1,055 research outputs found
An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations
We give a geometric approach to proving know regularity and existence
theorems for the 2D Navier-Stokes Equations. We feel this point of view is
instructive in better understanding the dynamics. The technique is inspired by
constructions in the Dynamical Systems.Comment: 15 Page
On cross-ratio distortion and Schwarz derivative
We prove asymptotic estimates for the cross-ratio distortion with respect to
a smooth or holomorphic function in terms of its Schwarz derivative.Comment: the spelling of the name `Schwarz' correcte
Herman's Theory Revisited
We prove that a -smooth orientation-preserving circle
diffeomorphism with rotation number in Diophantine class ,
, is -smoothly conjugate to a rigid
rotation. We also derive the most precise version of Denjoy's inequality for
such diffeomorphisms.Comment: 10 page
Thermodynamic formalism for field driven Lorentz gases
We analytically determine the dynamical properties of two dimensional field
driven Lorentz gases within the thermodynamic formalism. For dilute gases
subjected to an iso-kinetic thermostat, we calculate the topological pressure
as a function of a temperature-like parameter \ba up to second order in the
strength of the applied field. The Kolmogorov-Sinai entropy and the topological
entropy can be extracted from a dynamical entropy defined as a Legendre
transform of the topological pressure. Our calculations of the Kolmogorov-Sinai
entropy exactly agree with previous calculations based on a Lorentz-Boltzmann
equation approach. We give analytic results for the topological entropy and
calculate the dimension spectrum from the dynamical entropy function.Comment: 9 pages, 5 figure
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
Random matrices: Universality of local eigenvalue statistics up to the edge
This is a continuation of our earlier paper on the universality of the
eigenvalues of Wigner random matrices. The main new results of this paper are
an extension of the results in that paper from the bulk of the spectrum up to
the edge. In particular, we prove a variant of the universality results of
Soshnikov for the largest eigenvalues, assuming moment conditions rather than
symmetry conditions. The main new technical observation is that there is a
significant bias in the Cauchy interlacing law near the edge of the spectrum
which allows one to continue ensuring the delocalization of eigenvectors.Comment: 24 pages, no figures, to appear, Comm. Math. Phys. One new reference
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Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps
Ensemble averages of the sensitivity to initial conditions and the
entropy production per unit time of a {\it new} family of one-dimensional
dissipative maps, , and of the known
logistic-like maps, , are numerically studied, both
for {\it strong} (Lyapunov exponent ) and {\it weak} (chaos
threshold, i.e., ) chaotic cases. In all cases we verify that (i)
both and {\it linearly}
increase with time for (and only for) a special value of , ,
and (ii) the {\it slope} of {\it coincide},
thus interestingly extending the well known Pesin theorem. For strong chaos,
, whereas at the edge of chaos, .Comment: 5 pages, 5 figure
Anomalous diffusion in disordered multi-channel systems
We study diffusion of a particle in a system composed of K parallel channels,
where the transition rates within the channels are quenched random variables
whereas the inter-channel transition rate v is homogeneous. A variant of the
strong disorder renormalization group method and Monte Carlo simulations are
used. Generally, we observe anomalous diffusion, where the average distance
travelled by the particle, []_{av}, has a power-law time-dependence
[]_{av} ~ t^{\mu_K(v)}, with a diffusion exponent 0 \le \mu_K(v) \le 1.
In the presence of left-right symmetry of the distribution of random rates, the
recurrent point of the multi-channel system is independent of K, and the
diffusion exponent is found to increase with K and decrease with v. In the
absence of this symmetry, the recurrent point may be shifted with K and the
current can be reversed by varying the lane change rate v.Comment: 16 pages, 7 figure
A limit result for a system of particles in random environment
We consider an infinite system of particles in one dimension, each particle
performs independant Sinai's random walk in random environment. Considering an
instant , large enough, we prove a result in probability showing that the
particles are trapped in the neighborhood of well defined points of the lattice
depending on the random environment the time and the starting point of the
particles.Comment: 11 page
Distribution of the Oscillation Period in the Underdamped One Dimensional Sinai Model
We consider the Newtonian dynamics of a massive particle in a one dimemsional
random potential which is a Brownian motion in space. This is the zero
temperature nondamped Sinai model. As there is no dissipation the particle
oscillates between two turning points where its kinetic energy becomes zero.
The period of oscillation is a random variable fluctuating from sample to
sample of the random potential. We compute the probability distribution of this
period exactly and show that it has a power law tail for large period, P(T)\sim
T^{-5/3} and an essential singluarity P(T)\sim \exp(-1/T) as T\to 0. Our exact
results are confirmed by numerical simulations and also via a simple scaling
argument.Comment: 9 pages LateX, 2 .eps figure
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