arXiv subjects

Kabir Ramola

Publications and source records attributed to Kabir Ramola.

52 records · Page 3Linked to original sources

arXiv2018

Numerical test of the Edwards conjecture shows that all packings are equally probable at jamming

In the late 1980s, Sam Edwards proposed a possible statistical-mechanical framework to describe the properties of disordered granular materials. A key assumption underlying the theory was that all jammed packings are equally likely. In the intervening years it has never been possible to test this bold hypothesis directly. Here we present simulations that provide direct evidence that at the unjamming point, all packings of soft repulsive particles are equally likely, even though generically, jammed packings are not. Typically, jammed granular systems are observed precisely at the unjamming point since grains are not very compressible. Our results therefore support Edwards' original conjecture. We also present evidence that at unjamming the configurational entropy of the system is maximal.

cond-mat.soft↗

arXiv2017

Stress Response of Granular Systems

We develop a framework for stress response in two dimensional granular media, with and without friction, that respects vector force balance at the microscopic level. We introduce local gauge degrees of freedom that determine the response of contact forces between constituent grains on a given, disordered, contact network, to external perturbations. By mapping this response to the spectral properties of the graph Laplacian corresponding to the underlying contact network, we show that this naturally leads to spatial localization of forces. We present numerical evidence for localization using exact diagonalization studies of network Laplacians of soft disk packings. Finally, we discuss the role of other constraints, such as torque balance, in determining the stability of a granular packing to external perturbations.

cond-mat.dis-nn↗

arXiv2017

Gaps between avalanches in 1D Random Field Ising Models

We analyze the statistics of gaps ($ΔH$) between successive avalanches in one dimensional random field Ising models (RFIMs) in an external field $H$ at zero temperature. In the first part of the paper we study the nearest-neighbour ferromagnetic RFIM. We map the sequence of avalanches in this system to a non-homogeneous Poisson process with an $H$-dependent rate $ρ(H)$. We use this to analytically compute the distribution of gaps $P(ΔH)$ between avalanches as the field is increased monotonically from $-\infty$ to $+\infty$. We show that $P(ΔH)$ tends to a constant $\mathcal{C}(R)$ as $ΔH \to 0^+$, which displays a non-trivial behaviour with the strength of disorder $R$. We verify our predictions with numerical simulations. In the second part of the paper, motivated by avalanche gap distributions in driven disordered amorphous solids, we study a long-range antiferromagnetic RFIM. This model displays a gapped behaviour $P(ΔH) = 0$ up to a system size dependent offset value $ΔH_{\text{off}}$, and $P(ΔH) \sim (ΔH - ΔH_{\text{off}})^θ$ as $ΔH \to H_{\text{off}}^+$. We perform numerical simulations on this model and determine $θ\approx 0.95(5)$. We also discuss mechanisms which would lead to a non-zero exponent $θ$ for general spin models with quenched random fields.

cond-mat.dis-nn↗

arXiv2017

Scaling Theory for the Frictionless Unjamming Transition

We develop a scaling theory of the unjamming transition of soft frictionless disks in two dimensions by defining local areas, which can be uniquely assigned to each contact. These serve to define local order parameters, whose distribution exhibits divergences as the unjamming transition is approached. We derive scaling forms for these divergences from a mean-field approach that treats the local areas as non interacting entities, and demonstrate that these results agree remarkably well with numerical simulations. We find that the asymptotic behaviour of the scaling functions arises from the geometrical structure of the packing while the overall scaling with the compression energy depends on the force law. We use the scaling forms of the distributions to determine the scaling of the total grain area $A_G$, and the total number of contacts $N_C$.

