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Deepak Dhar

Publications and source records attributed to Deepak Dhar.

At least 55 records · Page 3Linked to original sources
arXiv2013

Memoryless nonlinear response: A simple mechanism for the 1/f noise

Discovering the mechanism underlying the ubiquity of $"1/f^α"$ noise has been a long--standing problem. The wide range of systems in which the fluctuations show the implied long--time correlations suggests the existence of some simple and general mechanism that is independent of the details of any specific system. We argue here that a {\it memoryless nonlinear response} suffices to explain the observed non--trivial values of $α$: a random input noisy signal $S(t)$ with a power spectrum varying as $1/f^{α'}$, when fed to an element with such a response function $R$ gives an output $R(S(t))$ that can have a power spectrum $1/f^α$ with $α< α'$. As an illustrative example, we show that an input Brownian noise ($α'=2$) acting on a device with a sigmoidal response function $R(S)= \sgn(S)|S|^x$, with $x<1$, produces an output with $α= 3/2 +x$, for $0 \leq x \leq 1/2$. Our discussion is easily extended to more general types of input noise as well as more general response functions.

cond-mat.stat-mech↗
arXiv2013

A sandpile model for proportionate growth

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, show proportionate growth, and sometimes having 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.

cond-mat.stat-mech↗
arXiv2013

A class of exactly solved assisted hopping models of active-absorbing state transitions on a line

We construct a class of assisted hopping models in one dimension in which a particle can move only if it does not lie in an otherwise empty interval of length greater than $n+1$. We determine the exact steady state by a mapping to a gas of defects with only on-site interaction. We show that this system undergoes a phase transition as a function of the density $ρ$ of particles, from a low-density phase with all particles immobile for $ρ\le ρ_c = \frac{1}{n+1}$, to an active state for $ρ> ρ_c$. The mean fraction of movable particles in the active steady state varies as $(ρ- ρ_c)^β$, for $ρ$ near $ρ_c$. We show that for the model with range $n$, the exponent $β=n$, and thus can be made arbitrarily large.

cond-mat.stat-mech↗
arXiv2013

The nematic-disordered phase transition in systems of long rigid rods on two dimensional lattices

We study the phase transition from a nematic phase to a high-density disordered phase in systems of long rigid rods of length $k$ on the square and triangular lattices. We use an efficient Monte Carlo scheme that partly overcomes the problem of very large relaxation times of nearly jammed configurations. The existence of a continuous transition is observed on both lattices for $k=7$. We study correlations in the high-density disordered phase, and we find evidence of a crossover length scale $ξ^* \gtrsim 1400$, on the square lattice. For distances smaller than $ξ^*$, correlations appear to decay algebraically. Our best estimates of the critical exponents differ from those of the Ising model, but we cannot rule out a crossover to Ising universality class at length scales $\gg ξ^*$. On the triangular lattice, the critical exponents are consistent with those of the two dimensional three-state Potts universality class.

cond-mat.stat-mech↗
arXiv2012

High-Activity Perturbation Expansion for the Hard Square Lattice Gas

We study a system of particles with nearest and next-nearest-neighbour exclusion on the square lattice (hard squares). This system undergoes a transition from a fluid phase at low density to a columnar ordered phase at high density. We develop a systematic high-activity perturbation expansion for the free energy per site about a state with perfect columnar order. We show that the different terms of the series can be regrouped to get a Mayer-like series for a polydisperse system of interacting vertical rods in which the $n$-th term is of order $z^{-(n+1)/2}$, where $z$ is the fugacity associated with each particle. We sum this series to get the exact expansion to order $1/z^{3/2}$.

cond-mat.stat-mech↗
arXiv2012

Modelling proportionate growth

An important question in biology is how the relative size of different organs is kept nearly constant during growth of an animal. This property, called proportionate growth, has received increased attention in recent years. We discuss our recent work on a simple model where this feature comes out quite naturally from local rules, without fine tuning any parameter. The patterns produced are composed of large distinguishable structures with sharp boundaries, all of which grow at the same rate, keeping their overall shapes unchanged.

