arXiv subjects

S. S. Manna

Publications and source records attributed to S. S. Manna.

At least 19 recordsLinked to original sources

arXiv2025

Colored Sandpile

After the introduction of sandpile model a number of different variants have been studied. In most of these models sand particles are indistinguishable. Here we have painted the sand particles using a few distinct colors, and restrict them to move in linear trajectories only along their assigned lattice axes, one axis reserved for one color. Different colored particles interact among themselves through the toppling of unstable sand columns. Consequently, the avalanches or in general the self-organization processes in the sandpile has no overall preferred direction, though the individual particles execute directed motion. For such non-abelian colored sandpiles the steady states are found to be different and also the avalanche size distributions. This sandpile so defined has a non-trivial spatial structure and belongs to a different universality class of sandpile models. Dynamics of a granular heap with grains of different colors and properties may be described using this sandpile.

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arXiv2025

Describing Self-organized Criticality as a continuous phase transition

Can the concept of self-organized criticality, exemplified by models such as the sandpile model, be described within the framework of continuous phase transitions? In this paper, we provide extensive numerical evidence supporting an affirmative answer. Specifically, we explore the BTW and Manna sandpile models as instances of percolation transitions from disordered to ordered phases. To facilitate this analysis, we introduce the concept of drop density, a continuously adjustable control variable that quantifies the average number of particles added to a site. By tuning this variable, we observe a transition in the sandpile from a sub-critical to a critical phase. Additionally, we define the scaled size of the largest avalanche occurring from the beginning of the sandpile as the order parameter for the self-organized critical transition and analyze its scaling behavior. Furthermore, we calculate the correlation length exponent and note its divergence as the critical point is approached. The finite size scaling analysis of the avalanche size distribution works quite well at the critical point of the BTW sandpile.

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arXiv2024

Q-factor: A measure of competition between the topper and the average in percolation and in SOC

We define the $Q$-factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability $p$ is increased, the $Q$-factor for the system size $L$ grows systematically to its maximum value $Q_{max}(L)$ at a specific value $p_{max}(L)$ and then gradually decays. Our numerical study of site percolation problems on the square, triangular and the simple cubic lattices exhibits that the asymptotic values of $p_{max}$ though close, are distinctly different from the corresponding percolation thresholds of these lattices. We have also shown using the scaling analysis that at $p_{max}$ the value of $Q_{max}(L)$ diverges as $L^d$ ($d$ denoting the dimension of the lattice) as the system size approaches to their asymptotic limit. We have further extended this idea to the non-equilibrium systems such as the sandpile model of self-organized criticality. Here, the $Q(ρ,L)$-factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches; $ρ$ being the drop density of the driving mechanism. This study has been prompted by some observations in Sociophysics.

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arXiv2023

Nonstationary but quasisteady states in Self-organized Criticality

The notion of Self-organized criticality (SOC) had been conceived to interpret the spontaneous emergence of long range correlations in nature. Since then many different models had been introduced to study SOC. All of them have few common features: externally driven dynamical systems self-organize themselves to non-equilibrium stationary states exhibiting fluctuations of all length scales as the signatures of criticality. In contrast, we have studied here in the framework of the sandpile model a system that has mass inflow but no outflow. There is no boundary, and particles cannot escape from the system by any means. Therefore, there is no current balance, and consequently it is not expected that the system would arrive at a stationary state. In spite of that, it is observed that the bulk of the system self-organizes to a quasi-steady state where the grain density is maintained at a nearly constant value. Power law distributed fluctuations of all length and time scales have been observed which are the signatures of criticality. Our detailed computer simulation study gives the set of critical exponents whose values are very close to their counter parts in the original sandpile model. This study indicates that (i) a physical boundary and (ii) the stationary state though sufficient but may not be the necessary criteria for achieving SOC.

