672 research outputs found
Limiting Distribution of Frobenius Numbers for
The purpose of this paper is to give a complete derivation of the limiting
distribution of large Frobenius numbers outlined in earlier work of J. Bourgain
and Ya. Sinai and fill some gaps formulated there as hypotheses.Comment: 13 page
Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials
It is demonstrated numerically that smooth three degrees of freedom
Hamiltonian systems which are arbitrarily close to three dimensional strictly
dispersing billiards (Sinai billiards) have islands of effective stability, and
hence are non-ergodic. The mechanism for creating the islands are corners of
the billiard domain.Comment: 6 pages, 8 figures, submitted to Chao
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
Correlations for pairs of periodic trajectories for open billiards
In this paper we prove two asymptotic estimates for pairs of closed
trajectories for open billiards similar to those established by Pollicott and
Sharp for closed geodesics on negatively curved compact surfaces. The first of
these estimates holds for general open billiards in any dimension. The more
intricate second estimate is established for open billiards satisfying the so
called Dolgopyat type estimates. This class of billiards includes all open
billiards in the plane and open billiards in satisfying some
additional conditions
Topics in chaotic dynamics
Various kinematical quantities associated with the statistical properties of
dynamical systems are examined: statistics of the motion, dynamical bases and
Lyapunov exponents. Markov partitons for chaotic systems, without any attempt
at describing ``optimal results''. The Ruelle principle is illustrated via its
relation with the theory of gases. An example of an application predicts the
results of an experiment along the lines of Evans, Cohen, Morriss' work on
viscosity fluctuations. A sequence of mathematically oriented problems
discusses the details of the main abstract ergodic theorems guiding to a proof
of Oseledec's theorem for the Lyapunov exponents and products of random
matricesComment: Plain TeX; compile twice; 30 pages; 140K Keywords: chaos,
nonequilibrium ensembles, Markov partitions, Ruelle principle, Lyapunov
exponents, random matrices, gaussian thermostats, ergodic theory, billiards,
conductivity, gas.
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
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
Critical density of a soliton gas
We quantify the notion of a dense soliton gas by establishing an upper bound
for the integrated density of states of the quantum-mechanical Schr\"odinger
operator associated with the KdV soliton gas dynamics. As a by-product of our
derivation we find the speed of sound in the soliton gas with Gaussian spectral
distribution function.Comment: 7 page
- …