arXiv subjects

Deepak Dhar

Publications and source records attributed to Deepak Dhar.

At least 91 records · Page 5Linked to original sources

arXiv2005

Linear and branched polymers on fractals

This is a pedagogical review of the subject of linear polymers on deterministic finitely ramified fractals. For these, one can determine the critical properties exactly by real-space renormalization group technique. We show how this is used to determine the critical exponents of self-avoiding walks on different fractals. The behavior of critical exponents for the $n$-simplex lattice in the limit of large $n$ is determined. We study self-avoiding walks when the fractal dimension of the underlying lattice is just below 2. We then consider the case of linear polymers with attractive interactions, which on some fractals leads to a collapse transition. The fractals also provide a setting where the adsorption of a linear chain near on attractive substrate surface and the zipping-unzipping transition of two mutually interacting chains can be studied analytically. We also discuss briefly the critical properties of branched polymers on fractals.

cond-mat.stat-mech↗

arXiv2005

Branched Polymers on the Given-Mandelbrot family of fractals

We study the average number A_n per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b, b > 1. The fractal dimension varies from $ log_{_2} 3$ to 2 as b is varied from 2 to infinity. We find that for all b > 2, A_n varies as $ λ^n exp(b n ^ψ)$, where $λ$ and $b$ are some constants, and $ 0 < ψ<1$. We determine the exponent $ψ$, and the size exponent $ν$ (average diameter of polymer varies as $n^ν$), exactly for all b > 2. This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev {\bf B 35} (1987) 4988].

cond-mat.stat-mech↗

arXiv2004

Probability distribution of residence-times of grains in sandpile models

We show that the probability distribution of the residence-times of sand grains in sandpile models, in the scaling limit, can be expressed in terms of the survival probability of a single diffusing particle in a medium with absorbing boundaries and space-dependent jump rates. The scaling function for the probability distribution of residence times is non-universal, and depends on the probability distribution according to which grains are added at different sites. We determine this function exactly for the 1-dimensional sandpile when grains are added randomly only at the ends. For sandpiles with grains are added everywhere with equal probability, in any dimension and of arbitrary shape, we prove that, in the scaling limit, the probability that the residence time greater than t is exp(-t/M), where M is the average mass of the pile in the steady state. We also study finite-size corrections to this function.

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arXiv2003

Steady State and Relaxation Spectrum of the Oslo Rice-pile

We show that the one-dimensional Oslo rice-pile model is a special case of the abelian distributed processors model. The exact steady state of the model is determined. We show that the time evolution operator W for the system satisfies the equation W^{n+1} = W^n where n = L(L+1)/2 for a pile with L sites. This is used to prove that W has only one eigenvalue 1 corresponding to the steady state, and all other eigenvalues are exactly zero. Also, all connected time-dependent correlation functions in the steady state of the pile are exactly zero for time difference greater that n. Generalization to other abelian critical height models where the critical thresholds are randomly reset after each toppling is briefly discussed.

cond-mat↗

arXiv2003

Convex lattice polygons of fixed area with perimeter dependent weights

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight t^m to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s^{-theta_{conv}} exp[K s^(1/2)] for large s and t less than a critical threshold t_c, where K is a t-dependent constant, and theta_{conv} is a critical exponent which does not change with t. We find theta_{conv} is 1/4 for the square lattice, but -1/4 for the hexagonal lattice. The reason for this unexpected non-universality of theta_{conv} is traced to existence of sharp corners in the asymptotic shape of these polygons.