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arXiv2022

Island and lake size distributions in Gradient Percolation

The well-known problem of gradient percolation has been revisited to study the probability distribution of island sizes. It is observed that as the ordinary percolation, this distribution is also described by a power-law decaying function but the associated critical exponents are found to be different. Because of the underlying gradient for the occupation probability, the average value of the island sizes also has a gradient. The variation of the average island size with the probability of occupation along the gradient has been studied together with its scaling analysis. Further, we have introduced and studied the gradient bond percolation and by studying the island size distribution statistics, we have obtained very similar results. We have also studied the characteristics of the diffusion profile of the particle system on a lattice that is initially half filled and half empty. Here also we observe the same value for the island size probability distribution exponent. Finally, the same study has been repeated for the nonlinear gradient percolation and the value of the island size distribution exponent is found to be a function of the strength of the nonlinear parameter.

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arXiv2022

Near universal values of social inequality indices in self-organized critical models

We have studied few social inequality measures associated with the sub-critical dynamical features (measured in terms of the avalanche size distributions) of four self-organized critical models while the corresponding systems approach their respective stationary critical states. It has been observed that these inequality measures (specifically the Gini and Kolkata indices) exhibit nearly universal values though the models studied here are widely different, namely the Bak-Tang-Wiesenfeld sandpile, the Manna sandpile and the quenched Edwards-Wilkinson interface, and the fiber bundle interface. These observations suggest that the self-organized critical systems have broad similarity in terms of these inequality measures. A comparison with similar earlier observations in the data of socio-economic systems with unrestricted competitions suggest the emergent inequality as a result of the possible proximity to the self-organized critical states.

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arXiv2020

Bond percolation between $k$ separated points on a square lattice

We consider a percolation process in which $k$ points separated by a distance proportional to system size $L$ simultaneously connect together ($k>1$), or a single point at the center of a system connects to the boundary ($k=1$), through adjacent connected points of a single cluster. These processes yield new thresholds $\overline p_{ck}$ defined as the average value of $p$ at which the desired connections first occur. These thresholds are not sharp as the distribution of values of $p_{ck}$ for individual samples remains broad in the limit of $L \to \infty$. We study $\overline p_{ck}$ for bond percolation on the square lattice, and find that $\overline p_{ck}$ are above the normal percolation threshold $p_c = 1/2$ and represent specific supercritical states. The $\overline p_{ck}$ can be related to integrals over powers of the function $P_\infty(p)$ equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of $P_\infty(p)$ on $L\times L$ systems that, for $L \to \infty$, $\overline p_{c1} = 0.51755(5)$, $\overline p_{c2} = 0.53219(5)$, $\overline p_{c3} = 0.54456(5)$, and $\overline p_{c4} = 0.55527(5).$ The percolation thresholds $\overline p_{ck}$ remain the same, even when the $k$ points are randomly selected within the lattice. We show that the finite-size corrections scale as $L^{-1/ν_k}$ where $ν_k = ν/(k β+1)$, with $β=5/36$ and $ν=4/3$ being the ordinary percolation critical exponents, so that $ν_1= 48/41$, $ν_2 = 24/23$, $ν_3 = 16/17$, $ν_4 = 6/7$, etc. We also study three-point correlations in the system, and show how for $p>p_c$, the correlation ratio goes to 1 (no net correlation) as $L \to \infty$, while at $p_c$ it reaches the known value of 1.022.

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arXiv2019

Brittle to quasibrittle transition in a compound fiber bundle

The brittle to quasibrittle transition has been studied for a compound of two different kinds of fibrous materials, having distinct difference in their breaking strengths under the framework of the fiber bundle model. A random fiber bundle model has been devised with a bimodal distribution of the breaking strengths of the individual fibers. The bimodal distribution is assumed to be consisting of two symmetrically placed rectangular probability distributions of strengths $p$ and $1 - p$, each of width $d$, and separated by a gap $2s$. Different properties of the transition have been studied varying these three parameters and using the well known equal load sharing dynamics. Our study exhibits a brittle to quasibrittle transition at the critical width $d_c(s,p) = p(1/2 - s)/(1 + p)$ confirmed by our numerical results.

