arXiv subjects

Kabir Ramola

Publications and source records attributed to Kabir Ramola.

At least 37 records · Page 2Linked to original sources

arXiv2023

Green's Functions For Random Resistor Networks

We analyze random resistor networks through a study of lattice Green's functions in arbitrary dimensions. We develop a systematic disorder perturbation expansion to describe the weak disorder regime of such a system. We use this formulation to compute ensemble averaged nodal voltages and bond currents in a hierarchical fashion. We verify the validity of this expansion with direct numerical simulations of a square lattice with resistances at each bond exponentially distributed. Additionally, we construct a formalism to recursively obtain the exact Green's functions for finitely many disordered bonds. We provide explicit expressions for lattices with up to four disordered bonds, which can be used to predict nodal voltage distributions for arbitrarily large disorder strengths. Finally, we introduce a novel order parameter that measures the overlap between the bond current and the optimal path (the path of least resistance), for a given resistance configuration, which helps to characterize the weak and strong disorder regimes of the system.

cond-mat.dis-nn↗

arXiv2022

First contact breaking distributions in strained disordered crystals

We derive exact probability distributions for the strain ($ε$) at which the first stress drop event occurs in uniformly strained disordered crystals, with quenched disorder introduced through polydispersity in particle sizes. We characterize these first stress drop events numerically as well as theoretically, and identify them with the first contact breaking event in the system. Our theoretical results are corroborated with numerical simulations of quasistatic volumetric strain applied to disordered near-crystalline configurations of athermal soft particles. We develop a general technique to determine the $distribution$ of strains at which the first stress drop events occur, through an exact mapping between the cumulative distribution of first contact breaking events and the volume of a convex polytope whose dimension is determined by the number of defects $N_d$ in the system. An exact numerical computation of this polytope volume for systems with small numbers of defects displays a remarkable match with the distribution of strains generated through direct numerical simulations. Finally, we derive the distribution of strains at which the first stress drop occurs, assuming that individual contact breaking events are uncorrelated, which accurately reproduces distributions obtained from direct numerical simulations.

cond-mat.soft↗

arXiv2022

Active Random Walks in One and Two Dimensions

We investigate active lattice walks: biased continuous time random walks which perform orientational diffusion between lattice directions in one and two spatial dimensions. We study the occupation probability of an arbitrary site on the lattice in one and two dimensions, and derive exact results in the continuum limit. Next, we compute the large deviation free energy function in both one and two dimensions, which we use to compute the moments and the cumulants of the displacements exactly at late times. Our exact results demonstrate that the cross-correlations between the motion in the $x$ and $y$ directions in two dimensions persist in the large deviation function. We also demonstrate that the large deviation function of an active particle with diffusion displays two regimes, with differing diffusive behaviors. We verify our analytic results with kinetic Monte Carlo simulations of an active lattice walker in one and two dimensions.

cond-mat.stat-mech↗

arXiv2022

Tensor Electromagnetism and Emergent Elasticity in Jammed Solids

The theory of mechanical response and stress transmission in disordered, jammed solids poses several open questions of how non-periodic networks -- apparently indistinguishable from a snapshot of a fluid -- sustain shear. We present a stress-only theory of emergent elasticity for a non-thermal amorphous assembly of grains in a jammed solid, where each grain is subjected to mechanical constraints of force and torque balance. These grain-level constraints lead to the Gauss's law of an emergent $U(1)$ tensor electromagnetism, which then accounts for the mechanical response of such solids. This formulation of amorphous elasticity has several immediate consequences. The mechanical response maps exactly to the static, dielectric response of this tensorial electromagnetism with the polarizability of the medium mapping to emergent elastic moduli. External forces act as vector electric charges whereas the tensorial magnetic fields are sourced by momentum density. The dynamics in the electric and magnetic sectors, naturally translate into the dynamics of the rigid jammed network and ballistic particle motion respectively. The theoretical predictions for both stress-stress correlations and responses are borne out by the results of numerical simulations of frictionless granular packings in the static limit of the theory in both 2D and 3D.

