal lower and upper bounds for the dissipation time in terms of various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
Entropy of semiclassical measures for nonpositively curved surfaces
We study the asymptotic properties of eigenfunctions of the Laplacian in the
case of a compact Riemannian surface of nonpositive sectional curvature. We
show that the Kolmogorov-Sinai entropy of a semiclassical measure for the
geodesic flow is bounded from below by half of the Ruelle upper bound. We
follow the same main strategy as in the Anosov case (arXiv:0809.0230). We focus
on the main differences and refer the reader to (arXiv:0809.0230) for the
details of analogous lemmas.Comment: 20 pages. This note provides a detailed proof of a result announced
in appendix A of a previous work (arXiv:0809.0230, version 2
Radius of curvature approach to the Kolmogorov-Sinai entropy of dilute hard particles in equilibrium
We consider the Kolmogorov-Sinai entropy for dilute gases of hard disks
or spheres. This can be expanded in density as , with the diameter of the sphere or disk,
the density, and the dimensionality of the system. We estimate the
constant by solving a linear differential equation for the approximate
distribution of eigenvalues of the inverse radius of curvature tensor. We
compare the resulting values of both to previous estimates and to existing
simulation results, finding very good agreement with the latter. Also, we
compare the distribution of eigenvalues of the inverse radius of curvature
tensor resulting from our calculations to new simulation results. For most of
the spectrum the agreement between our calculations and the simulations again
is very good.Comment: 12 pages, 4 figure
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