18,126 research outputs found
Fractals, Multifractals and the Science of Complexity
We discuss the formation of stochastic fractals and multifractals using the
kinetic equation of fragmentation approach. We also discuss the potential
application of this sequential breaking and attempt to explain how nature
creats fractals.Comment: 3 Pages, LaTeX, no figure, Submitted to New Scientis
Fractal formation and ordering in Random Sequential Adsorption
We study the kinetics of random sequential adsorption of a mixture of
particles with continuous distribution of sizes for different deposition rules.
It appears in the long time limit the resulting system can be described using
the fractal concept. We reveal that the fractal dimension increases with the
degree of increasing order.Comment: 3 pages, LaTeX, no figure, Submitted to Phys. Rev. Lett. as a
Comment. (Two Tables for numerical survey of fractal dimensions and some
minor corrections are added
Formation of a New Class of Random Fractals in Fragmentation with Mass Loss
We consider the fragmentation process with mass loss and discuss self-similar
properties of the arising structure both in time and space focusing on
dimensional analysis. This exhibits a spectrum of mass exponents ,
whose exact numerical values are given for which or has the dimension of particle size distribution function c(x,t) where z is
the kinetic exponent. We also give explicit scaling solution for special case.
Finally, we identify a new class of fractals ranging from random to non-random
and show that the fractal dimension increases with increasing order and a
transition to strictly self-similar pattern occurs when randomness is
completely seized.Comment: 5 pages, Latex, no Figures, bibliography updated and minor
corrections to text in this versio
Emergence of fractal in aggregation with stochastic self-replication
We propose and investigate a simple model which describes the kinetics of
aggregation of Brownian particles with stochastic self-replication. An exact
solution and the scaling theory are presented alongside numerical simulation
which fully support all theoretical findings. In particular, we show
analytically that the particle size distribution function exhibits dynamic
scaling and we verified it numerically using the idea of data-collapse.
Besides, the conditions under which the resulting system emerges as a fractal
are found, the fractal dimension of the system is given and the relationship
between this fractal dimension and a conserved quantity is pointed out.Comment: 8 pages, 8 figure
Extensive analytical and numerical investigation of the kinetic and stochastic Cantor set
We investigate, both analytically and numerically, the kinetic and stochastic
counterpart of the triadic Cantor set. The generator that divides an interval
either into three equal pieces or into three pieces randomly and remove the
middle third is applied to only one interval, picked with probability
proportional to its size, at each generation step in the kinetic and stochastic
Cantor set respectively. We show that the fractal dimension of the kinetic
Cantor set coincides with that of its classical counterpart despite the
apparent differences in the spatial distribution of the intervals. For the
stochastic Cantor set, however, we find that the resulting set has fractal
dimension which is less than its classical value . Nonetheless, in all three cases we show that the sum of the
th power, being the fractal dimension of the respective set, of all
the intervals at all time is equal to one or the size of the initiator
regardless of whether it is recursive, kinetic or stochastic Cantor set.
Besides, we propose exact algorithms for both the variants which can capture
the complete dynamics described by the rate equation used to solve the
respective model analytically. The perfect agreement between our analytical and
numerical simulation is a clear testament to that.Comment: 8 pages, 8 figure
New universality class in percolation on multifractal scale-free planar stochastic lattice
We investigate site percolation on a weighted planar stochastic lattice
(WPSL) which is a multifractal and whose dual is a scale-free network.
Percolation is typically characterized by percolation threshold and by a
set of critical exponents , , which describe the critical
behavior of percolation probability , mean cluster size
and the correlation length .
Besides, the exponent characterizes the cluster size distribution
function and the fractal dimension the spanning
cluster. We obtain an exact value for and for all these exponents. Our
results suggest that the percolation on WPSL belong to a new universality class
as its exponents do not share the same value as for all the existing planar
lattices.Comment: 5 pages, 5 figure
Universality class of site and bond percolation on multi-multifractal scale-free planar stochastic lattice
In this article, we investigate both site and bond percolation on a weighted
planar stochastic lattice (WPSL) which is a multi-multifractal and whose dual
is a scale-free network. The characteristic properties of percolation is that
it exhibits threshold phenomena as we find sudden or abrupt jump in spanning
probability across accompanied by the divergence of some other observable
quantities which is reminiscent of continuous phase transition. Indeed,
percolation is characterized by the critical behavior of percolation strength
, mean cluster size and the
system size which are known as the equivalent
counterpart of the order parameter, susceptibility and correlation length
respectively. Moreover, the cluster size distribution function and the mass-length relation of the spanning cluster
also provide useful characterization of the percolation process. We obtain an
exact value for and for all the exponents such as and . We find that, except , all the exponents are exactly the
same in both bond and site percolation despite the significant difference in
the definition of cluster and other quantities. Our results suggest that the
percolation on WPSL belongs to a new universality class as its exponents do not
share the same value as for all the existing planar lattices and like other
cases its site and bond belong to the same universality class.Comment: 12 pages, 7 figures, 1 tabl
Redefinition of site percolation in light of entropy and the second law of thermodynamics
In this article, we revisit random site and bond percolation in square
lattice focusing primarily on the behavior of entropy and order parameter. In
the case of traditional site percolation, we find that both the quantities are
zero at revealing that the system is in the perfectly ordered and in the
disordered state at the same time. Moreover, we find that entropy with ,
which is the equivalent counterpart of temperature, first increases and then
decreases again but we know that entropy with temperature cannot decrease.
However, bond percolation does not suffer from either of these two problems. To
overcome this we propose a new definition for site percolation where we occupy
sites to connect bonds and we measure cluster size by the number of bonds
connected by occupied sites. This resolves all the problems without affecting
any of the existing known results.Comment: 7 pages, 7 captioned figure
On the Kinetics of Multi-dimensional Fragmentation
We present two classes of exact solutions to a geometric model which
describes the kinetics of fragmentation of -dimensional hypercuboid-shaped
objects. The first class of exact solutions is described by a fragmentation
rate and daughter distribution function
b({x_1},..,{x_d} | {{x_{1}^{\p}}},...,{{x_{d}^{\p}}})= {{(\a_1 +
2)x_1^{\a_1}}\over{x_1^{\p(\a_1+1)}}}...{{(\a_d+2)x_d^{\a_d}}\over
{x_d^{\p(\a_d+1)}}}. The second class of exact solutions is described by a
fragmentation rate a({x_1},...,{x_d}) =
{{{x_1}^{\a_1}}...{{x_d}^{\a_d}}/{2^d}} and a daughter distribution function
b({x_1},..,{x_d} | {{x_{1}^{\p}}},...,{{x_{d}^{\p}}}) = {2^d}{\d(x_1 -
{{x_{1}^\p}}/2)...\d(x_d - {{x_{d}^\p}}/2)}. Each class of exact solutions is
analyzed in detail for the presence of scaling solutions and the occurrence of
shattering transitions; the results of these analyses are also presented.Comment: 18 Pages, LaTe
Can Smoluchowski equation account for gelation transition?
We revisit the scaling theory of the Smoluchowski equation with special
emphasis on the dimensional analysis to derive the scaling ansatz and to give
an insightful foundation to it. It has long been argued that the homogeneity
exponent of the aggregation kernel divides the aggregation process
into two regimes (i) nongelling and (ii) gelling.
However, our findings contradict with this result. In particular, we find that
the Smoluchowski equation is valid if and only if . We show that
beyond this limit i.e. at , it breaks down and hence it fails to
describe a gelation transition. This also happens to be accompanied by
violation of scaling.Comment: 6 pages LaTeX, no figur
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