1,417 research outputs found
High-Dimensional Diffusive Growth
We consider a model of aggregation, both diffusion-limited and ballistic,
based on the Cayley tree. Growth is from the leaves of the tree towards the
root, leading to non-trivial screening and branch competition effects. The
model exhibits a phase transition between ballistic and diffusion-controlled
growth, with non-trivial corrections to cluster size at the critical point.
Even in the ballistic regime, cluster scaling is controlled by extremal
statistics due to the branching structure of the Cayley tree; it is the
extremal nature of the fluctuations that enables us to solve the model.Comment: 5 pages, 3 figures; reference adde
Multifractal Dimensions for Branched Growth
A recently proposed theory for diffusion-limited aggregation (DLA), which
models this system as a random branched growth process, is reviewed. Like DLA,
this process is stochastic, and ensemble averaging is needed in order to define
multifractal dimensions. In an earlier work [T. C. Halsey and M. Leibig, Phys.
Rev. A46, 7793 (1992)], annealed average dimensions were computed for this
model. In this paper, we compute the quenched average dimensions, which are
expected to apply to typical members of the ensemble. We develop a perturbative
expansion for the average of the logarithm of the multifractal partition
function; the leading and sub-leading divergent terms in this expansion are
then resummed to all orders. The result is that in the limit where the number
of particles n -> \infty, the quenched and annealed dimensions are {\it
identical}; however, the attainment of this limit requires enormous values of
n. At smaller, more realistic values of n, the apparent quenched dimensions
differ from the annealed dimensions. We interpret these results to mean that
while multifractality as an ensemble property of random branched growth (and
hence of DLA) is quite robust, it subtly fails for typical members of the
ensemble.Comment: 82 pages, 24 included figures in 16 files, 1 included tabl
Exact Multifractal Spectra for Arbitrary Laplacian Random Walks
Iterated conformal mappings are used to obtain exact multifractal spectra of
the harmonic measure for arbitrary Laplacian random walks in two dimensions.
Separate spectra are found to describe scaling of the growth measure in time,
of the measure near the growth tip, and of the measure away from the growth
tip. The spectra away from the tip coincide with those of conformally invariant
equilibrium systems with arbitrary central charge , with related
to the particular walk chosen, while the scaling in time and near the tip
cannot be obtained from the equilibrium properties.Comment: 4 pages, 3 figures; references added, minor correction
Diffusion-limited aggregation as branched growth
I present a first-principles theory of diffusion-limited aggregation in two
dimensions. A renormalized mean-field approximation gives the form of the
unstable manifold for branch competition, following the method of Halsey and
Leibig [Phys. Rev. A {\bf 46}, 7793 (1992)]. This leads to a result for the
cluster dimensionality, D \approx 1.66, which is close to numerically obtained
values. In addition, the multifractal exponent \tau(3) = D in this theory, in
agreement with a proposed `electrostatic' scaling law.Comment: 13 pages, one figure not included (available by request, by ordinary
mail), Plain Te
Branched Growth with Walkers
Diffusion-limited aggregation has a natural generalization to the
"-models", in which random walkers must arrive at a point on the
cluster surface in order for growth to occur. It has recently been proposed
that in spatial dimensionality , there is an upper critical
above which the fractal dimensionality of the clusters is D=1. I compute the
first order correction to for , obtaining . The
methods used can also determine multifractal dimensions to first order in
.Comment: 6 pages, 1 figur
Transfer across Random versus Deterministic Fractal Interfaces
A numerical study of the transfer across random fractal surfaces shows that
their responses are very close to the response of deterministic model
geometries with the same fractal dimension. The simulations of several
interfaces with prefractal geometries show that, within very good
approximation, the flux depends only on a few characteristic features of the
interface geometry: the lower and higher cut-offs and the fractal dimension.
Although the active zones are different for different geometries, the electrode
reponses are very nearly the same. In that sense, the fractal dimension is the
essential "universal" exponent which determines the net transfer.Comment: 4 pages, 6 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
Exact Multifractal Exponents for Two-Dimensional Percolation
The harmonic measure (or diffusion field or electrostatic potential) near a
percolation cluster in two dimensions is considered. Its moments, summed over
the accessible external hull, exhibit a multifractal spectrum, which I
calculate exactly. The generalized dimensions D(n) as well as the MF function
f(alpha) are derived from generalized conformal invariance, and are shown to be
identical to those of the harmonic measure on 2D random walks or self-avoiding
walks. An exact application to the anomalous impedance of a rough percolative
electrode is given. The numerical checks are excellent. Another set of exact
and universal multifractal exponents is obtained for n independent
self-avoiding walks anchored at the boundary of a percolation cluster. These
exponents describe the multifractal scaling behavior of the average nth moment
of the probabity for a SAW to escape from the random fractal boundary of a
percolation cluster in two dimensions.Comment: 5 pages, 3 figures (in colors
Two-Dimensional Copolymers and Exact Conformal Multifractality
We consider in two dimensions the most general star-shaped copolymer, mixing
random (RW) or self-avoiding walks (SAW) with specific interactions thereof.
Its exact bulk or boundary conformal scaling dimensions in the plane are all
derived from an algebraic structure existing on a random lattice (2D quantum
gravity). The multifractal dimensions of the harmonic measure of a 2D RW or SAW
are conformal dimensions of certain star copolymers, here calculated exactly as
non rational algebraic numbers. The associated multifractal function f(alpha)
are found to be identical for a random walk or a SAW in 2D. These are the first
examples of exact conformal multifractality in two dimensions.Comment: 4 pages, 2 figures, revtex, to appear in Phys. Rev. Lett., January
199
Velocity Correlations in Dense Gravity Driven Granular Chute Flow
We report numerical results for velocity correlations in dense,
gravity-driven granular flow down an inclined plane. For the grains on the
surface layer, our results are consistent with experimental measurements
reported by Pouliquen. We show that the correlation structure within planes
parallel to the surface persists in the bulk. The two-point velocity
correlation function exhibits exponential decay for small to intermediate
values of the separation between spheres. The correlation lengths identified by
exponential fits to the data show nontrivial dependence on the averaging time
\dt used to determine grain velocities. We discuss the correlation length
dependence on averaging time, incline angle, pile height, depth of the layer,
system size and grain stiffness, and relate the results to other length scales
associated with the rheology of the system. We find that correlation lengths
are typically quite small, of the order of a particle diameter, and increase
approximately logarithmically with a minimum pile height for which flow is
possible, \hstop, contrary to the theoretical expectation of a proportional
relationship between the two length scales.Comment: 21 pages, 16 figure
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