arXiv subjects

Tridib Sadhu

Publications and source records attributed to Tridib Sadhu.

44 records · Page 3Linked to original sources

arXiv2011

Pattern formation in fast-growing sandpiles

We study the patterns formed by adding $N$ sand-grains at a single site on an initial periodic background in the Abelian sandpile models, and relaxing the configuration. When the heights at all sites in the initial background are low enough, one gets patterns showing proportionate growth, with the diameter of the pattern formed growing as $N^{1/d}$ for large $N$, in $d$-dimensions. On the other hand, if sites with maximum stable height in the starting configuration form an infinite cluster, we get avalanches that do not stop. In this paper, we describe our unexpected finding of an interesting class of backgrounds in two dimensions, that show an intermediate behavior: For any $N$, the avalanches are finite, but the diameter of the pattern increases as $N^α$, for large $N$, with $1/2 < α\leq 1$. Different values of $α$ can be realized on different backgrounds, and the patterns still show proportionate growth. The non-compact nature of growth simplifies their analysis significantly. We characterize the asymptotic pattern exactly for one illustrative example with $α=1$.

cond-mat.stat-mech↗

arXiv2011

Long-range steady state density profiles induced by localized drive

We show that the presence of a localized drive in an otherwise diffusive system results in steady-state density and current profiles that decay algebraically to their global average value, away from the drive in two or higher dimensions. An analogy to an electrostatic problem is established, whereby the density profile induced by a driving bond maps onto the electrostatic potential due to an electric dipole located along the bond. The dipole strength is proportional to the drive, and is determined self-consistently by solving the electrostatic problem. The profile resulting from a localized configuration of more than one driving bond can be straightforwardly determined by the superposition principle of electrostatics. This picture is shown to hold even in the presence of exclusion interaction between particles.

cond-mat.stat-mech↗

arXiv2010

Effect of Noise on Patterns Formed by Growing Sandpiles

We consider patterns generated by adding large number of sand grains at a single site in an abelian sandpile model with a periodic initial configuration, and relaxing. The patterns show proportionate growth. We study the robustness of these patterns against different types of noise, \textit{viz.}, randomness in the point of addition, disorder in the initial periodic configuration, and disorder in the connectivity of the underlying lattice. We find that the patterns show a varying degree of robustness to addition of a small amount of noise in each case. However, introducing stochasticity in the toppling rules seems to destroy the asymptotic patterns completely, even for a weak noise. We also discuss a variational formulation of the pattern selection problem in growing abelian sandpiles.

cond-mat.stat-mech↗

arXiv2010

A self-organized critical model of rearranging hydrogen-bonded network in ice

A dynamical four-vertex model of rearranging hydrogen-bonded network in ice, with a dynamics similar to the Grotthuss mechanism, is proposed. The model qualitatively explains the unusually high proton-conductivity in ice. It also serves as an interesting example of self-organized criticality with a non-conservative dynamics. The model is solved on a linear chain and its steady state is determined. Using numerical simulations, the model is further studied on a square lattice, and it is observed that it relaxes by three different avalanches, each of which follows a simple finite-size scaling.

cond-mat.stat-mech↗

arXiv2009

Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks

Adding sand grains at a single site in Abelian sandpile models produces beautiful but complex patterns. We study the effect of sink sites on such patterns. Sinks change the scaling of the diameter of the pattern with the number $N$ of sand grains added. For example, in two dimensions, in presence of a sink site, the diameter of the pattern grows as $\sqrt{(N/\log N)}$ for large $N$, whereas it grows as $\sqrt{N}$ if there are no sink sites. In presence of a line of sink sites, this rate reduces to $N^{1/3}$. We determine the growth rates for these sink geometries along with the case when there are two lines of sink sites forming a wedge, and its generalization to higher dimensions. We characterize one such asymptotic patterns on the two-dimensional F-lattice with a single source adjacent to a line of sink sites, in terms of position of different spatial features in the pattern. For this lattice, we also provide an exact characterization of the pattern with two sources, when the line joining them is along one of the axes.

cond-mat.stat-mech↗

arXiv2008

Steady state of Stochastic Sandpile Models

We study the steady state of the abelian sandpile models with stochastic toppling rules. The particle addition operators commute with each other, but in general these operators need not be diagonalizable. We use their abelian algebra to determine their eigenvalues, and the Jordan block structure. These are then used to determine the probability of different configurations in the steady state. We illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example, for systems of size $\le12$, and also study the density profile in the steady state.

cond-mat.stat-mech↗

arXiv2008

Pattern formation in growing sandpiles

Adding grains at a single site on a flat substrate in the Abelian sandpile models produce beautiful complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional square lattice with directed edges (each site has two arrows directed inward and two outward), starting with a periodic background with half the sites occupied. The size of the pattern formed scales with the number of grains added $N$ as $\sqrt{N}$. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern.

cond-mat.stat-mech↗

arXiv2007

Emergence of quasi-units in the one dimensional Zhang model

We study the Zhang model of sandpile on a one dimensional chain of length $L$, where a random amount of energy is added at a randomly chosen site at each time step. We show that in spite of this randomness in the input energy, the probability distribution function of energy at a site in the steady state is sharply peaked, and the width of the peak decreases as $ {L}^{-1/2}$ for large $L$. We discuss how the energy added at one time is distributed among different sites by topplings with time. We relate this distribution to the time-dependent probability distribution of the position of a marked grain in the one dimensional Abelian model with discrete heights. We argue that in the large $L$ limit, the variance of energy at site $x$ has a scaling form $L^{-1}g(x/L)$, where $g(ξ)$ varies as $\log(1/ξ)$ for small $ξ$, which agrees very well with the results from numerical simulations.

cond-mat.stat-mech↗