153 research outputs found
Numerical simulation of stochastic vortex tangles
We present the results of simulation of the chaotic dynamics of quantized
vortices in the bulk of superfluid He II.
Evolution of vortex lines is calculated on the base of the Biot-Savart law.
The dissipative effects appeared from the interaction with the normal
component, or/and from relaxation of the order parameter are taken into
account. Chaotic dynamics appears in the system via a random forcing, e.i. we
use the Langevin approach to the problem. In the present paper we require the
correlator of the random force to satisfy the fluctuation-disspation relation,
which implies that thermodynamic equilibrium should be reached. In the paper we
describe the numerical methods for integration of stochastic differential
equation (including a new algorithm for reconnection processes), and we present
the results of calculation of some characteristics of a vortex tangle such as
the total length, distribution of loops in the space of their length, and the
energy spectrum.Comment: 8 pages, 5 figure
Evolution of a Network of Vortex Loops in HeII. Exact Solution of the "Rate Equation"
Evolution of a network of vortex loops in HeII due to the fusion and
breakdown of vortex loops is studied. We perform investigation on the base of
the ''rate equation'' for the distribution function of number of loops
of length proposed by Copeland with coauthors. By using the special ansatz
in the ''collision'' integral we have found the exact power-like solution of
''kinetic equation'' in stationary case. That solution is the famous
equilibrium distribution obtained earlier in
numerical calculations. Our result, however, is not equilibrium, but on the
contrary, it describes the state with two mutual fluxes of the length (or
energy) in space of the vortex loop sizes. Analyzing this solution we drew
several results on the structure and dynamics of the vortex tangle in the
superfluid turbulent helium. In particular, we obtained that the mean radius of
the curvature is of order of interline space. We also obtain that the decay of
the vortex tangle obeys the Vinen equation, obtained earlier
phenomenologically. We evaluate also the full rate of reconnection events.
PACS-number 67.40Comment: 4 pages, submitted to PR
Kinetics of a Network of Vortex Loops in He II and a Theory of Superfluid Turbulence
A theory is developed to describe the superfluid turbulence on the base of
kinetics of the merging and splitting vortex loops. Because of very frequent
reconnections the vortex loops (as a whole) do not live long enough to perform
any essential evolution due to the deterministic motion. On the contrary, they
rapidly merge and split, and these random recombination processes prevail over
other slower dynamic processes. To develop quantitative description we take the
vortex loops to have a Brownian structure with the only degree of freedom,
which is the length of the loop. We perform investigation on the base of
the Boltzmann type kinetic equation for the distribution function of
number of loops with length . By use of the special ansatz in the collision
integral we have found the exact power-like solution to kinetic equation in the
stationary case. This solution is not (thermodynamically) equilibrium, but on
the contrary, it describes the state with two mutual fluxes of the length (or
energy) in space of sizes of the vortex loops. The term flux means just
redistribution of length (or energy) among the loops of different sizes due to
reconnections. Analyzing this solution we drew several results on the structure
and dynamics of the vortex tangle in the turbulent superfluid helium. In
particular, we evaluated the mean radius of the curvature and the full rate of
the reconnection events. We also studied the evolution of the full length of
vortex loops per unit volume-the so-called vortex line density. It is shown
this evolution to obey the famous Vinen equation. The properties of the Vinen
equation from the point of view of the developed approach had been discussed.Comment: 34 pages, 9 Postscript figures, [aps,preprint,12pt]{revtex4
Parametric generation of second sound in superfluid helium: linear stability and nonlinear dynamics
We report the experimental studies of a parametric excitation of a second
sound (SS) by a first sound (FS) in a superfluid helium in a resonance cavity.
The results on several topics in this system are presented: (i) The linear
properties of the instability, namely, the threshold, its temperature and
geometrical dependencies, and the spectra of SS just above the onset were
measured. They were found to be in a good quantitative agreement with the
theory. (ii) It was shown that the mechanism of SS amplitude saturation is due
to the nonlinear attenuation of SS via three wave interactions between the SS
waves. Strong low frequency amplitude fluctuations of SS above the threshold
were observed. The spectra of these fluctuations had a universal shape with
exponentially decaying tails. Furthermore, the spectral width grew continuously
with the FS amplitude. The role of three and four wave interactions are
discussed with respect to the nonlinear SS behavior. The first evidence of
Gaussian statistics of the wave amplitudes for the parametrically generated
wave ensemble was obtained. (iii) The experiments on simultaneous pumping of
the FS and independent SS waves revealed new effects. Below the instability
threshold, the SS phase conjugation as a result of three-wave interactions
between the FS and SS waves was observed. Above the threshold two new effects
were found: a giant amplification of the SS wave intensity and strong resonance
oscillations of the SS wave amplitude as a function of the FS amplitude.
Qualitative explanations of these effects are suggested.Comment: 73 pages, 23 figures. to appear in Phys. Rev. B, July 1 st (2001
Identification of Kelvin waves: numerical challenges
Kelvin waves are expected to play an essential role in the energy dissipation
for quantized vortices. However, the identification of these helical
distortions is not straightforward, especially in case of vortex tangle. Here
we review several numerical methods that have been used to identify Kelvin
waves within the vortex filament model. We test their validity using several
examples and estimate whether these methods are accurate enough to verify the
correct Kelvin spectrum. We also illustrate how the correlation dimension is
related to different Kelvin spectra and remind that the 3D energy spectrum E(k)
takes the form 1/k in the high-k region, even in the presence of Kelvin waves.Comment: 6 pages, 5 figures. The final publication is available at
http://www.springerlink.co
Structured Sparsity: Discrete and Convex approaches
Compressive sensing (CS) exploits sparsity to recover sparse or compressible
signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity
is also used to enhance interpretability in machine learning and statistics
applications: While the ambient dimension is vast in modern data analysis
problems, the relevant information therein typically resides in a much lower
dimensional space. However, many solutions proposed nowadays do not leverage
the true underlying structure. Recent results in CS extend the simple sparsity
idea to more sophisticated {\em structured} sparsity models, which describe the
interdependency between the nonzero components of a signal, allowing to
increase the interpretability of the results and lead to better recovery
performance. In order to better understand the impact of structured sparsity,
in this chapter we analyze the connections between the discrete models and
their convex relaxations, highlighting their relative advantages. We start with
the general group sparse model and then elaborate on two important special
cases: the dispersive and the hierarchical models. For each, we present the
models in their discrete nature, discuss how to solve the ensuing discrete
problems and then describe convex relaxations. We also consider more general
structures as defined by set functions and present their convex proxies.
Further, we discuss efficient optimization solutions for structured sparsity
problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure
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