cond-mat.stat-mech↗
arXiv2012

Power spectrum of mass and activity fluctuations in a sandpile

We consider a directed abelian sandpile on a strip of size $2\times n$, driven by adding a grain randomly at the left boundary after every $T$ time-steps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeroes in the ternary-base representation of the position of a random walker on a ring of size $3^n$. We find that while the fluctuations of mass have a power spectrum that varies as $1/f$ for frequencies in the range $ 3^{-2n} \ll f \ll 1/T$, the activity fluctuations in the same frequency range have a power spectrum that is linear in $f$.

cond-mat.stat-mech↗
arXiv2012

Non-perturbative corrections to mean-field behavior: spherical model on spider-web graph

We consider the spherical model on a spider-web graph. This graph is effectively infinite-dimensional, similar to the Bethe lattice, but has loops. We show that these lead to non-trivial corrections to the simple mean-field behavior. We first determine all normal modes of the coupled springs problem on this graph, using its large symmetry group. In the thermodynamic limit, the spectrum is a set of $δ$-functions, and all the modes are localized. The fractional number of modes with frequency less than $ω$ varies as $\exp (-C/ω)$ for $ω$ tending to zero, where $C$ is a constant. For an unbiased random walk on the vertices of this graph, this implies that the probability of return to the origin at time $t$ varies as $\exp(- C' t^{1/3})$, for large $t$, where $C'$ is a constant. For the spherical model, we show that while the critical exponents take the values expected from the mean-field theory, the free-energy per site at temperature $T$, near and above the critical temperature $T_c$, also has an essential singularity of the type $\exp[ -K {(T - T_c)}^{-1/2}]$.

cond-mat.stat-mech↗
arXiv2011

Resonating valence bond wavefunctions and classical interacting dimer models

We relate properties of nearest-neighbour resonating valence bond (nnRVB) wavefunctions for $SU(g)$ spin systems on two dimensional bipartite lattices to those of fully-packed classical dimer models with potential energy $V$ on the same lattice. We define a cluster expansion of $V$ in terms of $n$-body potentials $V_n$, which are recursively determined from the nnRVB wavefunction on {\em finite subgraphs} of the original lattice. The magnitude of the $n$-body interaction $V_n$ ($n>1$) is of order ${\mathcal O}(g^{-(n-1)})$ for small $g^{-1}$, while $V_1$ reduces to a constant due to the fully-packed nature of the model. At leading non-trivial order on the square lattice, the interacting dimer model only has two-body interactions $V_2(g)$ that favour two parallel dimers on elementary plaquettes. Setting $g=2$ and using the results of earlier work on this interacting dimer model, we find that the long-distance behaviour of the bond-energy correlation function is dominated by an oscillatory term that decays as $ 1/|\vec{r}|^α$ with $α\approx 1.22$ for SU(2) spins. This result is in remarkable quantitative agreement with earlier direct numerical studies of the corresponding wavefunction, which give $α\approx 1.20$.

cond-mat.str-el↗
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

Glassy dynamics and hysteresis in a linear system of orientable hard rods

We study the dynamics of a one-dimensional fluid of orientable hard rectangles with a non-coarse-grained microscopic mechanism of facilitation. The length occupied by a rectangle depends on its orientation, which is coupled to an external field. The equilibrium properties of our model are essentially those of the Tonks gas, but at high densities, the orientational degrees of freedom become effectively frozen due to jamming. This is a simple analytically tractable model of glassy phase. Under a cyclic variation of the pressure, hysteresis is observed. Following a pressure quench, the orientational persistence exhibits a two-stage decay characteristic of glassy systems.

cond-mat.dis-nn↗
arXiv2011

Bethe approximation for a system of hard rigid rods: the random locally tree-like layered lattice