cond-mat.soft↗

arXiv2021

Annealing effects of multidirectional oscillatory shear in model glass formers

We study the effects of cyclic, athermal quasi-static shear on a model glass-forming system in three dimensions. We utilize the three available orthogonal shear planes, namely $XY, YZ \text{ and } XZ$ to better explore the energy landscape. Using measurements of the stroboscopic $(γ= 0)$ energy, we study the effects of using an orthogonal shear direction to perturb unidirectional steady-states. We find that that each sequence of the unidirectional protocol leads to compaction with the universal, $ΔE \sim N^{-1}$ behavior as a function of the number of cycles, $N$. Additionally we find that cyclic shear utilizing multiple shear planes presents hierarchical compaction, producing progressively lower steady state energies compared to a protocol involving unidirectional cyclic shear alone. Furthermore, with the periodicity of the stroboscopic energy as reference, we show that it is possible to achieve steady state limit-cycles of tunable periodicities using different combinations of the three orthogonal strain directions. We find that such protocols exhibit better annealing as compared to protocols with steady states created using unidirectional shear. Importantly, we find a non-trivial trend in the annealing energy and the period of the steady-state limit-cycle, with an aperiodic protocol appearing to produce the most well annealed states. Finally, we compare the phase diagram of the average steady state energy $\langle E_{\text{S.S.}} \rangle$, as a function of the shearing amplitude $γ_{\max}$, using unidirectional and multidirectional protocols, and find that the universal features are preserved.

cond-mat.dis-nn↗

arXiv2021

Universal non-Debye low-frequency vibrations in sheared amorphous solids

We study energy minimized configurations of amorphous solids with a simple shear degree of freedom. We show that the low-frequency regime of the vibrational density of states of structural glass formers is crucially sensitive to the stress-ensemble from which the configurations are sampled. In both two and three dimensions, a shear-stabilized ensemble displays a $D(ω_{\min}) \sim ω^{5}_{\min}$ regime, as opposed to the $ω^{4}_{\min}$ regime observed under unstrained conditions. We also study an ensemble of two dimensional, strained amorphous solids near a plastic event. We show that the minimum eigenvalue distribution at a strain $γ$ near the plastic event occurring at $γ_{P}$, displays a collapse when scaled by $\sqrt{γ_P - γ}$, and with the number of particles as $N^{-0.22}$. Notably, at low-frequencies, this scaled distribution displays a robust $D(ω_{\min}) \sim ω^{6}_{\min}$ power-law regime, which survives in the large $N$ limit. Finally, we probe the universal properties of this ensemble through a characterization of the second and third eigenvalues of the Hessian matrix near a plastic event.

cond-mat.dis-nn↗

arXiv2021

Emergent power-law interactions in disordered crystals

We derive exact results for the fluctuations in energy produced by microscopic disorder in near-crystalline athermal systems. Our formalism captures the heterogeneity in the elastic energy of polydispersed soft disks in energy-minimized configurations. We use this to predict the distribution of interaction energy between two defects in a disordered background. We show this interaction energy displays an average power-law behaviour $\langle δE \rangle \sim Δ^{-4}$ at large distances $Δ$ between the defects. These interactions upon disorder average also display the sixfold symmetry of the underlying reference crystal. Additionally, we show that the fluctuations in the interaction energy encode the athermal correlations introduced by the disordered background. We verify our predictions with energy minimized configurations of polydispersed soft disks in two dimensions.

cond-mat.soft↗

arXiv2021

Displacement correlations in disordered athermal networks

We derive exact results for correlations in the displacement fields $\{ δ\vec{r} \} \equiv \{ δr_{μ= x,y} \}$ in near-crystalline athermal systems in two dimensions. We analyze the displacement correlations produced by different types of microscopic disorder, and show that disorder at the microscopic scale gives rise to long-range correlations with a dependence on the system size $L$ given by $\langle δr_μ δr_ν \rangle \sim c_{μν}(r/L,θ)$. In addition, we show that polydispersity in the constituent particle sizes and random bond disorder give rise to a logarithmic system size scaling, with $c_{μν}(ρ,θ) \sim \text{const}_{μν} - \text{a}_{μν}(θ)\log ρ+ \text{b}_{μν}(θ) ρ^{2} $ for $ρ~(=r/L) \to 0$. This scaling is different for the case of displacement correlations produced by random external forces at each vertex of the network, given by $c^{f}_{μν}(ρ,θ) \sim \text{const}^{f}_{μν} -( \text{a}^{f}_{μν}(θ) + \text{b}^{f}_{μν}(θ) \log ρ) ρ^2 $. Additionally, we find that correlations produced by polydispersity and the correlations produced by disorder in bond stiffness differ in their symmetry properties. Finally, we also predict the displacement correlations for a model of polydispersed soft disks subject to external pinning forces, that involve two different types of microscopic disorder. We verify our theoretical predictions using numerical simulations of polydispersed soft disks with random spring contacts in two dimensions.