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arXiv2003

Distribution of Transverse Distances in Directed Animals

We relate $ϕ(\bf{x},s)$, the average number of sites at a transverse distance $\bf{x}$ in the directed animals with $s$ sites in $d$ transverse dimensions, to the two-point correlation function of a lattice gas with nearest neighbor exclusion in $d$ dimensions. For large $s$, $ϕ(\bf{x},s)$ has the scaling form $\frac{s}{R_s^d} f(|\bf{x}|/R_s)$, where $R_s$ is the root mean square radius of gyration of animals of $s$ sites. We determine the exact scaling function for $d =1$ to be $f(r) = \frac{\sqrtπ}{2 \sqrt{3}}erfc(r/\sqrt{3})$. We also show that $ϕ(\bf{x}=0,s)$ can be determined in terms of the animals number generating function of the directed animals.

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arXiv2002

Generic Sandpile Models Have Directed Percolation Exponents

We study sandpile models with stochastic toppling rules and having sticky grains so that with a non-zero probability no toppling occurs, even if the local height of pile exceeds the threshold value. Dissipation is introduced by adding a small probability of particle loss at each toppling. Generically, for models with a preferred direction, the avalanche exponents are those of critical directed percolation clusters. For undirected models, avalanche exponents are those of directed percolation clusters in one higher dimension.

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arXiv2002

Pico-canonical ensembles: A theoretical description of metastable states

We define restricted ensembles, called pico-canonical ensembles, for a statistical-mechanical description of the metastable and glassy phases. In this approach, time-evolution is Markovian, with temperature dependent rates. Below a particular glass-temperature, the system is strongly non-ergodic, and the phase space breaks up into a large number of mutually disconnected sectors. Averages are calculated over states within one such sector, and then averaged over sectors. As a soluble example, we calculate these explicitly for a one dimensional lattice gas with nearest neighbor couplings.

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arXiv2002

Hysteresis in the Random Field Ising Model and Bootstrap Percolation

We study hysteresis in the random-field Ising model with an asymmetric distribution of quenched fields, in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap percolation, and show that the characteristic length for self-averaging $L^*$ increases as $exp(exp (J/Δ))$ in 2d, and as $exp(exp(exp(J/Δ)))$ in 3d, for disorder strength $Δ$ much less than the exchange coupling J. For system size $1 << L < L^*$, the coercive field $h_{coer}$ varies as $2J - Δ\ln \ln L$ for the square lattice, and as $2J - Δ\ln \ln \ln L$ on the cubic lattice. Its limiting value is 0 for L tending to infinity, both for square and cubic lattices. For lattices with coordination number 3, the limiting magnetization shows no jump, and $h_{coer}$ tends to J.

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arXiv2001

Percolation Systems away from the Critical Point

This article reviews some effects of disorder in percolation systems even away from the critical density p_c. For densities below p_c, the statistics of large clusters defines the animals problem. Its relation to the directed animals problem and the Lee-Yang edge singularity problem is described. Rare compact clusters give rise to Griffiths singuraties in the free energy of diluted ferromagnets, and lead to a very slow relaxation of magnetization. In biassed diffusion on percolation clusters, trapping in dead-end branches leads to asymptotic drift velocity becoming zero for strong bias, and very slow relaxation of velocity near the critical bias field.

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arXiv2001

The Adsorption and Collapse Transitions in a Linear Polymer Chain near an Attractive Wall

We deduce the qualitative phase diagram of a long flexible neutral polymer chain immersed in a poor solvent near an attracting surface using phenomenological arguments. The actual positions of the phase boundaries are estimated numerically from series expansion up to 19 sites of a self-attracting self avoiding walk in three dimensions. In two dimensions, we calculate analytically phase boundaries in some cases for a partially directed model. Both the numerical as well as analytical results corroborate the proposed qualitative phase diagram.

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arXiv2001

Probability distribution of the sizes of largest erased-loops in loop-erased random walks

We have studied the probability distribution of the perimeter and the area of the k-th largest erased-loop in loop-erased random walks in two-dimensions for k = 1 to 3. For a random walk of N steps, for large N, the average value of the k-th largest perimeter and area scales as N^{5/8} and N respectively. The behavior of the scaled distribution functions is determined for very large and very small arguments. We have used exact enumeration for N <= 20 to determine the probability that no loop of size greater than l (ell) is erased. We show that correlations between loops have to be taken into account to describe the average size of the k-th largest erased-loops. We propose a one-dimensional Levy walk model which takes care of these correlations. The simulations of this simpler model compare very well with the simulations of the original problem.

