1,561 research outputs found

### Aerodynamic properties of fractal grains: Implications for the primordial solar nebula

Under conditions in the primordial solar nebula and dense interstellar clouds, small grains have low relative velocities. This is the condition for efficient sticking and formation of fractal aggregates. A calculation of the ratio of cross section, sigma, to number of primary particles, N, for fractal clusters yielded 1n sigma/N = 0.2635 + 0.5189N sup (-0.1748). This ratio decreases slowly with N and approaches a constant for large N. Under the usual assumption of collisions producing spherical compact, uniform density aggregates, sigma/N varies as N sup -1/3 and decreases rapidly. Fractal grains are therefore much more closely coupled to the gas than are compact aggregates. This has a significant effect on the aerodynamic behavior of aggregates and consequently on their evolution and that of the nebula

### Smoluchowski's equation for cluster exogenous growth

We introduce an extended Smoluchowski equation describing coagulation
processes for which clusters of mass s grow between collisions with
$ds/dt=As^\beta$. A physical example, dropwise condensation is provided, and
its collision kernel K is derived. In the general case, the gelation criterion
is determined. Exact solutions are found and scaling solutions are
investigated. Finally we show how these results apply to nucleation of discs on
a planeComment: Revtex, 4 pages (multicol.sty), 1 eps figures (uses epsfig

### Conformal approach to cylindrical DLA

We extend the conformal mapping approach elaborated for the radial Diffusion
Limited Aggregation model (DLA) to the cylindrical geometry. We introduce in
particular a complex function which allows to grow a cylindrical cluster using
as intermediate step a radial aggregate. The grown aggregate exhibits the same
self-affine features of the original cylindrical DLA. The specific choice of
the transformation allows us to study the relationship between the radial and
the cylindrical geometry. In particular the cylindrical aggregate can be seen
as a radial aggregate with particles of size increasing with the radius. On the
other hand the radial aggregate can be seen as a cylindrical aggregate with
particles of size decreasing with the height. This framework, which shifts the
point of view from the geometry to the size of the particles, can open the way
to more quantitative studies on the relationship between radial and cylindrical
DLA.Comment: 16 pages, 8 figure

### Is it really possible to grow isotropic on-lattice diffusion-limited aggregates?

In a recent paper (Bogoyavlenskiy V A 2002 \JPA \textbf{35} 2533), an
algorithm aiming to generate isotropic clusters of the on-lattice
diffusion-limited aggregation (DLA) model was proposed. The procedure consists
of aggregation probabilities proportional to the squared number of occupied
sites ($k^2$). In the present work, we analyzed this algorithm using the noise
reduced version of the DLA model and large scale simulations. In the noiseless
limit, instead of isotropic patterns, a $45^\circ$ ($30^\circ$) rotation in the
anisotropy directions of the clusters grown on square (triangular) lattices was
observed. A generalized algorithm, in which the aggregation probability is
proportional to $k^\nu$, was proposed. The exponent $\nu$ has a nonuniversal
critical value $\nu_c$, for which the patterns generated in the noiseless limit
exhibit the original (axial) anisotropy for $\nu<\nu_c$ and the rotated one
(diagonal) for $\nu>\nu_c$. The values $\nu_c = 1.395\pm0.005$ and $\nu_c =
0.82\pm 0.01$ were found for square and triangular lattices, respectively.
Moreover, large scale simulations show that there are a nontrivial relation
between noise reduction and anisotropy direction. The case $\nu=2$ (\bogo's
rule) is an example where the patterns exhibit the axial anisotropy for small
and the diagonal one for large noise reduction.Comment: 12 pages, 8 figure

### Amalgams of Inverse Semigroups and C*-algebras

An amalgam of inverse semigroups [S,T,U] is full if U contains all of the
idempotents of S and T. We show that for a full amalgam [S,T,U], the C*-algebra
of the inverse semigroup amaglam of S and T over U is the C*-algebraic amalgam
of C*(S) and C*(T) over C*(U). Using this result, we describe certain
amalgamated free products of C*-algebras, including finite-dimensional
C*-algebras, the Toeplitz algebra, and the Toeplitz C*-algebras of graphs

