308 research outputs found
Rotating superfluids in anharmonic traps: From vortex lattices to giant vortices
We study a superfluid in a rotating anharmonic trap and explicate a rigorous
proof of a transition from a vortex lattice to a giant vortex state as the
rotation is increased beyond a limiting speed determined by the interaction
strength. The transition is characterized by the disappearance of the vortices
from the annulus where the bulk of the superfluid is concentrated due to
centrifugal forces while a macroscopic phase circulation remains. The analysis
is carried out within two-dimensional Gross-Pitaevskii theory at large coupling
constant and reveals significant differences between 'soft' anharmonic traps
(like a quartic plus quadratic trapping potential) and traps with a fixed
boundary: In the latter case the transition takes place in a parameter regime
where the size of vortices is very small relative to the width of the annulus
whereas in 'soft' traps the vortex lattice persists until the width of the
annulus becomes comparable to the vortex cores. Moreover, the density profile
in the annulus where the bulk is concentrated is, in the 'soft' case,
approximately gaussian with long tails and not of the Thomas-Fermi type like in
a trap with a fixed boundary.Comment: Published version. Typos corrected, references adde
Energy and Vorticity in Fast Rotating Bose-Einstein Condensates
We study a rapidly rotating Bose-Einstein condensate confined to a finite
trap in the framework of two-dimensional Gross-Pitaevskii theory in the strong
coupling (Thomas-Fermi) limit. Denoting the coupling parameter by 1/\eps^2
and the rotational velocity by , we evaluate exactly the next to
leading order contribution to the ground state energy in the parameter regime
|\log\eps|\ll \Omega\ll 1/(\eps^2|\log\eps|) with \eps\to 0. While the TF
energy includes only the contribution of the centrifugal forces the next order
corresponds to a lattice of vortices whose density is proportional to the
rotational velocity.Comment: 19 pages, LaTeX; typos corrected, clarifying remarks added, some
rearrangements in the tex
Bosons in Rapid Rotation
Some recent progress in the mathematical physics of rapidly rotating, dilute
Bose gases in anharmonic traps is reviewed.Comment: 16 pages, lecture given at a symposium in honor of Michel
Dubois-Violette, Orsay, April 200
Ginzburg-Landau vortex dynamics with pinning and strong applied currents
We study a mixed heat and Schr\"odinger Ginzburg-Landau evolution equation on
a bounded two-dimensional domain with an electric current applied on the
boundary and a pinning potential term. This is meant to model a superconductor
subjected to an applied electric current and electromagnetic field and
containing impurities. Such a current is expected to set the vortices in
motion, while the pinning term drives them toward minima of the pinning
potential and "pins" them there. We derive the limiting dynamics of a finite
number of vortices in the limit of a large Ginzburg-Landau parameter, or \ep
\to 0, when the intensity of the electric current and applied magnetic field
on the boundary scale like \lep. We show that the limiting velocity of the
vortices is the sum of a Lorentz force, due to the current, and a pinning
force. We state an analogous result for a model Ginzburg-Landau equation
without magnetic field but with forcing terms. Our proof provides a unified
approach to various proofs of dynamics of Ginzburg-Landau vortices.Comment: 48 pages; v2: minor errors and typos correcte
Convergence of Ginzburg-Landau functionals in 3-d superconductivity
In this paper we consider the asymptotic behavior of the Ginzburg- Landau
model for superconductivity in 3-d, in various energy regimes. We rigorously
derive, through an analysis via {\Gamma}-convergence, a reduced model for the
vortex density, and we deduce a curvature equation for the vortex lines. In a
companion paper, we describe further applications to superconductivity and
superfluidity, such as general expressions for the first critical magnetic
field H_{c1}, and the critical angular velocity of rotating Bose-Einstein
condensates.Comment: 45 page
Vortex density models for superconductivity and superfluidity
We study some functionals that describe the density of vortex lines in
superconductors subject to an applied magnetic field, and in Bose-Einstein
condensates subject to rotational forcing, in quite general domains in 3
dimensions. These functionals are derived from more basic models via
Gamma-convergence, here and in a companion paper. In our main results, we use
