12,457 research outputs found
Microscopic Selection of Fluid Fingering Pattern
We study the issue of the selection of viscous fingering patterns in the
limit of small surface tension. Through detailed simulations of anisotropic
fingering, we demonstrate conclusively that no selection independent of the
small-scale cutoff (macroscopic selection) occurs in this system. Rather, the
small-scale cutoff completely controls the pattern, even on short time scales,
in accord with the theory of microscopic solvability. We demonstrate that
ordered patterns are dynamically selected only for not too small surface
tensions. For extremely small surface tensions, the system exhibits chaotic
behavior and no regular pattern is realized.Comment: 6 pages, 5 figure
Nonlinear lattice model of viscoelastic Mode III fracture
We study the effect of general nonlinear force laws in viscoelastic lattice
models of fracture, focusing on the existence and stability of steady-state
Mode III cracks. We show that the hysteretic behavior at small driving is very
sensitive to the smoothness of the force law. At large driving, we find a Hopf
bifurcation to a straight crack whose velocity is periodic in time. The
frequency of the unstable bifurcating mode depends on the smoothness of the
potential, but is very close to an exact period-doubling instability. Slightly
above the onset of the instability, the system settles into a exactly
period-doubled state, presumably connected to the aforementioned bifurcation
structure. We explicitly solve for this new state and map out its
velocity-driving relation
From the area under the Bessel excursion to anomalous diffusion of cold atoms
Levy flights are random walks in which the probability distribution of the
step sizes is fat-tailed. Levy spatial diffusion has been observed for a
collection of ultra-cold Rb atoms and single Mg+ ions in an optical lattice.
Using the semiclassical theory of Sisyphus cooling, we treat the problem as a
coupled Levy walk, with correlations between the length and duration of the
excursions. The problem is related to the area under Bessel excursions,
overdamped Langevin motions that start and end at the origin, constrained to
remain positive, in the presence of an external logarithmic potential. In the
limit of a weak potential, the Airy distribution describing the areal
distribution of the Brownian excursion is found. Three distinct phases of the
dynamics are studied: normal diffusion, Levy diffusion and, below a certain
critical depth of the optical potential, x~ t^{3/2} scaling. The focus of the
paper is the analytical calculation of the joint probability density function
from a newly developed theory of the area under the Bessel excursion. The
latter describes the spatiotemporal correlations in the problem and is the
microscopic input needed to characterize the spatial diffusion of the atomic
cloud. A modified Montroll-Weiss (MW) equation for the density is obtained,
which depends on the statistics of velocity excursions and meanders. The
meander, a random walk in velocity space which starts at the origin and does
not cross it, describes the last jump event in the sequence. In the anomalous
phases, the statistics of meanders and excursions are essential for the
calculation of the mean square displacement, showing that our correction to the
MW equation is crucial, and points to the sensitivity of the transport on a
single jump event. Our work provides relations between the statistics of
velocity excursions and meanders and that of the diffusivity.Comment: Supersedes arXiv: 1305.008
Scaling Green-Kubo relation and application to three aging systems
The Green-Kubo formula relates the spatial diffusion coefficient to the
stationary velocity autocorrelation function. We derive a generalization of the
Green-Kubo formula valid for systems with long-range or nonstationary
correlations for which the standard approach is no longer valid. For the
systems under consideration, the velocity autocorrelation function asymptotically exhibits a certain scaling behavior and
the diffusion is anomalous . We
show how both the anomalous diffusion coefficient and exponent
can be extracted from this scaling form. Our scaling Green-Kubo relation thus
extends an important relation between transport properties and correlation
functions to generic systems with scale invariant dynamics. This includes
stationary systems with slowly decaying power law correlations as well as aging
systems, whose properties depend on the the age of the system. Even for systems
that are stationary in the long time limit, we find that the long time
diffusive behavior can strongly depend on the initial preparation of the
system. In these cases, the diffusivity is not unique and we
determine its values for a stationary respectively nonstationary initial state.
We discuss three applications of the scaling Green-Kubo relation: Free
diffusion with nonlinear friction corresponding to cold atoms diffusing in
optical lattices, the fractional Langevin equation with external noise recently
suggested to model active transport in cells and the L\'evy walk with numerous
applications, in particular blinking quantum dots. These examples underline the
wide applicability of our approach, which is able to treat very different
mechanisms of anomalous diffusion.Comment: 16 pages, 6 figures, 1 tabl
The Universal Gaussian in Soliton Tails
We show that in a large class of equations, solitons formed from generic
initial conditions do not have infinitely long exponential tails, but are
truncated by a region of Gaussian decay. This phenomenon makes it possible to
treat solitons as localized, individual objects. For the case of the KdV
equation, we show how the Gaussian decay emerges in the inverse scattering
formalism.Comment: 4 pages, 2 figures, revtex with eps
Two-finger selection theory in the Saffman-Taylor problem
We find that solvability theory selects a set of stationary solutions of the
Saffman-Taylor problem with coexistence of two \it unequal \rm fingers
advancing with the same velocity but with different relative widths
and and different tip positions. For vanishingly small
dimensionless surface tension , an infinite discrete set of values of the
total filling fraction and of the relative
individual finger width are selected out of a
two-parameter continuous degeneracy. They scale as
and . The selected values of differ from
those of the single finger case. Explicit approximate expressions for both
spectra are given.Comment: 4 pages, 3 figure
Dynamics of Conformal Maps for a Class of Non-Laplacian Growth Phenomena
Time-dependent conformal maps are used to model a class of growth phenomena
limited by coupled non-Laplacian transport processes, such as nonlinear
diffusion, advection, and electro-migration. Both continuous and stochastic
dynamics are described by generalizing conformal-mapping techniques for viscous
fingering and diffusion-limited aggregation, respectively. A general notion of
time in stochastic growth is also introduced. The theory is applied to
simulations of advection-diffusion-limited aggregation in a background
potential flow. A universal crossover in morphology is observed from
diffusion-limited to advection-limited fractal patterns with an associated
crossover in the growth rate, controlled by a time-dependent effective Peclet
number. Remarkably, the fractal dimension is not affected by advection, in
spite of dramatic increases in anisotropy and growth rate, due to the
persistence of diffusion limitation at small scales.Comment: 4 pages, 2 figures (six color plates
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