arXiv subjects

M. K. Hassan

Publications and source records attributed to M. K. Hassan.

30 records · Page 2Linked to original sources

arXiv2004

Scale Invariant Fractal and Slow Dynamics in Nucleation and Growth Processes

We propose a stochastic counterpart of the classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) model to describe the nucleation-and-growth phenomena of a stable phase (S-phase). We report that for growth velocity of S-phase $v=s(t)/t$ where $s(t)$ is the mean value of the interval size $x$ of metastable phase (M-phase) and for $v=x/τ(x)$ where $τ(x)$ is the mean nucleation time, the system exhibits a power law decay of M-phase. We also find that the resulting structure exhibits self-similarity and can be best described as a fractal. Interestingly, the fractal dimension $d_f$ helps generalising the exponent $(1+d_f)$ of the power-law decay. However, when either $v=v_0$ (constant) or $v=σ/t$ ($σ$ is a constant) the decay is exponential and it is accompanied by the violation of scaling.

cond-mat.stat-mech↗

arXiv2002

An analytic model for a cooperative ballistic deposition in one dimension

We formulate a model for a cooperative ballistic deposition (CBD) process whereby the incoming particles are correlated with the ones already adsorbed via attractive force. The strength of the correlation is controlled by a tunable parameter $a$ that interpolates the classical car parking problem at $a=0$, the ballistic deposition at $a=1$ and the CBD model at $a>1$. The effects of the correlation in the CBD model are as follows. The jamming coverage $q(a)$ increases with the strength of attraction $a$ due to an ever increasing tendency of cluster formation. The system almost reaches the closest packing structure as $a\to\infty$ but never forms a percolating cluster which is typical to 1D system. In the large $a$ regime, the mean cluster size $k$ increases as $a^{1/2}$. Furthermore, the asymptotic approach towards the closest packing is purely algebraic both with $a$ as $q(\infty)-q(a) \sim a^{-1/2}$ and with $k$ as $q(\infty)-q(k) \sim k^{-1}$ where $q(\infty)\simeq 1$.

cond-mat.stat-mech↗

arXiv2002

Jamming and asymptotic behavior in competitive random parking of bidisperse cars

We propose a generalized car parking problem where either a car of size $σ$ or of size $mσ$ ($m>1$) is sequentially parked on a line with probability $q$ and $(1-q)$, respectively. The free parameter $q$ interpolates between the classical car parking problem at either extreme ($q=0$ and $q=1$) and the competitive random sequential adsorption of a binary mixture in between. We find that the coverage in the jamming limit for a mixture always exceeds the value obtained for the uni-sized case. The introduction of a bidisperse mixture results in the slow approach ($\sim t^{-1}$) to the jamming limit by the smaller species while the larger species reach their asymptotic values exponentially fast $\sim t^{-1}e^{-(m-1)qt}$.

cond-mat.stat-mech↗

arXiv2002

Can randomness alone tune the fractal dimension?

We present a generalized stochastic Cantor set by means of a simple {\it cut and delete process} and discuss the self-similar properties of the arising geometric structure. To increase the flexibility of the model, two free parameters, $m$ and $b$, are introduced which tune the relative strength of the two processes and the degree of randomness respectively. In doing so, we have identified a new set with a wide spectrum of subsets produced by tuning either $m$ or $b$. Measuring the size of the resulting set in terms of fractal dimension, we show that the fractal dimension increases with increasing order and reaches its maximum value when the randomness is completely ceased.

cond-mat.stat-mech↗

arXiv2001

Jamming coverage in competitive random sequential adsorption of binary mixture

We propose a generalized car parking problem where cars of two different sizes are sequentially parked on a line with a given probability $q$. The free parameter $q$ interpolates between the classical car parking problem of only one car size and the competitive random sequential adsorption (CRSA) of a binary mixture. We give an exact solution to the CRSA rate equations and find that the final coverage, the jamming limit, of the line is always larger for a binary mixture than for the uni-sized case. The analytical results are in good agreement with our direct numerical simulations of the problem.

cond-mat.stat-mech↗

arXiv2001

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 $λ$ of the aggregation kernel divides the aggregation process into two regimes (i) $λ\leq 1$ nongelling and (ii) $λ>1$ gelling. However, our findings contradict with this result. In particular, we find that the Smoluchowski equation is valid if and only if $λ<1$. We show that beyond this limit i.e. at $λ\geq 1$, it breaks down and hence it fails to describe a gelation transition. This also happens to be accompanied by violation of scaling.

cond-mat.stat-mech↗

arXiv2001

Transition from random to ordered fractals in fragmentation of particles in an open system

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 $θ$, whose exact numerical values are given for which $x^{-θ}$ or $t^{θz}$ has the dimension of particle size distribution function $ψ(x,t)$, where $z$ is the kinetic exponent. We obtained conditions for which the scaling and fragmentation process altogether break down and 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 completely ceases.

cond-mat.stat-mech↗

arXiv2001

Competitive random sequential adsorption of point and fixed-sized particles: analytical results

We study the kinetics of competitive random sequential adsorption (RSA) of particles of binary mixture of points and fixed-sized particles within the mean-field approach. The present work is a generalization of the random car parking problem in the sense that it considers the case when either a car of fixed size is parked with probability q or the parking space is partitioned into two smaller spaces with probability (1-q) at each time event. This allows an interesting interplay between the classical RSA problem at one extreme (q=1), and the kinetics of fragmentation processes at the other extreme (q=0). We present exact analytical results for coverage for a whole range of q values, and physical explanations are given for different aspects of the problem. In addition, a comprehensive account of the scaling theory, emphasizing on dimensional analysis, is presented, and the exact expression for the scaling function and exponents are obtained.

cond-mat.stat-mech↗

arXiv2000

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 $θ$, whose exact numerical values are given for which $x^{-θ}$ or $t^{θz}$ 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.

cond-mat.stat-mech↗

arXiv1996

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.

cond-mat↗

arXiv1996

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.

cond-mat↗

arXiv1996

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.

cond-mat↗