2,147 research outputs found
Billiards with polynomial mixing rates
While many dynamical systems of mechanical origin, in particular billiards,
are strongly chaotic -- enjoy exponential mixing, the rates of mixing in many
other models are slow (algebraic, or polynomial). The dynamics in the latter
are intermittent between regular and chaotic, which makes them particularly
interesting in physical studies. However, mathematical methods for the analysis
of systems with slow mixing rates were developed just recently and are still
difficult to apply to realistic models. Here we reduce those methods to a
practical scheme that allows us to obtain a nearly optimal bound on mixing
rates. We demonstrate how the method works by applying it to several classes of
chaotic billiards with slow mixing as well as discuss a few examples where the
method, in its present form, fails.Comment: 39pages, 11 figue
Deterministic Walks in Quenched Random Environments of Chaotic Maps
This paper concerns the propagation of particles through a quenched random
medium. In the one- and two-dimensional models considered, the local dynamics
is given by expanding circle maps and hyperbolic toral automorphisms,
respectively. The particle motion in both models is chaotic and found to
fluctuate about a linear drift. In the proper scaling limit, the cumulative
distribution function of the fluctuations converges to a Gaussian one with
system dependent variance while the density function shows no convergence to
any function. We have verified our analytical results using extreme precision
numerical computations.Comment: 18 pages, 9 figure
Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas
We investigate analytically and numerically the spatial structure of the
non-equilibrium stationary states (NESS) of a point particle moving in a two
dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a
constant external electric field E as well as a Gaussian thermostat which keeps
the speed |v| constant. We show that despite the singular nature of the SRB
measure its projections on the space coordinates are absolutely continuous. We
further show that these projections satisfy linear response laws for small E.
Some of them are computed numerically. We compare these results with those
obtained from simple models in which the collisions with the obstacles are
replaced by random collisions.Similarities and differences are noted.Comment: 24 pages with 9 figure
Persistence effects in deterministic diffusion
In systems which exhibit deterministic diffusion, the gross parameter
dependence of the diffusion coefficient can often be understood in terms of
random walk models. Provided the decay of correlations is fast enough, one can
ignore memory effects and approximate the diffusion coefficient according to
dimensional arguments. By successively including the effects of one and two
steps of memory on this approximation, we examine the effects of
``persistence'' on the diffusion coefficients of extended two-dimensional
billiard tables and show how to properly account for these effects, using walks
in which a particle undergoes jumps in different directions with probabilities
that depend on where they came from.Comment: 7 pages, 7 figure
Multilevel ensemble Kalman filtering for spatio-temporal processes
We design and analyse the performance of a multilevel ensemble Kalman filter
method (MLEnKF) for filtering settings where the underlying state-space model
is an infinite-dimensional spatio-temporal process. We consider underlying
models that needs to be simulated by numerical methods, with discretization in
both space and time. The multilevel Monte Carlo (MLMC) sampling strategy,
achieving variance reduction through pairwise coupling of ensemble particles on
neighboring resolutions, is used in the sample-moment step of MLEnKF to produce
an efficient hierarchical filtering method for spatio-temporal models. Under
sufficient regularity, MLEnKF is proven to be more efficient for weak
approximations than EnKF, asymptotically in the large-ensemble and
fine-numerical-resolution limit. Numerical examples support our theoretical
findings.Comment: Version 1: 39 pages, 4 figures.arXiv admin note: substantial text
overlap with arXiv:1608.08558 . Version 2 (this version): 52 pages, 6
figures. Revision primarily of the introduction and the numerical examples
sectio
Rotating Leaks in the Stadium Billiard
The open stadium billiard has a survival probability, , that depends on
the rate of escape of particles through the leak. It is known that the decay of
is exponential early in time while for long times the decay follows a
power law. In this work we investigate an open stadium billiard in which the
leak is free to rotate around the boundary of the stadium at a constant
velocity, . It is found that is very sensitive to . For
certain values is purely exponential while for other values the
power law behaviour at long times persists. We identify three ranges of
values corresponding to three different responses of . It is
shown that these variations in are due to the interaction of the moving
leak with Marginally Unstable Periodic Orbits (MUPOs)
A simple piston problem in one dimension
We study a heavy piston that separates finitely many ideal gas particles
moving inside a one-dimensional gas chamber. Using averaging techniques, we
prove precise rates of convergence of the actual motions of the piston to its
averaged behavior. The convergence is uniform over all initial conditions in a
compact set. The results extend earlier work by Sinai and Neishtadt, who
determined that the averaged behavior is periodic oscillation. In addition, we
investigate the piston system when the particle interactions have been
smoothed. The convergence to the averaged behavior again takes place uniformly,
both over initial conditions and over the amount of smoothing.Comment: Accepted by Nonlinearity. 27 pages, 2 figure
Chaos and stability in a two-parameter family of convex billiard tables
We study, by numerical simulations and semi-rigorous arguments, a
two-parameter family of convex, two-dimensional billiard tables, generalizing
the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A
17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard
tables are continuously deformed from the integrable circular billiard to
different versions of completely-chaotic stadia. In particular, we conjecture
that a new class of ergodic billiard tables is obtained in certain regions of
the two-dimensional parameter space, when the billiards are close to skewed
stadia. We provide heuristic arguments supporting this conjecture, and give
numerical confirmation using the powerful method of Lyapunov-weighted dynamics.Comment: 19 pages, 13 figures. Submitted for publication. Supplementary video
available at http://sistemas.fciencias.unam.mx/~dsanders
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