226 research outputs found
An almost sure invariance principle for random walks in a space-time random environment
We consider a discrete time random walk in a space-time i.i.d. random
environment. We use a martingale approach to show that the walk is diffusive in
almost every fixed environment. We improve on existing results by proving an
invariance principle and considering environments with an annealed drift.
We also state an a.s. invariance principle for random walks in general random
environments whose hypothesis requires a subdiffusive bound on the variance of
the quenched mean, under an ergodic invariant measure for the environment
chain
On the blow-up of some complex solutions of the 3D NavierâStokes equations: theoretical predictions and computer simulations
We consider some complex-valued solutions of the NavierâStokes equations in R^3 for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the âfluidâ remains quiet
On the blow-up of some complex solutions of the 3D NavierâStokes equations: theoretical predictions and computer simulations
We consider some complex-valued solutions of the NavierâStokes equations in R^3 for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the âfluidâ remains quiet
One-Dimensional Hard-Rod Caricature of Hydrodynamics: Navier-Stokes Correction
One-dimensional system of hard-rod particles of length a is studied in the hydrodynamical limit. The Navier-Stokes correction to Euler's equation is found for an initial locally-equilibrium family of states of constant density Ï Ï” [0,a^(-1)). The correction is given, at t~0, by the non-linear second-order differential operator (Bf)(q,v) = (a^2/2)(â/âq)[â«dw|v-w|f(q,w)(â/âq)f(q,v) - f(q,v)â«dw|v-w|(â/âq)f(q,w)](1-Ïa)^(-1) where f(q,v) is the (hydrodynamical) density at a point q Ï” R^1 of the species of particles with velocity v Ï” R^1
Navier-Stokes equations on the flat cylinder with vorticity production on the boundary
We study the two-dimensional Navier-Stokes system on a flat cylinder with the
usual Dirichlet boundary conditions for the velocity field u. We formulate the
problem as an infinite system of ODE's for the natural Fourier components of
the vorticity, and the boundary conditions are taken into account by adding a
vorticity production at the boundary. We prove equivalence to the original
Navier-Stokes system and show that the decay of the Fourier modes is
exponential for any positive time in the periodic direction, but it is only
power-like in the other direction.Comment: 25 page
On the convergence to statistical equilibrium for harmonic crystals
We consider the dynamics of a harmonic crystal in dimensions with
components, arbitrary, , and study the distribution of
the solution at time . The initial measure has a
translation-invariant correlation matrix, zero mean, and finite mean energy
density. It also satisfies a Rosenblatt- resp. Ibragimov-Linnik type mixing
condition. The main result is the convergence of to a Gaussian measure
as . The proof is based on the long time asymptotics of the Green's
function and on Bernstein's ``room-corridors'' method
Hyperbolic billiards with nearly flat focusing boundaries. I
The standard Wojtkowski-Markarian-Donnay-Bunimovich technique for the
hyperbolicity of focusing or mixed billiards in the plane requires the diameter
of a billiard table to be of the same order as the largest ray of curvature
along the focusing boundary. This is due to the physical principle that is used
in the proofs, the so-called defocusing mechanism of geometrical optics. In
this paper we construct examples of hyperbolic billiards with a focusing
boundary component of arbitrarily small curvature whose diameter is bounded by
a constant independent of that curvature. Our proof employs a nonstardard cone
bundle that does not solely use the familiar dispersing and defocusing
mechanisms.Comment: 21 pages, 9 figure
Quantum stochastic equation for test particle interacting with dilute Bose gas
We use the stochastic limit method to study long time quantum dynamics of a
test particle interacting with a dilute Bose gas. The case of arbitrary
form-factors and an arbitrary, not necessarily equilibrium, quasifree low
density state of the Bose gas is considered. Starting from microscopic dynamics
we derive in the low density limit a quantum white noise equation for the
evolution operator. This equation is equivalent to a quantum stochastic
equation driven by a quantum Poisson process with intensity , where is
the one-particle matrix. The novelty of our approach is that the equations
are derived directly in terms of correlators, without use of a Fock-antiFock
(or Gel'fand-Naimark-Segal) representation. Advantages of our approach are the
simplicity of derivation of the limiting equation and that the algebra of the
master fields and the Ito table do not depend on the initial state of the Bose
gas. The notion of a causal state is introduced. We construct master fields
(white noise and number operators) describing the dynamics in the low density
limit and prove the convergence of chronological (causal) correlators of the
field operators to correlators of the master fields in the causal state.Comment: 21 pages, LaTeX, published version (few improvements
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