We study the Bethe approximation for a system of long rigid rods of fixed length k, with only excluded volume interaction. For large enough k, this system undergoes an isotropic-nematic phase transition as a function of density of the rods. The Bethe lattice, which is conventionally used to derive the self-consistent equations in the Bethe approximation, is not suitable for studying the hard-rods system, as it does not allow a dense packing of rods. We define a new lattice, called the random locally tree-like layered lattice, which allows a dense packing of rods, and for which the approximation is exact. We find that for a 4-coordinated lattice, k-mers with k>=4 undergo a continuous phase transition. For even coordination number q>=6, the transition exists only for k >= k_{min}(q), and is first order.

cond-mat.stat-mech↗
arXiv2011

Cooperation amongst competing agents in minority games

We study a variation of the minority game. There are N agents. Each has to choose between one of two alternatives everyday, and there is reward to each member of the smaller group. The agents cannot communicate with each other, but try to guess the choice others will make, based only the past history of number of people choosing the two alternatives. We describe a simple probabilistic strategy using which the agents acting independently, can still maximize the average number of people benefitting every day. The strategy leads to a very efficient utilization of resources, and the average deviation from the maximum possible can be made of order $(N^ε)$, for any $ε>0$. We also show that a single agent does not expect to gain by not following the strategy.

q-fin.TR↗
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

Spin-1 Kitaev model in one dimension

We study a one-dimensional version of the Kitaev model on a ring of size N, in which there is a spin S > 1/2 on each site and the Hamiltonian is J \sum_i S^x_i S^y_{i+1}. The cases where S is integer and half-odd-integer are qualitatively different. We show that there is a Z_2 valued conserved quantity W_n for each bond (n,n+1) of the system. For integer S, the Hilbert space can be decomposed into 2^N sectors, of unequal sizes. The number of states in most of the sectors grows as d^N, where d depends on the sector. The largest sector contains the ground state, and for this sector, for S=1, d =(\sqrt{5}+1)/2. We carry out exact diagonalization for small systems. The extrapolation of our results to large N indicates that the energy gap remains finite in this limit. In the ground state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term λ\sum_n W_n, and show that this has gapless excitations in the range λ^c_1 \leq λ\leq λ^c_2. We use the variational wave functions to study how the ground state energy and the defect density vary near the two critical points λ^c_1 and λ^c_2.

cond-mat.stat-mech↗
arXiv2010

Classical Heisenberg spins on a hexagonal lattice with Kitaev couplings

We analyse the low temperature properties of a system of classical Heisenberg spins on a hexagonal lattice with Kitaev couplings. For a lattice of 2N sites with periodic boundary conditions, we show that the ground states form an (N+1) dimensional manifold. We show that the ensemble of ground states is equivalent to that of a solid-on-solid model with continuously variable heights and nearest neighbour interactions, at a finite temperature. For temperature T tending to zero, all ground states have equal weight, and there is no order-by-disorder in this model. We argue that the bond-energy bond-energy correlations at distance R decay as 1/R^2 at zero temperature. This is verified by Monte Carlo simulations. We also discuss the relation to the quantum spin-S Kitaev model for large S, and obtain lower and upper bounds on the ground state energy of the quantum model.

cond-mat.stat-mech↗
arXiv2010

Restricted equilibrium ensembles: Exact equation of state of a model glass

We investigate the thermodynamic properties of a toy model of glasses: a hard-core lattice gas with nearest neighbor interaction in one dimension. The time-evolution is Markovian, with nearest-neighbor and next-nearest neighbor hoppings, and the transition rates are assumed to satisfy detailed balance condition, but the system is non-ergodic below a glass temperature. Below this temperature, the system is in restricted thermal equilibrium, where both the number of sectors, and the number of accessible states within a sector grow exponentially with the size of the system. Using partition functions that sum only over dynamically accessible states within a sector, and then taking a quenched average over the sectors, we determine the exact equation of state of this system.

cond-mat.stat-mech↗