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 $\theta$, whose exact numerical values are given for which $x^{-\theta}$ or $t^{\theta z}$ 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

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

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 $d_f=0.56155$ which is less than its classical value $d_f={{\ln 2}\over{\ln 3}}$. Nonetheless, in all three cases we show that the sum of the $d_f$th power, $d_f$ 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 $[0,1]$ 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 $p_c$ and by a set of critical exponents $\beta$, $\gamma$, $\nu$ which describe the critical behavior of percolation probability $P(p)\sim (p_c-p)^\beta$, mean cluster size $S\sim (p_c-p)^{-\gamma}$ and the correlation length $\xi\sim (p_c-p)^{-\nu}$. Besides, the exponent $\tau$ characterizes the cluster size distribution function $n_s(p_c)\sim s^{-\tau}$ and the fractal dimension $d_f$ the spanning cluster. We obtain an exact value for $p_c$ 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 $p_c$ 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 $P(p)\sim (p_c-p)^\beta$, mean cluster size $S\sim (p_c-p)^{-\gamma}$ and the system size $L\sim (p_c-p)^{-\nu}$ which are known as the equivalent counterpart of the order parameter, susceptibility and correlation length respectively. Moreover, the cluster size distribution function $n_s(p_c)\sim s^{-\tau}$ and the mass-length relation $M\sim L^{d_f}$ of the spanning cluster also provide useful characterization of the percolation process. We obtain an exact value for $p_c$ and for all the exponents such as $\beta, \nu, \gamma, \tau$ and $d_f$. We find that, except $p_c$, 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 $p=0$ revealing that the system is in the perfectly ordered and in the disordered state at the same time. Moreover, we find that entropy with $1-p$, 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 $d$-dimensional hypercuboid-shaped objects. The first class of exact solutions is described by a fragmentation rate $a({x_1},...,{x_d}) = 1$ 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 $\lambda$ of the aggregation kernel divides the aggregation process into two regimes (i) $\lambda\leq 1$ nongelling and (ii) $\lambda>1$ gelling. However, our findings contradict with this result. In particular, we find that the Smoluchowski equation is valid if and only if $\lambda<1$. We show that beyond this limit i.e. at $\lambda\geq 1$, 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