Deepak Dhar

The Abelian sandpile and related models

The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality. This paper presents a brief review of known results about the model. The Abelian group structure of the algebra of operators allows an exact calculation of many of its properties. In particular, when there is a preferred direction, one can calculate all the critical exponents characterizing the distribution of avalanche-sizes in all dimensions. For the undirected case, the model is related to q → 0 state Potts model. This enables exact calculation of some exponents in two dimensions, and there are some conjectures about others. We also discuss a generalization of the model to a network of communicating reactive processors. This includes sandpile models with stochastic toppling rules as a special case. We also consider a nondashAbelian stochastic variant, which lies in a different universality class, related to directed percolation.

Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model

We establish an equivalence between the undirected Abelian sandpile model and the q→0 limit of the q-state Potts model. The equivalence is valid for arbitrary finite graphs. Two-dimensional Abelian sandpile models, thus, correspond to a conformal field theory with central charge c = −2. The equivalence also gives a Monte Carlo algorithm to generate random spanning trees. We study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the Abelian sandpile model. The average number of sites burnt upto time t varies at ta. In two dimensions our numerically determined value of a agrees with the theoretical prediction a = 85. We relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes.

Eulerian Walkers as a Model of Self-Organized Criticality

We propose a new model of self-organized criticality. A particle is dropped at random on a lattice and moves along directions specified by arrows at each site. As it moves, it changes the direction of the arrows according to fixed rules. On closed graphs these walks generate Euler circuits. On open graphs, the particle eventually leaves the system, and a new particle is then added. The operators corresponding to particle addition generate an Abelian group, same as the group for the Abelian sandpile model on the graph. We determine the critical steady state and some critical exponents exactly, using this equivalence.

Algebraic aspects of Abelian sandpile models

The Abelian sandpile models feature a finite Abelian group G generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of G as a product of cyclic groups G=Z(d1)*Z(d2)*Z(d3)...*Z(dg), where g is the least number of generators of G, and di is a multiple of di+1. The structure of G is determined in terms of the toppling matrix Delta . We construct scalar functions, linear in the height variables of the pile, that are invariant under toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an L*L square lattice, we show that g=L. In this case, we observe that the system has non-trivial symmetries, transcending the obvious symmetries of the square, namely those coming from the action of the cyclotomic Galois group GalL of the 2(L+1)th roots of unity (which operates on the set of eigenvalues of h). These eigenvalues are algebraic integers, the product of which is the order mod G mod . With the help of GalL we are able to group the eigenvalues into certain subsets the products of which are separately integers, and thus obtain an explicit factorization of mod G mod . We also use GalL to define other simpler sets of toppling invariants.

Lattices of effectively nonintegral dimensionality

We construct a class of lattice systems that have effectively nonintegral dimensionality. A reasonable definition of effective dimensionality applicable to lattice systems is proposed and the effective dimensionalities of these lattices are determined. The renormalization procedure is used to determine the critical behavior of the classical XY model and the Fortuin–Kasteleyn cluster model on the truncated tetrahedron lattice which is shown to have the effective dimensionality 2 log3 /log5. It is found that no phase transition occurs at any finite temperature.

Abelian sandpile model on the Bethe lattice

The authors study Bak, Tang and Wiesenfelds Abelian sandpile model (1987) of self-organised criticality on the Bethe lattice. Exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained. They also determine the probability distribution of the number of distinct sites that topple at least once, the number of toplings at the origin and the total number of toplings in an avalanche. The probability that an avalanche consists of more than n toplings varies as n-1/2 for large n. The probability that its duration exceeds T decreases as 1/T for large T. These exponents are the same as for the critical percolation clusters in mean field theory.

Height correlations in the Abelian sandpile model

The authors study the distribution of heights in the self-organized critical state of the Abelian sandpile model on a d-dimensional hypercubic lattice. They calculate analytically the concentration of sites having minimum allowed value in the critical state. They also calculate, in the critical state, the probability that the heights, at two sites separated by a distance r, would both have minimum values and show that the lowest-order r-dependent term in it varies as r-2d for large r.

Continuous dissolution of structure during the unfolding of a small protein

The unfolding kinetics of many small proteins appears to be first order, when measured by ensemble-averaging probes such as fluorescence and circular dichroism. For one such protein, monellin, it is shown here that hidden behind this deceptive simplicity is a complexity that becomes evident with the use of experimental probes that are able to discriminate between different conformations in an ensemble of structures. In this study, the unfolding of monellin has been probed by measurement of the changes in the distributions of 4 different intramolecular distances, using a multisite, time-resolved fluorescence resonance energy transfer methodology. During the course of unfolding, the protein molecules are seen to undergo slow and continuous, diffusive swelling. The swelling process can be modeled as the slow diffusive swelling of a Rouse-like chain with some additional noncovalent, intramolecular interactions. Here, we show that specific structure is lost during the swelling process gradually, and not in an all-or-none manner, during unfolding.

Electric field of a six-needle array electrode used in drug and DNA delivery in vivo: analytical versus numerical solution

We present an analytical solution for the electrical potential and field established by a six-needle array electroporation electrode, which is used in vivo for cancer treatment and DNA delivery. The analytical solution closely matches the numerical solution obtained with the finite element method: the mean error is less than 0.6 %.

Self-avoiding random walks: some exactly soluble cases

We use the exact renormalization group equations to determine the asymptotic behavior of long self‐avoiding random walks on some pseudolattices. The lattices considered are the truncated 3‐simplex, the truncated 4‐simplex, and the modified rectangular lattices. The total number of random walks C_n, the number of polygons P_n of perimeter n, and the mean square end to end distance 〈R^2_n〉 are assumed to be asymptotically proportional to μ^nn^(γ−1), μ^nn^(α−3), and n^(2ν) respectively for large n, where n is the total length of the walk. The exact values of the connectivity constant μ, and the critical exponents λ, α, ν are determined for the three lattices. We give an example of two lattice systems that have the same effective nonintegral dimensionality 3/2 but different values of the critical exponents γ, α, and ν.