90 research outputs found
Contour dynamics model for electric discharges
A contour dynamics model for electrical discharges is obtained and analyzed.
The model is deduced as the asymptotic limit of the minimal streamer model for
the propagation of electric discharges, in the limit of small electron
diffusion. The dispersion relation for a non planar 2-D discharge is
calculated. The development and propagation of finger-like patterns are studied
and their main features quantified.Comment: 4 pages, 2 fi
Electric discharge contour dynamics model: the effects of curvature and finite conductivity
In this paper we present the complete derivation of the effective contour
model for electrical discharges which appears as the asymptotic limit of the
minimal streamer model for the propagation of electric discharges, when the
electron diffusion is small. It consists of two integro-differential equations
defined at the boundary of the plasma region: one for the motion and a second
equation for the net charge density at the interface. We have computed explicit
solutions with cylindrical symmetry and found the dispersion relation for small
symmetry-breaking perturbations in the case of finite resistivity. We implement
a numerical procedure to solve our model in general situations. As a result we
compute the dispersion relation for the cylindrical case and compare it with
the analytical predictions. Comparisons with experimental data for a 2-D
positive streamers discharge are provided and predictions confirmed.Comment: 23 pages, 3 figure
Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
We show that unstable fingering patterns of two dimensional flows of viscous
fluids with open boundary are described by a dispersionless limit of the KdV
hierarchy. In this framework, the fingering instability is linked to a known
instability leading to regularized shock solutions for nonlinear waves, in
dispersive media. The integrable structure of the flow suggests a dispersive
regularization of the finite-time singularities.Comment: Published versio
Diffusion-Limited Aggregation on Curved Surfaces
We develop a general theory of transport-limited aggregation phenomena
occurring on curved surfaces, based on stochastic iterated conformal maps and
conformal projections to the complex plane. To illustrate the theory, we use
stereographic projections to simulate diffusion-limited-aggregation (DLA) on
surfaces of constant Gaussian curvature, including the sphere () and
pseudo-sphere (), which approximate "bumps" and "saddles" in smooth
surfaces, respectively. Although curvature affects the global morphology of the
aggregates, the fractal dimension (in the curved metric) is remarkably
insensitive to curvature, as long as the particle size is much smaller than the
radius of curvature. We conjecture that all aggregates grown by conformally
invariant transport on curved surfaces have the same fractal dimension as DLA
in the plane. Our simulations suggest, however, that the multifractal
dimensions increase from hyperbolic () geometry, which
we attribute to curvature-dependent screening of tip branching.Comment: 4 pages, 3 fig
Logarithmic diffusion and porous media equations: a unified description
In this work we present the logarithmic diffusion equation as a limit case
when the index that characterizes a nonlinear Fokker-Planck equation, in its
diffusive term, goes to zero. A linear drift and a source term are considered
in this equation. Its solution has a lorentzian form, consequently this
equation characterizes a super diffusion like a L\'evy kind. In addition is
obtained an equation that unifies the porous media and the logarithmic
diffusion equations, including a generalized diffusion equation in fractal
dimension. This unification is performed in the nonextensive thermostatistics
context and increases the possibilities about the description of anomalous
diffusive processes.Comment: 5 pages. To appear in Phys. Rev.
Laplacian Growth and Whitham Equations of Soliton Theory
The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in
the case of zero surface tension is proven to be equivalent to an integrable
systems of Whitham equations known in soliton theory. The Whitham equations
describe slowly modulated periodic solutions of integrable hierarchies of
nonlinear differential equations. Through this connection the Laplacian growth
is understood as a flow in the moduli space of Riemann surfaces.Comment: 33 pages, 7 figures, typos corrected, new references adde
Generic critical points of normal matrix ensembles
The evolution of the degenerate complex curve associated with the ensemble at
a generic critical point is related to the finite time singularities of
Laplacian Growth. It is shown that the scaling behavior at a critical point of
singular geometry is described by the first Painlev\'e
transcendent. The regularization of the curve resulting from discretization is
discussed.Comment: Based on a talk given at the conference on Random Matrices, Random
Processes and Integrable Systems, CRM Montreal, June 200
A New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation
We present a new class of exact solutions for the so-called {\it Laplacian
Growth Equation} describing the zero-surface-tension limit of a variety of 2D
pattern formation problems. Contrary to common belief, we prove that these
solutions are free of finite-time singularities (cusps) for quite general
initial conditions and may well describe real fingering instabilities. At long
times the interface consists of N separated moving Saffman-Taylor fingers, with
``stagnation points'' in between, in agreement with numerous observations. This
evolution resembles the N-soliton solution of classical integrable PDE's.Comment: LaTeX, uuencoded postscript file
Random Matrices in 2D, Laplacian Growth and Operator Theory
Since it was first applied to the study of nuclear interactions by Wigner and
Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a
field of its own within applied mathematics, and is now essential to many parts
of theoretical physics, from condensed matter to high energy. The fundamental
results obtained so far rely mostly on the theory of random matrices in one
dimension (the dimensionality of the spectrum, or equilibrium probability
density). In the last few years, this theory has been extended to the case
where the spectrum is two-dimensional, or even fractal, with dimensions between
1 and 2. In this article, we review these recent developments and indicate some
physical problems where the theory can be applied.Comment: 88 pages, 8 figure
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