636 research outputs found
Separating Solution of a Quadratic Recurrent Equation
In this paper we consider the recurrent equation
for with and given. We give conditions
on that guarantee the existence of such that the sequence
with tends to a finite positive limit as .Comment: 13 pages, 6 figures, submitted to J. Stat. Phy
Proving The Ergodic Hypothesis for Billiards With Disjoint Cylindric Scatterers
In this paper we study the ergodic properties of mathematical billiards
describing the uniform motion of a point in a flat torus from which finitely
many, pairwise disjoint, tubular neighborhoods of translated subtori (the so
called cylindric scatterers) have been removed. We prove that every such system
is ergodic (actually, a Bernoulli flow), unless a simple geometric obstacle for
the ergodicity is present.Comment: 24 pages, AMS-TeX fil
Universality of Cluster Dynamics
We have studied the kinetics of cluster formation for dynamical systems of
dimensions up to interacting through elastic collisions or coalescence.
These systems could serve as possible models for gas kinetics, polymerization
and self-assembly. In the case of elastic collisions, we found that the cluster
size probability distribution undergoes a phase transition at a critical time
which can be predicted from the average time between collisions. This enables
forecasting of rare events based on limited statistical sampling of the
collision dynamics over short time windows. The analysis was extended to
L-normed spaces () to allow for some amount of
interpenetration or volume exclusion. The results for the elastic collisions
are consistent with previously published low-dimensional results in that a
power law is observed for the empirical cluster size distribution at the
critical time. We found that the same power law also exists for all dimensions
, 2D L norms, and even for coalescing collisions in 2D. This
broad universality in behavior may be indicative of a more fundamental process
governing the growth of clusters
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
Shocks and Universal Statistics in (1+1)-Dimensional Relativistic Turbulence
We propose that statistical averages in relativistic turbulence exhibit
universal properties. We consider analytically the velocity and temperature
differences structure functions in the (1+1)-dimensional relativistic
turbulence in which shock waves provide the main contribution to the structure
functions in the inertial range. We study shock scattering, demonstrate the
stability of the shock waves, and calculate the anomalous exponents. We comment
on the possibility of finite time blowup singularities.Comment: 37 pages, 7 figure
Weighted Fixed Points in Self-Similar Analysis of Time Series
The self-similar analysis of time series is generalized by introducing the
notion of scenario probabilities. This makes it possible to give a complete
statistical description for the forecast spectrum by defining the average
forecast as a weighted fixed point and by calculating the corresponding a
priori standard deviation and variance coefficient. Several examples of
stock-market time series illustrate the method.Comment: two additional references are include
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.
The entropy of ``strange'' billiards inside n-simplexes
In the present work we investigate a new type of billiards defined inside of
--simplex regions. We determine an invariant ergodic (SRB) measure of the
dynamics for any dimension. In using symbolic dynamics, the (KS or metric)
entropy is computed and we find that the system is chaotic for all cases .Comment: 8 pages, uuencoded compressed postscript fil
Evolution to a singular measure and two sums of Lyapunov exponents
We consider dissipative dynamical systems represented by a smooth
compressible flow in a finite domain. The density evolves according to the
continuity (Liouville) equation. For a general, non-degenerate flow the result
of the infinite time evolution of an initially smooth density is a singular
measure. We give a condition for the non-degeneracy which allows to decide for
a given flow whether the infinite time limit is singular. The condition uses a
Green-Kubo type formula for the space-averaged sum of forward and
backward-in-time Lyapunov exponents. We discuss how the sums determine the
fluctuations of the entropy production rate in the SRB state and give examples
of computation of the sums for certain velocity fields.Comment: 4 pages, published versio
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