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arXiv2019

Band structure in collective motion with quenched range of interaction

A variant of the well known Vicsek model of the collective motion of a group of agents has been studied where the range of interactions are spatially quenched and non-overlapping. To define such interactions, the underlying two dimensional space is discretized and is divided into the primitive cells of an imaginary square lattice. At any arbitrary time instant, all agents within one cell mutually interact with one another. Therefore, when an agent crosses the boundary of a cell, and moves to a neighboring cell, only then its influence is spread to the adjacent cell. Tuning the strength of the scalar noise $η$ it has been observed that the system makes a discontinuous transition from a random diffusive phase to an ordered phase through a critical noise strength $η_c$ where directed bands with high agent densities appear. Unlike the original Vicsek model here a host of different types of bands has been observed with different angles of orientation and different wrapping numbers. More interestingly, two mutually crossed independent sets of simultaneously moving bands are also observed. A prescription for the detailed characterization of different types of bands have been formulated.

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arXiv2018

Jamming and percolation properties of random sequential adsorption with relaxation

The random sequential adsorption (RSA) model is a classical model in Statistical Physics for adsorption on two-dimensional surfaces. Objects are deposited sequentially at random and adsorb irreversibly on the landing site, provided that they do not overlap any previously adsorbed object. The kinetics of adsorption ceases when no more objects can be adsorbed (jamming state). Here, we investigate the role of post-relaxation on the jamming state and percolation properties of RSA of dimers on a two-dimensional lattice. We consider that, if the deposited dimer partially overlaps with a previously adsorbed one, a sequence of dimer displacements may occur to accommodate the new dimer. The introduction of this simple relaxation dynamics leads to a more dense jamming state than the one obtained with RSA without relaxation. We also consider the anisotropic case, where one dimer orientation is favored over the other, finding a non-monotonic dependence of the jamming coverage on the strength of anisotropy. We find that the density of adsorbed dimers at which percolation occurs is reduced with relaxation, but the value depends on the strength of anisotropy.

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arXiv2017

Colored Percolation

A model named `Colored Percolation' has been introduced with its infinite number of versions in two dimensions. The sites of a regular lattice are randomly occupied with probability $p$ and are then colored by one of the $n$ distinct colors using uniform probability $q = 1/n$. Denoting different colors by the letters of the Roman alphabet, we have studied different versions of the model like $AB, ABC, ABCD, ABCDE, ...$ etc. Here, only those lattice bonds having two different colored atoms at the ends are defined as connected. The percolation thresholds $p_c(n)$ asymptotically converges to its limiting value of $p_c$ as $1/n$. The model has been generalized by introducing a preference towards a subset of colors when $m$ out of $n$ colors are selected with probability $q/m$ each and rest of the colors are selected with probability $(1 - q)/(n - m)$. It has been observed that $p_c(q,m)$ depends non-trivially on $q$ and has a minimum at $q_{min} = m/n$. In another generalization the fractions of bonds between similar and dissimilar colored atoms have been treated as independent parameters. Phase diagrams in this parameter space have been drawn exhibiting percolating and non-percolating phases.

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arXiv2017

Double Transition in a Model of Oscillating Percolation

Two distinct transition points have been observed in a problem of lattice percolation studied using a system of pulsating discs. Sites on a regular lattice are occupied by circular discs whose radii vary sinusoidally within $[0,R_0]$ starting from a random distribution of phase angles. A lattice bond is said to be connected when its two end discs overlap with each other. Depending on the difference of the phase angles of these discs a bond may be termed as dead or live. While a dead bond can never be connected, a live bond is connected at least once in a complete time period. Two different time scales can be associated with such a system, leading to two transition points. Namely, a percolation transition occurs at $R_{0c} =0.908$ when a spanning cluster of connected bonds emerges in the system. Here, information propagates across the system instantly, i.e., with infinite speed. Secondly, there exists another transition point $R_0^* = 0.5907$ where the giant cluster of live bonds spans the lattice. In this case the information takes finite time to propagate across the system through the dynamical evolution of finite size clusters. This passage time diverges as $R_0 \to R_0^*$ from above. Both the transitions exhibit the critical behavior of ordinary percolation transition. The entire scenario is robust with respect to the distribution of frequencies of the individual discs. This study may be relevant in the context of wireless sensor networks.

