72 research outputs found

### Deterministic diffusion in flower shape billiards

We propose a flower shape billiard in order to study the irregular parameter
dependence of chaotic normal diffusion. Our model is an open system consisting
of periodically distributed obstacles of flower shape, and it is strongly
chaotic for almost all parameter values. We compute the parameter dependent
diffusion coefficient of this model from computer simulations and analyze its
functional form by different schemes all generalizing the simple random walk
approximation of Machta and Zwanzig. The improved methods we use are based
either on heuristic higher-order corrections to the simple random walk model,
on lattice gas simulation methods, or they start from a suitable Green-Kubo
formula for diffusion. We show that dynamical correlations, or memory effects,
are of crucial importance to reproduce the precise parameter dependence of the
diffusion coefficent.Comment: 8 pages (revtex) with 9 figures (encapsulated postscript

### Spectral Statistics in the Quantized Cardioid Billiard

The spectral statistics in the strongly chaotic cardioid billiard are
studied. The analysis is based on the first 11000 quantal energy levels for odd
and even symmetry respectively. It is found that the level-spacing distribution
is in good agreement with the GOE distribution of random-matrix theory. In case
of the number variance and rigidity we observe agreement with the random-matrix
model for short-range correlations only, whereas for long-range correlations
both statistics saturate in agreement with semiclassical expectations.
Furthermore the conjecture that for classically chaotic systems the normalized
mode fluctuations have a universal Gaussian distribution with unit variance is
tested and found to be in very good agreement for both symmetry classes. By
means of the Gutzwiller trace formula the trace of the cosine-modulated heat
kernel is studied. Since the billiard boundary is focusing there are conjugate
points giving rise to zeros at the locations of the periodic orbits instead of
exclusively Gaussian peaks.Comment: 20 pages, uu-encoded ps.Z-fil

### Approximating multi-dimensional Hamiltonian flows by billiards

Consider a family of smooth potentials $V_{\epsilon}$, which, in the limit
$\epsilon\to0$, become a singular hard-wall potential of a multi-dimensional
billiard. We define auxiliary billiard domains that asymptote, as
$\epsilon\to0$ to the original billiard, and provide asymptotic expansion of
the smooth Hamiltonian solution in terms of these billiard approximations. The
asymptotic expansion includes error estimates in the $C^{r}$ norm and an
iteration scheme for improving this approximation. Applying this theory to
smooth potentials which limit to the multi-dimensional close to ellipsoidal
billiards, we predict when the separatrix splitting persists for various types
of potentials

### Viscosity in the escape-rate formalism

We apply the escape-rate formalism to compute the shear viscosity in terms of
the chaotic properties of the underlying microscopic dynamics. A first passage
problem is set up for the escape of the Helfand moment associated with
viscosity out of an interval delimited by absorbing boundaries. At the
microscopic level of description, the absorbing boundaries generate a fractal
repeller. The fractal dimensions of this repeller are directly related to the
shear viscosity and the Lyapunov exponent, which allows us to compute its
values. We apply this method to the Bunimovich-Spohn minimal model of viscosity
which is composed of two hard disks in elastic collision on a torus. These
values are in excellent agreement with the values obtained by other methods
such as the Green-Kubo and Einstein-Helfand formulas.Comment: 16 pages, 16 figures (accepted in Phys. Rev. E; October 2003

### The optimal sink and the best source in a Markov chain

It is well known that the distributions of hitting times in Markov chains are
quite irregular, unless the limit as time tends to infinity is considered. We
show that nevertheless for a typical finite irreducible Markov chain and for
nondegenerate initial distributions the tails of the distributions of the
hitting times for the states of a Markov chain can be ordered, i.e., they do
not overlap after a certain finite moment of time.
If one considers instead each state of a Markov chain as a source rather than
a sink then again the states can generically be ordered according to their
efficiency. The mechanisms underlying these two orderings are essentially
different though.Comment: 12 pages, 1 figur

### A strong pair correlation bound implies the CLT for Sinai Billiards

For Dynamical Systems, a strong bound on multiple correlations implies the
Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is
derived for dynamically Holder continuous observables of dispersing Billiards.
Here we weaken the regularity assumption and subsequently show that the bound
on multiple correlations follows directly from the bound on pair correlations.
Thus, a strong bound on pair correlations alone implies the CLT, for a wider
class of observables. The result is extended to Anosov diffeomorphisms in any
dimension.Comment: 13 page

### Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

We consider the billiard dynamics in a non-compact set of R^d that is
constructed as a bi-infinite chain of translated copies of the same
d-dimensional polytope. A random configuration of semi-dispersing scatterers is
placed in each copy. The ensemble of dynamical systems thus defined, one for
each global realization of the scatterers, is called `quenched random Lorentz
tube'. Under some fairly general conditions, we prove that every system in the
ensemble is hyperbolic and almost every system is recurrent, ergodic, and
enjoys some higher chaotic properties.Comment: Final version for J. Stat. Phys., 18 pages, 4 figure

### Slow relaxation in weakly open vertex-splitting rational polygons

The problem of splitting effects by vertex angles is discussed for
nonintegrable rational polygonal billiards. A statistical analysis of the decay
dynamics in weakly open polygons is given through the orbit survival
probability. Two distinct channels for the late-time relaxation of type
1/t^delta are established. The primary channel, associated with the universal
relaxation of ''regular'' orbits, with delta = 1, is common for both the closed
and open, chaotic and nonchaotic billiards. The secondary relaxation channel,
with delta > 1, is originated from ''irregular'' orbits and is due to the
rationality of vertices.Comment: Key words: Dynamics of systems of particles, control of chaos,
channels of relaxation. 21 pages, 4 figure

### Tunable Lyapunov exponent in inverse magnetic billiards

The stability properties of the classical trajectories of charged particles
are investigated in a two dimensional stadium-shaped inverse magnetic domain,
where the magnetic field is zero inside the stadium domain and constant
outside. In the case of infinite magnetic field the dynamics of the system is
the same as in the Bunimovich billiard, i.e., ergodic and mixing. However, for
weaker magnetic fields the phase space becomes mixed and the chaotic part
gradually shrinks. The numerical measurements of the Lyapunov exponent
(performed with a novel method) and the integrable/chaotic phase space volume
ratio show that both quantities can be smoothly tuned by varying the external
magnetic field. A possible experimental realization of the arrangement is also
discussed.Comment: 4 pages, 6 figure

### Coin Tossing as a Billiard Problem

We demonstrate that the free motion of any two-dimensional rigid body
colliding elastically with two parallel, flat walls is equivalent to a billiard
system. Using this equivalence, we analyze the integrable and chaotic
properties of this new class of billiards. This provides a demonstration that
coin tossing, the prototypical example of an independent random process, is a
completely chaotic (Bernoulli) problem. The related question of which billiard
geometries can be represented as rigid body systems is examined.Comment: 16 pages, LaTe

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