750,790 research outputs found
Pinning Induced Fluctuations on Driven Vortices
We use a simple model to study the long time fluctuations induced by random
pinning on the motion of driven non--interacting vortices. We find that vortex
motion seen from the co--moving frame is diffusive and anisotropic, with
velocity dependent diffusion constants. Longitudinal and transverse diffusion
constants cross at a characteristic velocity where diffusion is isotropic. The
diffusion front is elongated in the direction of the drive at low velocities
and elongated in the transverse direction at large velocities. We find that the
mobility in the driven direction is always larger than the transverse mobility,
and becomes isotropic only in the large velocity limit.Comment: 4 pages, 3 figs, Vortex IV Proceedings, Sep. 3-9, 2005, Crete, Greec
Pattern formation for reactive species undergoing anisotropic diffusion
Turing instabilities for a two species reaction-diffusion systems is studied
under anisotropic diffusion. More specifically, the diffusion constants which
characterize the ability of the species to relocate in space are direction
sensitive. Under this working hypothesis, the conditions for the onset of the
instability are mathematically derived and numerically validated. Patterns
which closely resemble those obtained in the classical context of isotropic
diffusion, develop when the usual Turing condition is violated, along one of
the two accessible directions of migration. Remarkably, the instability can
also set in when the activator diffuses faster than the inhibitor, along the
direction for which the usual Turing conditions are not matched
Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion
We consider elliptic diffusion problems with a random anisotropic diffusion
coefficient, where, in a notable direction given by a random vector field, the
diffusion strength differs from the diffusion strength perpendicular to this
notable direction. The Karhunen-Lo\`eve expansion then yields a parametrisation
of the random vector field and, therefore, also of the solution of the elliptic
diffusion problem. We show that, given regularity of the elliptic diffusion
problem, the decay of the Karhunen-Lo\`eve expansion entirely determines the
regularity of the solution's dependence on the random parameter, also when
considering this higher spatial regularity. This result then implies that
multilevel collocation and multilevel quadrature methods may be used to lessen
the computation complexity when approximating quantities of interest, like the
solution's mean or its second moment, while still yielding the expected rates
of convergence. Numerical examples in three spatial dimensions are provided to
validate the presented theory
False discovery rate analysis of brain diffusion direction maps
Diffusion tensor imaging (DTI) is a novel modality of magnetic resonance
imaging that allows noninvasive mapping of the brain's white matter. A
particular map derived from DTI measurements is a map of water principal
diffusion directions, which are proxies for neural fiber directions. We
consider a study in which diffusion direction maps were acquired for two groups
of subjects. The objective of the analysis is to find regions of the brain in
which the corresponding diffusion directions differ between the groups. This is
attained by first computing a test statistic for the difference in direction at
every brain location using a Watson model for directional data. Interesting
locations are subsequently selected with control of the false discovery rate.
More accurate modeling of the null distribution is obtained using an empirical
null density based on the empirical distribution of the test statistics across
the brain. Further, substantial improvements in power are achieved by local
spatial averaging of the test statistic map. Although the focus is on one
particular study and imaging technology, the proposed inference methods can be
applied to other large scale simultaneous hypothesis testing problems with a
continuous underlying spatial structure.Comment: Published in at http://dx.doi.org/10.1214/07-AOAS133 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Confinement without boundaries: Anisotropic diffusion on the surface of a cylinder
Densely packed systems of thermal particles in curved geometries are
frequently encountered in biological and microfluidic systems. In 2D systems,
at sufficiently high surface coverage, diffusive motion is widely known to be
strongly affected by physical confinement, e.g., by the walls. In this Letter,
we explore the effects of confinement by shape, not rigid boundaries, on the
diffusion of particles by confining them to the surface of a cylinder. We find
that both the magnitude and the directionality of lateral diffusion is strongly
influenced by the radius of the cylinder. An anisotropy between diffusion in
the longitudinal and circumferential direction of the cylinder develops. We
demonstrate that the origin of this effect lies in the fact that screw-like
packings of mono- and oligodisperse discs on the surface of a cylinder induce
preferential collective motions in the circumferential direction, but also show
that even in polydisperse systems lacking such order an intrinsic finite size
confinement effect increases diffusivity in the circumferential direction
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