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arXiv2016

Percolation model with an additional source of disorder

The ranges of transmission of the mobiles in a Mobile Ad-hoc Network are not uniform in reality. They are affected by the temperature fluctuation in air, obstruction due to the solid objects, even the humidity difference in the environment, etc. How the varying range of transmission of the individual active elements affects the global connectivity in the network may be an important practical question to ask. Here a new model of percolation phenomena, with an additional source of disorder, has been introduced for a theoretical understanding of this problem. As in ordinary percolation, sites of a square lattice are occupied randomly with the probability $p$. Each occupied site is then assigned a circular disc of random value $R$ for its radius. A bond is defined to be occupied if and only if the radii $R_1$ and $R_2$ of the discs centered at the ends satisfy certain pre-defined condition. In a very general formulation, one divides the $R_1 - R_2$ plane into two regions by an arbitrary closed curve. One defines that a point within one region represents an occupied bond, otherwise it is a vacant bond. Study of three different rules under this general formulation, indicates that the percolation threshold is always larger and varies continuously. This threshold has two limiting values, one is $p_c$(sq), the percolation threshold for the ordinary site percolation on the square lattice and the other being unity. The variation of the thresholds are characterized by exponents, which are not known in the literature. In a special case, all lattice sites are occupied by discs of random radii $R \in \{0,R_0\}$ and a percolation transition is observed with $R_0$ as the control variable, similar to the site occupation probability.

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arXiv2015

Fiber Bundle model with Highly Disordered Breaking Thresholds

We present a study of the fiber bundle model using equal load sharing dynamics where the breaking thresholds of the fibers are drawn randomly from a power law distribution of the form $p(b)\sim b^{-1}$ in the range $10^{-β}$ to $10^β$. Tuning the value of $β$ continuously over a wide range, the critical behavior of the fiber bundle has been studied both analytically as well as numerically. Our results are: (i) The critical load $σ_c(β,N)$ for the bundle of size $N$ approaches its asymptotic value $σ_c(β)$ as $σ_c(β,N) = σ_c(β)+AN^{-1/ν(β)}$ where $σ_c(β)$ has been obtained analytically as $σ_c(β) = 10^β/(2βe\ln10)$ for $β\geq β_u = 1/(2\ln10)$, and for $β<β_u$ the weakest fiber failure leads to the catastrophic breakdown of the entire fiber bundle, similar to brittle materials, leading to $σ_c(β) = 10^{-β}$; (ii) the fraction of broken fibers right before the complete breakdown of the bundle has the form $1-1/(2β\ln10)$; (iii) the distribution $D(Δ)$ of the avalanches of size $Δ$ follows a power law $D(Δ)\sim Δ^{-ξ}$ with $ξ= 5/2$ for $Δ\gg Δ_c(β)$ and $ξ= 3/2$ for $Δ\ll Δ_c(β)$, where the crossover avalanche size $Δ_c(β) = 2/(1-e10^{-2β})^2$.

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arXiv2014

Interacting particles in a periodically moving potential: Traveling wave and transport

We study a system of interacting particles in a periodically moving external potential, within the simplest possible description of paradigmatic symmetric exclusion process on a ring. The model describes diffusion of hardcore particles where the diffusion dynamics is locally modified at a uniformly moving defect site, mimicking the effect of the periodically moving external potential. The model, though simple, exhibits remarkably rich features in particle transport, such as polarity reversal and double peaks in particle current upon variation of defect velocity and particle density. By tuning these variables, the most efficient transport can be achieved in either direction along the ring. These features can be understood in terms of a traveling density wave propagating in the system. Our results could be experimentally tested, e.g., in a system of colloidal particles driven by a moving optical tweezer.