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arXiv2001

Persistence in a Stationary Time-series

We study the persistence in a class of continuous stochastic processes that are stationary only under integer shifts of time. We show that under certain conditions, the persistence of such a continuous process reduces to the persistence of a corresponding discrete sequence obtained from the measurement of the process only at integer times. We then construct a specific sequence for which the persistence can be computed even though the sequence is non-Markovian. We show that this may be considered as a limiting case of persistence in the diffusion process on a hierarchical lattice.

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arXiv2001

Distribution of Avalanche Sizes in the Hysteretic Response of Random Field Ising Model on a Bethe Lattice at Zero Temperature

We consider the zero-temperature single-spin-flip dynamics of the random-field Ising model on a Bethe lattice in the presence of an external field h. We derive the exact self-consistent equations to determine the distribution Prob(s) of avalanche sizes s, as the external field increases from large negative to positive values. We solve these equations explicitly for a rectangular distribution of the random fields for a linear chain and the Bethe lattice of coordination number z=3, and show that in these cases, Prob(s) decreases exponentially with s for large s for all h on the hysteresis loop. We found that for z >3 and for small disorder, the magnetization shows a first order discontinuity for several continuous and unimodel distributions of random fields. The avalanche distribution Prob(s) varies as s^{-3/2} for large s near the discontinuity.

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arXiv2001

An Exactly Solvable Anisotropic Directed Percolation Model in Three Dimensions

We solve exactly a special case of the anisotropic directed bond percolation problem in three dimensions, in which the occupation probability is 1 along two spatial directions, by mapping it to a five-vertex model. We determine the asymptotic shape of the ininite cluster and hence the direction dependent critical probability. The exponents characterising the fluctuations of the boundary of the wetted cluster in d-dimensions are related to those of the (d-2)-dimensional KPZ equation.

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arXiv2000

Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions

We show that in the loop-erased random walk problem, the exponent characterizing probability distribution of areas of erased loops is superuniversal. In d-dimensions, the probability that the erased loop has an area A varies as A^{-2} for large A, independent of d, for 2 <= d <= 4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d = 2 and 3 by large-scale Monte Carlo simulations. Our estimate of the fractal dimension z in two-dimensions is consistent with the known exact value 5/4. In three-dimensions, we get z = 1.6183 +- 0.0004. The exponent for the distribution of durations of avalanche in the three-dimensional abelian sandpile model is determined from this by using scaling relations.

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arXiv1999

Studying Self-Organized Criticality with Exactly Solved Models

This is a somewhat expanded version of the notes of a series of lectures given at Lausanne and Stellenbosch in 1998-99. They are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models : the q=0 state Potts model, Takayasu aggregation model, the voter model, spanning trees, Eulerian walkers model etc. It provides an overview of the known results, and explains the equivalence of these models. Some open questions are discussed in the concluding section.

cond-mat.stat-mech↗

arXiv1999

Some Results and a Conjecture for Manna's Stochastic Sandpile Model

We present some analytical results for the stochastic sandpile model, studied earlier by Manna. In this model, the operators corresponding to particle addition at different sites commute. The eigenvalues of operators satisfy a system of coupled polynomial equations. For an L X L square, we construct a nontrivial toppling invariant, and hence a ladder operator which acting on eigenvectors of evolution operator gives new eigenvectors with different eigenvalues. For periodic boundary conditions in one direction, one more toppling invariant can be constructed. We show that there are many forbidden subconfigurations, and only an exponentially small fraction of all stable configurations are recurrent. We obtain rigorous lower and upper bounds for the minimum number of particles in a recurrent configuration, and conjecture a formula for its exact value for finite-size rectangles.

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