635 research outputs found
Laplacian growth with separately controlled noise and anisotropy
Conformal mapping models are used to study competition of noise and
anisotropy in Laplacian growth. For that, a new family of models is introduced
with the noise level and directional anisotropy controlled independently.
Fractalization is observed in both anisotropic growth and the growth with
varying noise. Fractal dimension is determined from cluster size scaling with
its area. For isotropic growth we find d = 1.7, both at high and low noise. For
anisotropic growth with reduced noise the dimension can be as low as d = 1.5
and apparently is not universal. Also, we study fluctuations of particle areas
and observe, in agreement with previous studies, that exceptionally large
particles may appear during the growth, leading to pathologically irregular
clusters. This difficulty is circumvented by using an acceptance window for
particle areas.Comment: 13 pages, 15 figure
Regularized overlap and the chiral determinant
We study the relationship between the continuum overlap and its corresponding
chiral determinant, showing that the former amounts to an unregularised version
of the latter. We then construct a regularised continuum overlap, and consider
the chiral anomalies that follow therefrom. The relation between these
anomalies and the ones derived from the formal (i.e., unregularised) overlap is
elucidated.Comment: 14 pages, late
The Stress Transmission Universality Classes of Periodic Granular Arrays
The transmission of stress is analysed for static periodic arrays of rigid
grains, with perfect and zero friction. For minimal coordination number (which
is sensitive to friction, sphericity and dimensionality), the stress
distribution is soluble without reference to the corresponding displacement
fields. In non-degenerate cases, the constitutive equations are found to be
simple linear in the stress components. The corresponding coefficients depend
crucially upon geometrical disorder of the grain contacts.Comment: 7 pages, 1 figur
Hastings-Levitov aggregation in the small-particle limit
We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web
Randomly Crosslinked Macromolecular Systems: Vulcanisation Transition to and Properties of the Amorphous Solid State
As Charles Goodyear discovered in 1839, when he first vulcanised rubber, a
macromolecular liquid is transformed into a solid when a sufficient density of
permanent crosslinks is introduced at random. At this continuous equi- librium
phase transition, the liquid state, in which all macromolecules are
delocalised, is transformed into a solid state, in which a nonzero fraction of
macromolecules have spontaneously become localised. This solid state is a most
unusual one: localisation occurs about mean positions that are distributed
homogeneously and randomly, and to an extent that varies randomly from monomer
to monomer. Thus, the solid state emerging at the vulcanisation transition is
an equilibrium amorphous solid state: it is properly viewed as a solid state
that bears the same relationship to the liquid and crystalline states as the
spin glass state of certain magnetic systems bears to the paramagnetic and
ferromagnetic states, in the sense that, like the spin glass state, it is
diagnosed by a subtle order parameter.
In this review we give a detailed exposition of a theoretical approach to the
physical properties of systems of randomly, permanently crosslinked
macromolecules. Our primary focus is on the equilibrium properties of such
systems, especially in the regime of Goodyear's vulcanisation transition.Comment: Review Article, REVTEX, 58 pages, 3 PostScript figure
Simulating (electro)hydrodynamic effects in colloidal dispersions: smoothed profile method
Previously, we have proposed a direct simulation scheme for colloidal
dispersions in a Newtonian solvent [Phys.Rev.E 71,036707 (2005)]. An improved
formulation called the ``Smoothed Profile (SP) method'' is presented here in
which simultaneous time-marching is used for the host fluid and colloids. The
SP method is a direct numerical simulation of particulate flows and provides a
coupling scheme between the continuum fluid dynamics and rigid-body dynamics
through utilization of a smoothed profile for the colloidal particles.
Moreover, the improved formulation includes an extension to incorporate
multi-component fluids, allowing systems such as charged colloids in
electrolyte solutions to be studied. The dynamics of the colloidal dispersions
are solved with the same computational cost as required for solving
non-particulate flows. Numerical results which assess the hydrodynamic
interactions of colloidal dispersions are presented to validate the SP method.
The SP method is not restricted to particular constitutive models of the host
fluids and can hence be applied to colloidal dispersions in complex fluids
Scaling exponent of the maximum growth probability in diffusion-limited aggregation
An early (and influential) scaling relation in the multifractal theory of
Diffusion Limited Aggregation(DLA) is the Turkevich-Scher conjecture that
relates the exponent \alpha_{min} that characterizes the ``hottest'' region of
the harmonic measure and the fractal dimension D of the cluster, i.e.
D=1+\alpha_{min}. Due to lack of accurate direct measurements of both D and
\alpha_{min} this conjecture could never be put to serious test. Using the
method of iterated conformal maps D was recently determined as D=1.713+-0.003.
In this Letter we determine \alpha_{min} accurately, with the result
\alpha_{min}=0.665+-0.004. We thus conclude that the Turkevich-Scher conjecture
is incorrect for DLA.Comment: 4 pages, 5 figure
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