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arXiv2014

Space-filling Percolation

A region of two-dimensional space has been filled randomly with large number of growing circular discs allowing only a `slight' overlapping among them just before their growth stop. More specifically, each disc grows from a nucleation center that is selected at a random location within the uncovered region. The growth rate $δ$ is a continuously tunable parameter of the problem which assumes a specific value while a particular pattern of discs is generated. When a growing disc overlaps for the first time with at least another disc, it's growth is stopped and is said to be `frozen'. In this paper we study the percolation properties of the set of frozen discs. Using numerical simulations we present evidence for the following: (i) The Order Parameter appears to jump discontinuously at a certain critical value of the area coverage; (ii) the width of the window of the area coverage needed to observe a macroscopic jump in the Order Parameter tends to vanish as $δ\to 0$ and on the contrary (iii) the cluster size distribution has a power law decaying functional form. While the first two results are the signatures of a discontinuous transition, the third result is indicative of a continuous transition. Therefore we refer this transition as a sharp but continuous transition similar to what has been observed in the recently introduced Achlioptas process of Explosive Percolation. It is also observed that in the limit of $δ\to 0$, the critical area coverage at the transition point tends to unity, implying the limiting pattern is space-filling. In this limit, the fractal dimension of the pore space at the percolation point has been estimated to be $1.42(10)$ and the contact network of the disc assembly is found to be a scale-free network.

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arXiv2013

Cyclic and Coherent States in Flocks with Topological Distance

A simple model of the two dimensional collective motion of a group of mobile agents have been studied. Like birds, these agents travel in open free space where each of them interacts with the first $n$ neighbors determined by the topological distance with a free boundary condition. Using the same prescription for interactions used in the Vicsek model with scalar noise it has been observed that the flock, in absence of the noise, arrives at a number of interesting stationary states. In the `single sink state' the entire flock maintains perfect cohesion and coherence. In the `cyclic state' every agent executes a uniform circular motion, and the entire flock executes a pulsating dynamics i.e., expands and contracts periodically between a minimum and a maximum size of the flock. When refreshing rate of the interaction zone is the fastest, the entire flock gets fragmented into smaller clusters of different sizes. On introduction of scalar noise a crossover is observed when the agents cross over from a ballistic motion to a diffusive motion. Expectedly the crossover time is dependent on the strength of the noise $η$ and diverges as $η\to 0$. In simpler version the translational degrees of freedom of the agents are suppressed but their angular motion are retained. Here agents are the spins, placed at the sites of a square lattice with periodic boundary condition. Every spin interacts with its $n$ = 2, 3 or 4 nearest neighbors. In the stationary state the entire spin pattern moves as a whole when interactions are anisotropic with $n$ = 2 and 3; but it is completely frozen when the interaction is isotropic with $n=4$. These spin configurations have vortex-antivortex pairs whose density increases as the noise $η$ increases and follows an excellent finite-size scaling analysis.

physics.bio-ph↗

arXiv2013

Scaling forms for Relaxation Times of the Fiber Bundle model

Using extensive numerical analysis of the Fiber Bundle Model with Equal Load Sharing dynamics we studied the finite-size scaling forms of the relaxation times against the deviations of applied load per fiber from the critical point. Our most crucial result is we have not found any $\ln (N)$ dependence of the average relaxation time $\langle T(σ,N) \rangle$ in the precritical state. The other results are: (i) The critical load $σ_c(N)$ for the bundle of size $N$ approaches its asymptotic value $σ_c(\infty)$ as $σ_c(N) = σ_c(\infty) + AN^{-1/ν}$. (ii) Right at the critical point the average relaxation time $\langle T(σ_c(N),N) \rangle$ scales with the bundle size $N$ as: $\langle T(σ_c(N),N) \rangle \sim N^η$ and this behavior remains valid within a small window of size $|Δσ| \sim N^{-ζ}$ around the critical point. (iii) When $1/N < |Δσ| < 100N^{-ζ}$ the finite-size scaling takes the form: $\langle T(σ,N) \rangle / N^η \sim {\cal G}[\{σ_c(N)-σ\}N^ζ]$ so that in the limit of $N \to \infty$ one has $\langle T(σ) \rangle \sim (σ- σ_c)^{-τ}$. The high precision of our numerical estimates led us to verify that $ν= 3/2$, conjecture that $η= 1/3$, $ζ= 2/3$ and therefore $τ= 1/2$.

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