1,012 research outputs found
Singularities and nonhyperbolic manifolds do not coincide
We consider the billiard flow of elastically colliding hard balls on the flat
-torus (), and prove that no singularity manifold can even
locally coincide with a manifold describing future non-hyperbolicity of the
trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli
mixing property) of all such systems, i.e. the verification of the
Boltzmann-Sinai Ergodic Hypothesis.Comment: Final version, to appear in Nonlinearit
Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems
We consider a dynamical system with state space , a smooth, compact subset
of some , and evolution given by , , ;
is invertible and the time may be discrete, , , or continuous, . Here we show that starting with a
continuous positive initial probability density , with respect
to , the smooth volume measure induced on by Lebesgue measure on , the expectation value of , with respect to any
stationary (i.e. time invariant) measure , is linear in , . depends only on and vanishes
when is absolutely continuous wrt .Comment: 7 pages, plain TeX; [email protected],
[email protected], [email protected], to appear in Chaos: An
Interdisciplinary Journal of Nonlinear Science, Volume 8, Issue
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
Evolution of collision numbers for a chaotic gas dynamics
We put forward a conjecture of recurrence for a gas of hard spheres that
collide elastically in a finite volume. The dynamics consists of a sequence of
instantaneous binary collisions. We study how the numbers of collisions of
different pairs of particles grow as functions of time. We observe that these
numbers can be represented as a time-integral of a function on the phase space.
Assuming the results of the ergodic theory apply, we describe the evolution of
the numbers by an effective Langevin dynamics. We use the facts that hold for
these dynamics with probability one, in order to establish properties of a
single trajectory of the system. We find that for any triplet of particles
there will be an infinite sequence of moments of time, when the numbers of
collisions of all three different pairs of the triplet will be equal. Moreover,
any value of difference of collision numbers of pairs in the triplet will
repeat indefinitely. On the other hand, for larger number of pairs there is but
a finite number of repetitions. Thus the ergodic theory produces a limitation
on the dynamics.Comment: 4 pages, published versio
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