151 research outputs found
Singular Laplacian Growth
The general equations of motion for two dimensional Laplacian growth are
derived using the conformal mapping method. In the singular case, all
singularities of the conformal map are on the unit circle, and the map is a
degenerate Schwarz-Christoffel map. The equations of motion describe the
motions of these singularities. Despite the typical fractal-like outcomes of
Laplacian growth processes, the equations of motion are shown to be not
particularly sensitive to initial conditions. It is argued that the sensitivity
of this system derives from a novel cause, the non-uniqueness of solutions to
the differential system. By a mechanism of singularity creation, every solution
can become more complex, even in the absence of noise, without violating the
growth law. These processes are permitted, but are not required, meaning the
equation of motion does not determine the motion, even in the small.Comment: 8 pages, Latex, 4 figures, Submitted to Phys. Rev.
A reparametrization invariant surface ordering
We introduce a notion of a non-Abelian loop gauge field defined on points in
loop space. For this purpose we first find an infinite-dimensional tensor
product representation of the Lie algebra which is particularly suited for
fields on loop space. We define the non-Abelian Wilson surface as a `time'
ordered exponential in terms of this loop gauge field and show that it is
reparametrization invariant.Comment: 11 pages, clarifications and added ref
Five-dimensional N=4, SU(2) X U(1) Gauged Supergravity from Type IIB
We construct the complete and explicit non-linear Kaluza-Klein ansatz for
deriving the bosonic sector of N=4 SU(2)\times U(1) gauged five-dimensional
supergravity from the reduction of type IIB supergravity on S^5. This provides
the first complete example of such an S^5 reduction that includes non-abelian
gauge fields, and it allows any bosonic solution of the five-dimensional N=4
gauged theory to be embedded in D=10.Comment: latex, 12 page
Topology of Cell-Aggregated Planar Graphs
We present new algorithm for growth of non-clustered planar graphs by
aggregation of cells with given distribution of size and constraint of
connectivity k=3 per node. The emergent graph structures are controlled by two
parameters--chemical potential of the cell aggregation and the width of the
cell size distribution. We compute several statistical properties of these
graphs--fractal dimension of the perimeter, distribution of shortest paths
between pairs of nodes and topological betweenness of nodes and links. We show
how these topological properties depend on the control parameters of the
aggregation process and discuss their relevance for the conduction of current
in self-assembled nanopatterns.Comment: 8 pages, 5 figure
Hidden Symmetries and Integrable Hierarchy of the N=4 Supersymmetric Yang-Mills Equations
We describe an infinite-dimensional algebra of hidden symmetries of N=4
supersymmetric Yang-Mills (SYM) theory. Our derivation is based on a
generalization of the supertwistor correspondence. Using the latter, we
construct an infinite sequence of flows on the solution space of the N=4 SYM
equations. The dependence of the SYM fields on the parameters along the flows
can be recovered by solving the equations of the hierarchy. We embed the N=4
SYM equations in the infinite system of the hierarchy equations and show that
this SYM hierarchy is associated with an infinite set of graded symmetries
recursively generated from supertranslations. Presumably, the existence of such
nonlocal symmetries underlies the observed integrable structures in quantum N=4
SYM theory.Comment: 24 page
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
One-dimensional structures behind twisted and untwisted superYang-Mills theory
We give a one-dimensional interpretation of the four-dimensional twisted N=1
superYang-Mills theory on a Kaehler manifold by performing an appropriate
dimensional reduction. We prove the existence of a 6-generator superalgebra,
which does not possess any invariant Lagrangian but contains two different
subalgebras that determine the twisted and untwisted formulations of the N=1
superYang-Mills theory.Comment: 12 pages. Final version to appear in Lett. Math. Phys. with improved
notation and misprints correcte
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
Convergent Calculation of the Asymptotic Dimension of Diffusion Limited Aggregates: Scaling and Renormalization of Small Clusters
Diffusion Limited Aggregation (DLA) is a model of fractal growth that had
attained a paradigmatic status due to its simplicity and its underlying role
for a variety of pattern forming processes. We present a convergent calculation
of the fractal dimension D of DLA based on a renormalization scheme for the
first Laurent coefficient of the conformal map from the unit circle to the
expanding boundary of the fractal cluster. The theory is applicable from very
small (2-3 particles) to asymptotically large (n \to \infty) clusters. The
computed dimension is D=1.713\pm 0.003
Hierarchical population model with a carrying capacity distribution
A time- and space-discrete model for the growth of a rapidly saturating local
biological population is derived from a hierarchical random deposition
process previously studied in statistical physics. Two biologically relevant
parameters, the probabilities of birth, , and of death, , determine the
carrying capacity . Due to the randomness the population depends strongly on
position, , and there is a distribution of carrying capacities, .
This distribution has self-similar character owing to the imposed hierarchy.
The most probable carrying capacity and its probability are studied as a
function of and . The effective growth rate decreases with time, roughly
as in a Verhulst process. The model is possibly applicable, for example, to
bacteria forming a "towering pillar" biofilm. The bacteria divide on randomly
distributed nutrient-rich regions and are exposed to random local bactericidal
agent (antibiotic spray). A gradual overall temperature change away from
optimal growth conditions, for instance, reduces bacterial reproduction, while
biofilm development degrades antimicrobial susceptibility, causing stagnation
into a stationary state.Comment: 25 pages, 11 (9+2) figure
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