1,829 research outputs found
Log-periodic drift oscillations in self-similar billiards
We study a particle moving at unit speed in a self-similar Lorentz billiard
channel; the latter consists of an infinite sequence of cells which are
identical in shape but growing exponentially in size, from left to right. We
present numerical computation of the drift term in this system and establish
the logarithmic periodicity of the corrections to the average drift
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
Locally Perturbed Random Walks with Unbounded Jumps
In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively
scaled, simple symmetric random walk, weak convergence to the Brownian motion
holds even in the case of local impurities if . The extension of their
result to finite range random walks is straightforward. Here, however, we are
interested in the situation when the random walk has unbounded range.
Concretely we generalize the statement of \cite{SzT} to unbounded random walks
whose jump distribution belongs to the domain of attraction of the normal law.
We do this first: for diffusively scaled random walks on having finite variance; and second: for random walks with distribution
belonging to the non-normal domain of attraction of the normal law. This result
can be applied to random walks with tail behavior analogous to that of the
infinite horizon Lorentz-process; these, in particular, have infinite variance,
and convergence to Brownian motion holds with the superdiffusive scaling.Comment: 16 page
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
Stable regimes for hard disks in a channel with twisting walls
We study a gas of hard disks in a box with semi-periodic boundary
conditions. The unperturbed gas is hyperbolic and ergodic (these facts are
proved for N=2 and expected to be true for all ). We study various
perturbations by twisting the outgoing velocity at collisions with the walls.
We show that the dynamics tends to collapse to various stable regimes, however
we define the perturbations and however small they are.Comment: 30 pages, final version to appear in "Chaos
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
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
Random billiards with wall temperature and associated Markov chains
By a random billiard we mean a billiard system in which the standard specular
reflection rule is replaced with a Markov transition probabilities operator P
that, at each collision of the billiard particle with the boundary of the
billiard domain, gives the probability distribution of the post-collision
velocity for a given pre-collision velocity. A random billiard with
microstructure (RBM) is a random billiard for which P is derived from a choice
of geometric/mechanical structure on the boundary of the billiard domain. RBMs
provide simple and explicit mechanical models of particle-surface interaction
that can incorporate thermal effects and permit a detailed study of
thermostatic action from the perspective of the standard theory of Markov
chains on general state spaces.
We focus on the operator P itself and how it relates to the
mechanical/geometric features of the microstructure, such as mass ratios,
curvatures, and potentials. The main results are as follows: (1) we
characterize the stationary probabilities (equilibrium states) of P and show
how standard equilibrium distributions studied in classical statistical
mechanics, such as the Maxwell-Boltzmann distribution and the Knudsen cosine
law, arise naturally as generalized invariant billiard measures; (2) we obtain
some basic functional theoretic properties of P. Under very general conditions,
we show that P is a self-adjoint operator of norm 1 on an appropriate Hilbert
space. In a simple but illustrative example, we show that P is a compact
(Hilbert-Schmidt) operator. This leads to the issue of relating the spectrum of
eigenvalues of P to the features of the microstructure;(3) we explore the
latter issue both analytically and numerically in a few representative
examples;(4) we present a general algorithm for simulating these Markov chains
based on a geometric description of the invariant volumes of classical
statistical mechanics
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