these functionals to obtain descriptions of the critical applied magnetic field
(for superconductors) and forcing (for Bose-Einstein), above which ground
states exhibit nontrivial vorticity, as well as a characterization of the
vortex density in terms of a non local vector-valued generalization of the
classical obstacle problem.Comment: 34 page
Inhomogeneous Vortex Patterns in Rotating Bose-Einstein Condensates
We consider a 2D rotating Bose gas described by the Gross-Pitaevskii (GP)
theory and investigate the properties of the ground state of the theory for
rotational speeds close to the critical speed for vortex nucleation. While one
could expect that the vortex distribution should be homogeneous within the
condensate we prove by means of an asymptotic analysis in the strongly
interacting (Thomas-Fermi) regime that it is not. More precisely we rigorously
derive a formula due to Sheehy and Radzihovsky [Phys. Rev. A 70, 063620(R)
(2004)] for the vortex distribution, a consequence of which is that the vortex
distribution is strongly inhomogeneous close to the critical speed and
gradually homogenizes when the rotation speed is increased. From the
mathematical point of view, a novelty of our approach is that we do not use any
compactness argument in the proof, but instead provide explicit estimates on
the difference between the vorticity measure of the GP ground state and the
minimizer of a certain renormalized energy functional.Comment: 41 pages, journal ref: Communications in Mathematical Physics: Volume
321, Issue 3 (2013), Page 817-860, DOI : 10.1007/s00220-013-1697-
Critical Rotational Speeds for Superfluids in Homogeneous Traps
We present an asymptotic analysis of the effects of rapid rotation on the
ground state properties of a superfluid confined in a two-dimensional trap. The
trapping potential is assumed to be radial and homogeneous of degree larger
than two in addition to a quadratic term. Three critical rotational velocities
are identified, marking respectively the first appearance of vortices, the
creation of a `hole' of low density within a vortex lattice, and the emergence
of a giant vortex state free of vortices in the bulk. These phenomena have
previously been established rigorously for a `flat' trap with fixed boundary
but the `soft' traps considered in the present paper exhibit some significant
differences, in particular the giant vortex regime, that necessitate a new
approach. These differences concern both the shape of the bulk profile and the
size of vortices relative to the width of the annulus where the bulk of the
superfluid resides. Close to the giant vortex transition the profile is of
Thomas-Fermi type in `flat' traps, whereas it is gaussian for soft traps, and
the `last' vortices to survive in the bulk before the giant vortex transition
are small relative to the width of the annulus in the former case but of
comparable size in the latter.Comment: To appear in J. Math. Phys, published versio
The Effects of Negative Legacies on the Adjustment of Parentally Bereaved Children and Adolescents
This is a report of a qualitative analysis of a sample of bereaved families in which one parent died and in which children scored in the clinical range on the Child Behavior Check List. The purpose of this analysis was to learn more about the lives of these children. They were considered to be at risk of developing emotional and behavioral problems associated with the death. We discovered that many of these “high risk” children had a continuing bond with the deceased that was primarily negative and troubling for them in contrast to a comparison group of children not at risk from the same study. Five types of legacies, not mutually exclusive, were identified: health related, role related, personal qualities, legacy of blame, and an emotional legacy. Coping behavior on the part of the surviving parent seemed to make a difference in whether or not a legacy was experienced as negative
Analysis of Nematic Liquid Crystals with Disclination Lines
We investigate the structure of nematic liquid crystal thin films described
by the Landau--de Gennes tensor-valued order parameter with Dirichlet boundary
conditions of nonzero degree. We prove that as the elasticity constant goes to
zero a limiting uniaxial texture forms with disclination lines corresponding to
a finite number of defects, all of degree 1/2 or all of degree -1/2. We also
state a result on the limiting behavior of minimizers of the Chern-Simons-Higgs
model without magnetic field that follows from a similar proof.Comment: 40 pages, 1 figur
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