34 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
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
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
Dynamics of some piecewise smooth Fermi-Ulam Models
We find a normal form which describes the high energy dynamics of a class of
piecewise smooth Fermi-Ulam ping pong models; depending on the value of a
single real parameter, the dynamics can be either hyperbolic or elliptic. In
the first case we prove that the set of orbits undergoing Fermi acceleration
has zero measure but full Hausdorff dimension. We also show that for almost
every orbit the energy eventually falls below a fixed threshold. In the second
case we prove that, generically, we have stable periodic orbits for arbitrarily
high energies, and that the set of Fermi accelerating orbits may have infinite
measure.Comment: 22 pages, 4 figure
Khinchin theorem for integral points on quadratic varieties
We prove an analogue the Khinchin theorem for the Diophantine approximation
by integer vectors lying on a quadratic variety. The proof is based on the
study of a dynamical system on a homogeneous space of the orthogonal group. We
show that in this system, generic trajectories visit a family of shrinking
subsets infinitely often.Comment: 19 page
Heat conduction and Fourier's law in a class of many particle dispersing billiards
We consider the motion of many confined billiard balls in interaction and
discuss their transport and chaotic properties. In spite of the absence of mass
transport, due to confinement, energy transport can take place through binary
collisions between neighbouring particles. We explore the conditions under
which relaxation to local equilibrium occurs on time scales much shorter than
that of binary collisions, which characterize the transport of energy, and
subsequent relaxation to local thermal equilibrium. Starting from the
pseudo-Liouville equation for the time evolution of phase-space distributions,
we derive a master equation which governs the energy exchange between the
system constituents. We thus obtain analytical results relating the transport
coefficient of thermal conductivity to the frequency of collision events and
compute these quantities. We also provide estimates of the Lyapunov exponents
and Kolmogorov-Sinai entropy under the assumption of scale separation. The
validity of our results is confirmed by extensive numerical studies.Comment: fixed figs. 12 & 13. Version to appear in New J. Phy