S. B. Santra

Fluid-induced particle-size segregation in sheared granular assemblies

We perform a two-dimensional molecular-dynamics study of a model for sheared bidisperse granular systems under conditions of simple shear and Poiseuille flow. We propose a mechanism for particle-size segregation based on the observation that segregation occurs if the viscous length scale introduced by a liquid in the system is smaller than of the order of the particle size. We show that the ratio of shear rate to viscosity must be small if one wants to find size segregation. In this case the particles in the system arrange themselves in bands of big and small particles oriented along the direction of the flow. Similarly, in Poiseuille flow we find the formation of particle bands. Here, in addition, the variety of time scales in the flow leads to an aggregation of particles in the zones of low shear rate and can suppress size segregation in these regions. The results have been verified against simulations using a full Navier-Stokes description for the liquid.

Breakdown of universality in directed spiral percolation

Abstract.Directed spiral percolation (DSP), percolation under both directional and rotational constraints, is studied on the triangular lattice in two dimensions (2D). The results are compared with that of the 2D square lattice. Clusters generated in this model are generally rarefied and have chiral dangling ends on both the square and triangular lattices. It is found that the clusters are more compact and less anisotropic on the triangular lattice than on the square lattice. The elongation of the clusters is in a different direction than the imposed directional constraint on both the lattices. The values of some of the critical exponents and fractal dimension are found considerably different on the two lattices. The DSP model then exhibits a breakdown of universality in 2D between the square and triangular lattices. The values of the critical exponents obtained for the triangular lattice are not only different from that of the square lattice but also different form other percolation models.

Critical fluctuations and self-organized fractality in chemical reactions: Spontaneous gradient percolation in the etching of random solids

A simple model of the chemical attack of random solids by etching solutions of finite volumes has been studied in two dimensions. The etching species are consumed in the chemical reaction and the etchant strength at any time is calculated from the varying concentration of the solution. The occupation probability of solid surface sites by the etching liquid varies in time and space along with the evolution of the process. There is then a spontaneously established gradient of the chemical potential at the moving interface. The criticality of the system has been identified by the maximum etching speed and extremal fluctuations of the length of the liquid–solid interface. This occurs when the etching strength corresponds to the percolation threshold. This is the first example of the existence of critical fluctuations in the framework of chemical reactions. The results suggest that the system belongs to the same universality class as gradient percolation. The fact that the rate of interface production is apparently maximum at criticality suggests the possible existence of an underlying variational principle linking extremal surfaces to percolation.

Critical properties of a dissipative sandpile model on small-world networks.

A dissipative sandpile model is constructed and studied on small-world networks (SWNs). SWNs are generated by adding extra links between two arbitrary sites of a two-dimensional square lattice with different shortcut densities ϕ. Three regimes are identified: regular lattice (RL) for ϕ≲2(-12), SWN for 2(-12)<ϕ<0.1, and random network (RN) for ϕ≥0.1. In the RL regime, the sandpile dynamics is characterized by the usual Bak, Tang, and Weisenfeld (BTW)-type correlated scaling, whereas in the RN regime it is characterized by mean-field scaling. On SWNs, both scaling behaviors are found to coexist. Small compact avalanches below a certain characteristic size s(c) are found to belong to the BTW universality class, whereas large, sparse avalanches above s(c) are found to belong to the mean-field universality class. A scaling theory for the coexistence of two scaling forms on a SWN is developed and numerically verified. Though finite-size scaling is not valid for the dissipative sandpile model on RLs or on SWNs, it is found to be valid on RNs for the same model. Finite-size scaling on RNs appears to be an outcome of super diffusive sand transport and uncorrelated toppling waves.

Directed spiral site percolation on the square lattice

Abstract:A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold pc ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension df ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → pc with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution Ps(p) show power law behaviour with | p - pc| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of Ps(p). The results obtained are in good agreement with other model calculations.

Dissipative stochastic sandpile model on small-world networks: Properties of nondissipative and dissipative avalanches

A dissipative stochastic sandpile model is constructed and studied on small-world networks in one and two dimensions with different shortcut densities ϕ, where ϕ=0 represents regular lattice and ϕ=1 represents random network. The effect of dimension, network topology, and specific dissipation mode (bulk or boundary) on the the steady-state critical properties of nondissipative and dissipative avalanches along with all avalanches are analyzed. Though the distributions of all avalanches and nondissipative avalanches display stochastic scaling at ϕ=0 and mean-field scaling at ϕ=1, the dissipative avalanches display nontrivial critical properties at ϕ=0 and 1 in both one and two dimensions. In the small-world regime (2^{-12}≤ϕ≤0.1), the size distributions of different types of avalanches are found to exhibit more than one power-law scaling with different scaling exponents around a crossover toppling size s_{c}. Stochastic scaling is found to occur for ss_{c}. As different scaling forms are found to coexist in a single probability distribution, a coexistence scaling theory on small world network is developed and numerically verified.

Percolation under rotational constraint: a finite-size scaling study

A percolation model, spiral percolation, in which a rotational constraint is operative is studied by the finite-size scaling method. The critical percolation probability pc and the critical exponents v, beta , gamma , tau , sigma and also the fractal dimension D of the spanning cluster are determined. Evidence is obtained for a scaling form of the cluster distribution function.

STUDY OF FINITE SIZE EFFECTS ON DIRECTED SPIRAL PERCOLATION

Percolation under both directional and rotational constraints is studied numerically on the square lattice of different finite sizes L. The critical percolation threshold pc≈0.655 of the infinite network is determined by extrapolating the finite size data. The fractal dimension df of the infinite percolation clusters is found df≈1.72 from the finite size scaling, S∞~Ldf where S∞ is the mass of the infinite cluster. The critical exponents are estimated as a function of the system size L. It is seen that the results of smaller systems converge to that of the large systems. The results are then extrapolated to the infinite network. The extrapolated results for the infinite network are compared with Monte Carlo results on a single large lattice. A good agreement is found.

A simple model for gelation under shear

A simple model for gelation of monomers in a solution under shear has been studied at different concentrations in two dimensions. Different properties of the clusters like the length R and width W of the clusters, and the angle θ formed with the negative x-direction are determined as a function of time t. The cluster size distribution ns(t) is determined as a function of the cluster size s and time t.

Size distribution of different types of sites in Abelian sandpile avalanches

The avalanches in the Abelian sandpile model have a compact structure and consist of shells. The different shells consist solely of sites that have toppled a fixed number of times. For the outermost shell the number is one and increases successively by one from an outer shell to the next inner shell. In this paper, we determine the size-distribution exponents of various categories of sites: sites that have toppled at least once, at least twice, at least thrice, only once and only twice. A particular subclass of avalanches is identified which have no shell structure and consist solely of sites which have toppled only once. The size distribution exponent of this type of clusters is determined.