Tridib Sadhu

Large deviations in single-file diffusion.

We apply macroscopic fluctuation theory to study the diffusion of a tracer in a one-dimensional interacting particle system with excluded mutual passage, known as single-file diffusion. In the case of Brownian point particles with hard-core repulsion, we derive the cumulant generating function of the tracer position and its large deviation function. In the general case of arbitrary interparticle interactions, we express the variance of the tracer position in terms of the collective transport properties, viz., the diffusion coefficient and the mobility. Our analysis applies both for fluctuating (annealed) and fixed (quenched) initial configurations.

Pattern formation in growing sandpiles

Adding grains at a single site on a flat substrate in the Abelian sandpile models produces 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 model shows proportionate growth and the size of the pattern formed by adding N grains scales as . We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern.

Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks

AbstractAdding sand grains at a single site in the 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 the presence of a sink site, the diameter of the pattern grows as

Pattern formation in fast-growing sandpiles

\sqrt{(N/\log N)}

A sandpile model for proportionate growth

for large N, whereas it grows as

Steady State of Stochastic Sandpile Models

\sqrt{N}

Large deviation function of a tracer position in single file diffusion

if there are no sink sites. In the presence of a line of sink sites, this rate reduces to N1/3. We determine the growth rates for various sink geometries along with the case when there are two lines of sink sites forming a wedge, and generalizations to higher dimensions. We characterize the asymptotic pattern in the large N limit for one such case, the two-dimensional F-lattice with a single source adjacent to a line of sink sites. The characterization is done in terms of the positions 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 of the lattice.

Dynamical properties of single-file diffusion

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≤1. Different values of α can be realized on different backgrounds, and the patterns still show proportionate growth. The noncompact nature of growth simplifies their analysis significantly. We characterize the asymptotic pattern exactly for one illustrative example with α=1.

Melting of an Ising quadrant

An interesting feature of growth in animals is that different parts of the body grow at approximately the same rate. This property is called proportionate growth. In this paper, we review our recent work on patterns formed by adding N grains at a single site in the Abelian sandpile model. These simple models show very intricate patterns which show proportionate growth, and sometimes have a striking resemblance to natural forms. We give several examples of such patterns. We discuss the special cases where the asymptotic pattern can be determined exactly. The effect of noise in the background or in the rules on the patterns is also discussed.

Long-range correlations in a locally driven exclusion process.

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 ≤12, and also study the density profile in the steady state.