cond-mat.dis-nn↗

arXiv2017

Disordered Contact Networks in Jammed Packings of Frictionless Disks

We analyse properties of contact networks formed in packings of soft frictionless disks near the unjamming transition. We construct polygonal tilings and triangulations of the contact network that partitions space into convex regions which are either covered or uncovered. This allows us to characterize the local spatial structure of the packing near the transition using well-defined geometric objects. We construct bounds on the number of polygons and triangulation vectors that appear in such packings. We study these networks using simulations of bidispersed disks interacting via a one-sided linear spring potential. We find that several underlying geometric distributions are reproducible and display self averaging properties. We find that the total covered area is a reliable real space parameter that can serve as a substitute for the packing fraction. We find that the unjamming transition occurs at a fraction of covered area $A_G^{*} = 0.446(1)$. We determine scaling exponents of the excess covered area as the energy of the system approaches zero $E_G \to 0^+$, and the coordination number $\langle z_g \rangle$ approaches its isostatic value $ΔZ = \langle z_g \rangle - \langle z_g \rangle_{\rm iso} \to 0^{+}$. We find $ΔA_G \sim Δ{E_G}^{0.28(1)}$ and $ΔA_G \sim ΔZ^{1.00(1)}$, representing new structural critical exponents. We use the distribution functions of local areas to study the underlying geometric disorder in the packings. We find that a finite fraction of order $Ψ_O^* = 0.369(1)$ persists as the transition is approached.

cond-mat.dis-nn↗

arXiv2017

Shear-induced organization of forces in dense suspensions: signatures of discontinuous shear thickening

Dense suspensions can exhibit an abrupt change in their viscosity in response to increasing shear rate. The origin of this discontinuous shear thickening (DST) has been ascribed to the transformation of lubricated contacts to frictional, particle-on-particle contacts. Recent research on the flowing and jamming behavior of dense suspensions has explored the intersection of ideas from granular physics and Stokesian fluid dynamics to better understand this transition from lubricated to frictional rheology. DST is reminiscent of classical phase transitions, and a key question is how interactions between the microscopic constituents give rise to a macroscopic transition. In this paper, we extend a formalism that has proven to be successful in understanding shear jamming of dry grains to dense suspensions. Quantitative analysis of the collective evolution of the contact-force network accompanying the DST transition demonstrates clear changes in the distribution of microscopic variables, and leads to the identification of an "order parameter" characterizing DST.

cond-mat.soft↗

arXiv2015

Columnar order and Ashkin-Teller criticality in mixtures of hard-squares and dimers

We show that critical exponents of the transition to columnar order in a {\em mixture} of $2 \times 1$ dimers and $2 \times 2$ hard-squares on the square lattice {\em depends on the composition of the mixture} in exactly the manner predicted by the theory of Ashkin-Teller criticality, including in the hard-square limit. This result settles the question regarding the nature of the transition in the hard-square lattice gas. It also provides the first example of a polydisperse system whose critical properties depend on composition. Our ideas also lead to some interesting predictions for a class of frustrated quantum magnets that exhibit columnar ordering of the bond-energies at low temperature.

cond-mat.stat-mech↗

arXiv2015

Spatial Extent of Branching Brownian Motion

We study the one dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost ($X_{\max}\geq 0$) and leftmost ($X_{\min} \leq 0$) visited sites up to time $t$. At each time step the existing particles in the system either diffuse (with diffusion constant $D$), die (with rate $a$) or split into two particles (with rate $b$). We focus on the regime $b \leq a$ where these two extreme values $X_{\max}$ and $X_{\min}$ are strongly correlated. We show that at large time $t$, the joint probability distribution function (PDF) of the two extreme points becomes stationary $P(X,Y,t \to \infty) \to p(X,Y)$. Our exact results for $p(X,Y)$ demonstrate that the correlation between $X_{\max}$ and $X_{\min}$ is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF $p(ζ)$ of the (dimensionless) span $ζ= {(X_{\max} - X_{\min})}/{\sqrt{D/b}}$, which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior ${p}(ζ) \sim \frac{1}{2} \left(1 + Δ\right) ζ$ for small spans, with $Δ= \left(\frac{a}{b} -1\right)$. In the critical case ($Δ= 0$) this distribution has a non-trivial power law tail ${p}(ζ) \sim 8 π\sqrt{3} /ζ^3$ for large spans. On the other hand, in the subcritical case ($Δ> 0$), we show that the span distribution decays exponentially as ${p}(ζ) \sim (A^2/2) ζ\exp \left(- \sqrtΔ~ζ\right)$ for large spans, where $A$ is a non-trivial function of $Δ$ which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between $X_{\max}$ and $X_{\min}$. Finally we verify our results via direct Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2014