### Spatial scaling in fracture propagation in dilute systems

The geometry of fracture patterns in a dilute elastic network is explored
using molecular dynamics simulation. The network in two dimensions is subjected
to a uniform strain which drives the fracture to develop by the growth and
coalescence of the vacancy clusters in the network. For strong dilution, it has
been shown earlier that there exists a characteristic time $t_c$ at which a
dynamical transition occurs with a power law divergence (with the exponent $z$)
of the average cluster size. Close to $t_c$, the growth of the clusters is
scale-invariant in time and satisfies a dynamical scaling law. This paper shows
that the cluster growth near $t_c$ also exhibits spatial scaling in addition to
the temporal scaling. As fracture develops with time, the connectivity length
$\xi$ of the clusters increses and diverges at $t_c$ as $\xi \sim
(t_c-t)^{-\nu}$, with $\nu = 0.83 \pm 0.06$. As a result of the scale-invariant
growth, the vacancy clusters attain a fractal structure at $t_c$ with an
effective dimensionality $d_f \sim 1.85 \pm 0.05$. These values are independent
(within the limit of statistical error) of the concentration (provided it is
sufficiently high) with which the network is diluted to begin with. Moreover,
the values are very different from the corresponding values in qualitatively
similar phenomena suggesting a different universality class of the problem. The
values of $\nu$ and $d_f$ supports the scaling relation $z=\nu d_f$ with the
value of $z$ obtained before.Comment: A single ps file (6 figures included), 12 pages, to appear in Physica

### Diffusion-limited deposition with dipolar interactions: fractal dimension and multifractal structure

Computer simulations are used to generate two-dimensional diffusion-limited
deposits of dipoles. The structure of these deposits is analyzed by measuring
some global quantities: the density of the deposit and the lateral correlation
function at a given height, the mean height of the upper surface for a given
number of deposited particles and the interfacial width at a given height.
Evidences are given that the fractal dimension of the deposits remains constant
as the deposition proceeds, independently of the dipolar strength. These same
deposits are used to obtain the growth probability measure through Monte Carlo
techniques. It is found that the distribution of growth probabilities obeys
multifractal scaling, i.e. it can be analyzed in terms of its $f(\alpha)$
multifractal spectrum. For low dipolar strengths, the $f(\alpha)$ spectrum is
similar to that of diffusion-limited aggregation. Our results suggest that for
increasing dipolar strength both the minimal local growth exponent
$\alpha_{min}$ and the information dimension $D_1$ decrease, while the fractal
dimension remains the same.Comment: 10 pages, 7 figure

### 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

### Diffusion-limited deposition of dipolar particles

Deposits of dipolar particles are investigated by means of extensive Monte
Carlo simulations. We found that the effect of the interactions is described by
an initial, non-universal, scaling regime characterized by orientationally
ordered deposits. In the dipolar regime, the order and geometry of the clusters
depend on the strength of the interactions and the magnetic properties are
tunable by controlling the growth conditions. At later stages, the growth is
dominated by thermal effects and the diffusion-limited universal regime
obtains, at finite temperatures. At low temperatures the crossover size
increases exponentially as T decreases and at T=0 only the dipolar regime is
observed.Comment: 5 pages, 4 figure

### Aggregation in a mixture of Brownian and ballistic wandering particles

In this paper, we analyze the scaling properties of a model that has as
limiting cases the diffusion-limited aggregation (DLA) and the ballistic
aggregation (BA) models. This model allows us to control the radial and angular
scaling of the patterns, as well as, their gap distributions. The particles
added to the cluster can follow either ballistic trajectories, with probability
$P_{ba}$, or random ones, with probability $P_{rw}=1-P_{ba}$. The patterns were
characterized through several quantities, including those related to the radial
and angular scaling. The fractal dimension as a function of $P_{ba}$
continuously increases from $d_f\approx 1.72$ (DLA dimensionality) for
$P_{ba}=0$ to $d_f\approx 2$ (BA dimensionality) for $P_{ba}=1$. However, the
lacunarity and the active zone width exhibt a distinct behavior: they are
convex functions of $P_{ba}$ with a maximum at $P_{ba}\approx1/2$. Through the
analysis of the angular correlation function, we found that the difference
between the radial and angular exponents decreases continuously with increasing
$P_{ba}$ and rapidly vanishes for $P_{ba}>1/2$, in agreement with recent
results concerning the asymptotic scaling of DLA clusters.Comment: 7 pages, 6 figures. accepted for publication on PR

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