12,748 research outputs found
White Noise Limits for Inertial Particles in a Random Field
In this paper we present a rigorous analysis of a scaling limit related to the motion of an inertial particle in a Gaussian random field. The mathematical model comprises Stokes's law for the particle motion and an infinite dimensional Ornstein-Uhlenbeck process for the fluid velocity field. The scaling limit studied leads to a white noise limit for the fluid velocity, which balances particle inertia and the friction term. Strong convergence methods are used to justify the limiting equations. The rigorously derived limiting equations are of physical interest for the concrete problem under investigation and facilitate the study of two-point motions in the white noise limit. Furthermore, the methodology developed may also prove useful in the study of various other asymptotic problems for stochastic differential equations in infinite dimensions
Analysis of White Noise Limits for Stochastic Systems with Two Fast Relaxation Times
In this paper we present a rigorous asymptotic analysis for stochastic
systems with two fast relaxation times. The mathematical model analyzed in this
paper consists of a Langevin equation for the particle motion with
time-dependent force constructed through an infinite dimensional Gaussian noise
process. We study the limit as the particle relaxation time as well as the
correlation time of the noise tend to zero and we obtain the limiting equations
under appropriate assumptions on the Gaussian noise. We show that the limiting
equation depends on the relative magnitude of the two fast time scales of the
system. In particular, we prove that in the case where the two relaxation times
converge to zero at the same rate there is a drift correction, in addition to
the limiting It\^{o} integral, which is not of Stratonovich type. If, on the
other hand, the colored noise is smooth on the scale of particle relaxation
then the drift correction is the standard Stratonovich correction. If the noise
is rough on this scale then there is no drift correction. Strong (i.e.
pathwise) techniques are used for the proof of the convergence theorems.Comment: 35 pages, 0 figures, To appear in SIAM J. MM
Motion of inertial particles in Gaussian fields driven by an infinite-dimensional fractional Brownian motion
We study the motion of an inertial particle in a fractional Gaussian random
field. The motion of the particle is described by Newton's second law, where
the force is proportional to the difference between a background fluid velocity
and the particle velocity. The fluid velocity satisfies a linear stochastic
partial differential equation driven by an infinite-dimensional fractional
Brownian motion with arbitrary Hurst parameter H in (0,1). The usefulness of
such random velocity fields in simulations is that we can create random
velocity fields with desired statistical properties, thus generating artificial
images of realistic turbulent flows. This model captures also the clustering
phenomenon of preferential concentration, observed in real world and numerical
experiments, i.e. particles cluster in regions of low vorticity and high strain
rate. We prove almost sure existence and uniqueness of particle paths and give
sufficient conditions to rewrite this system as a random dynamical system with
a global random pullback attractor. Finally, we visualize the random attractor
through a numerical experiment.Comment: 30 pages, 1 figur
How to estimate the differential acceleration in a two-species atom interferometer to test the equivalence principle
We propose a scheme for testing the weak equivalence principle (Universality
of Free Fall) using an atom-interferometric measurement of the local
differential acceleration between two atomic species with a large mass ratio as
test masses. A apparatus in free fall can be used to track atomic free-fall
trajectories over large distances. We show how the differential acceleration
can be extracted from the interferometric signal using Bayesian statistical
estimation, even in the case of a large mass and laser wavelength difference.
We show that this statistical estimation method does not suffer from
acceleration noise of the platform and does not require repeatable experimental
conditions. We specialize our discussion to a dual potassium/rubidium
interferometer and extend our protocol with other atomic mixtures. Finally, we
discuss the performances of the UFF test developed for the free-fall (0-g)
airplane in the ICE project (\verb"http://www.ice-space.fr"
Statistical model for collisions and recollisions of inertial particles in mixing flows
Finding a quantitative description of the rate of collisions between small
particles suspended in mixing flows is a long-standing problem. Here we
investigate the validity of a parameterisation of the collision rate for
identical particles subject to Stokes force, based on results for relative
velocities of heavy particles that were recently obtained within a statistical
model for the dynamics of turbulent aerosols. This model represents the
turbulent velocity fluctuations by Gaussian random functions. We find that the
parameterisation gives quantitatively good results in the limit where the \lq
ghost-particle approximation' applies. The collision rate is a sum of two
contributions due to \lq caustics' and to \lq clustering'. Within the
statistical model we compare the relative importance of these two collision
mechanisms. The caustic formation rate is high when the particle inertia
becomes large, and we find that caustics dominate the collision rate as soon as
they form frequently. We compare the magnitude of the caustic contribution to
the collision rate to the formation rate of caustics.Comment: 9 pages, 4 figures, final version as publishe
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