Fluctuations of random matrix products and 1D Dirac equation with random mass

We study the fluctuations of certain random matrix products $Π_N=M_N\cdots M_2M_1$ of $\mathrm{SL}(2,\mathbb{R})$, describing localisation properties of the one-dimensional Dirac equation with random mass. In the continuum limit, i.e. when matrices $M_n$'s are close to the identity matrix, we obtain convenient integral representations for the variance $Γ_2=\lim_{N\to\infty}\mathrm{Var}(\ln||Π_N||)/N$. The case studied exhibits a saturation of the variance at low energy $\varepsilon$ along with a vanishing Lyapunov exponent $Γ_1=\lim_{N\to\infty}\ln||Π_N||/N$, leading to the behaviour $Γ_2/Γ_1\sim\ln(1/|\varepsilon|)\to\infty$ as $\varepsilon\to0$. Our continuum description sheds new light on the Kappus-Wegner (band center) anomaly.

cond-mat.dis-nn↗

arXiv2014

Branching Brownian Motion Conditioned on Particle Numbers

We study analytically the order and gap statistics of particles at time $t$ for the one dimensional branching Brownian motion, conditioned to have a fixed number of particles at $t$. The dynamics of the process proceeds in continuous time where at each time step, every particle in the system either diffuses (with diffusion constant $D$), dies (with rate $d$) or splits into two independent particles (with rate $b$). We derive exact results for the probability distribution function of $g_k(t) = x_k(t) - x_{k+1}(t)$, the distance between successive particles, conditioned on the event that there are exactly $n$ particles in the system at a given time $t$. We show that at large times these conditional distributions become stationary $P(g_k, t \to \infty|n) = p(g_k|n)$. We show that they are characterised by an exponential tail $p(g_k|n) \sim \exp[-\sqrt{\frac{|b - d|}{2 D}} ~g_k]$ for large gaps in the subcritical ($b < d$) and supercritical ($b > d$) phases, and a power law tail $p(g_k) \sim 8\left(\frac{D}{b}\right){g_k}^{-3}$ at the critical point ($b = d$), independently of $n$ and $k$. Some of these results for the critical case were announced in a recent letter [K. Ramola, S. N. Majumdar and G. Schehr, Phys. Rev. Lett. 112, 210602 (2014)].

cond-mat.stat-mech↗

arXiv2014

Universal Order and Gap Statistics of Critical Branching Brownian Motion

We study the order statistics of one dimensional branching Brownian motion in which particles either diffuse (with diffusion constant $D$), die (with rate $d$) or split into two particles (with rate $b$). At the critical point $b=d$ which we focus on, we show that, at large time $t$, the particles are collectively bunched together. We find indeed that there are two length scales in the system: (i) the diffusive length scale $\sim \sqrt{Dt}$ which controls the collective fluctuations of the whole bunch and (ii) the length scale of the gap between the bunched particles $\sim \sqrt{D/b}$. We compute the probability distribution function $P(g_k,t|n)$ of the $k$th gap $g_k = x_k - x_{k+1}$ between the $k$th and $(k+1)$th particles given that the system contains exactly $n>k$ particles at time $t$. We show that at large $t$, it converges to a stationary distribution $P(g_k,t\to \infty|n) = p(g_k|n)$ with an algebraic tail $p(g_k|n) \sim 8(D/b) g_k^{-3}$, for $g_k \gg 1$, independent of $k$ and $n$. We verify our predictions with Monte Carlo simulations.

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↗

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↗

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↗

arXiv2008

Fragment Formation in Biased Random Walks

We analyse a biased random walk on a 1D lattice with unequal step lengths. Such a walk was recently shown to undergo a phase transition from a state containing a single connected cluster of visited sites to one with several clusters of visited sites (fragments) separated by unvisited sites at a critical probability p_c, [PRL 99, 180602 (2007)]. The behaviour of rho(l), the probability of formation of fragments of length l is analysed. An exact expression for the generating function of rho(l) at the critical point is derived. We prove that the asymptotic behaviour is of the form rho(l) ~ 3/[l(log l)^2].

cond-mat.stat-mech↗