cond-mat.stat-mech↗

arXiv2021

Long-range correlations in pinned athermal networks

We derive exact results for displacement fields that develop as a response to external pinning forces in two dimensional athermal networks. For a triangular lattice arrangement of particles interacting through soft potentials, we develop a Green's function formalism which we use to derive exact results for displacement fields produced by localized external forces. We show that in the continuum limit the displacement fields decay as $1/r$ at large distances $r$ away from a force dipole. Finally, we extend our formulation to study correlations in the displacement fields produced by the external pinning forces. We show that uncorrelated pinned forces at each vertex give rise to long-range correlations in displacements in athermal systems, with a non-trivial system size dependence. We verify our predictions with numerical simulations of athermal networks in two dimensions.

cond-mat.soft↗

arXiv2021

Disorder Perturbation Expansion for Athermal Crystals

We introduce a perturbation expansion for athermal systems that allows an exact determination of displacement fields away from the crystalline state as a response to disorder. We show that the displacement fields in energy minimized configurations of particles interacting through central potentials with microscopic disorder, can be obtained as a series expansion in the strength of the disorder. We introduce a hierarchy of force balance equations that allows an order-by-order determination of the displacement fields, with the solutions at lower orders providing sources for the higher order solutions. This allows the simultaneous force balance equations to be solved, within a hierarchical perturbation expansion to arbitrary accuracy. We present exact results for an isotropic defect introduced into the crystalline ground state at linear order and second order in our expansion. We show that the displacement fields produced by the defect display interesting self-similar properties at every order. We derive a $|δr| \sim 1/r$ and $|δf| \sim 1/r^2$ decay for the displacement fields and excess forces at large distances $r$ away from the defect. Finally we derive non-linear corrections introduced by the interactions between defects at second order in our expansion. We verify our exact results with displacement fields obtained from energy minimized configurations of soft disks.

cond-mat.soft↗

arXiv2020

Light and heavy particles on a fluctuating surface: Bunchwise balance, irreducible sequences and local density-height correlations

We study the early time and coarsening dynamics in the Light-Heavy model, a system consisting of two species of particles ($light$ and $heavy$) coupled to a fluctuating surface (described by tilt fields). The dynamics of particles and tilts are coupled through local update rules, and are known to lead to different ordered and disordered steady state phases depending on the microscopic rates. We introduce a generalized balance mechanism in non-equilibrium systems, namely $bunchwise~balance$, in which incoming and outgoing transition currents are balanced between groups of configurations. This allows us to exactly determine the steady state in a subspace of the phase diagram of this model. We introduce the concept of $irreducible~sequences$ of interfaces and bends in this model. These sequences are non-local, and we show that they provide a coarsening length scale in the ordered phases at late times. Finally, we propose a $local$ correlation function ($\mathcal{S}$) that has a direct relation to the number of irreducible sequences, and is able to distinguish between several phases of this system through its coarsening properties. Starting from a totally disordered initial configuration, $\mathcal{S}$ displays an initial linear rise and a broad maximum. As the system evolves towards the ordered steady states, $\mathcal{S}$ further exhibits power law decays at late times that encode coarsening properties of the approach to the ordered phases. Focusing on early time dynamics, we posit coupled mean-field evolution equations governing the particles and tilts, which at short times are well approximated by a set of linearized equations, which we solve analytically. Beyond a timescale set by a lattice cutoff and preceding the onset of coarsening, our linearized theory predicts the existence of an intermediate power-law stretch, which we also find in simulations of the ordered regime of the system.

cond-mat.stat-mech↗

arXiv2020

Singularities in Hessian element distributions of amorphous media

We show that the distribution of elements $H$ in the Hessian matrices associated with amorphous materials exhibit singularities $P(H) \sim {\lvert H \rvert}^γ$ with an exponent $γ< 0$, as $\lvert H \rvert \to 0$. We exploit the rotational invariance of the underlying disorder in amorphous structures to derive these exponents exactly for systems interacting via radially symmetric potentials. We show that $γ$ depends only on the degree of smoothness $n$ of the potential of interaction between the constituent particles at the cut-off distance, independent of the details of interaction in both two and three dimensions. We verify our predictions with numerical simulations of models of structural glass formers. Finally, we show that such singularities affect the stability of amorphous solids, through the distributions of the minimum eigenvalue of the Hessian matrix.

cond-mat.dis-nn↗

arXiv2020

Emergent Elasticity in Amorphous Solids

The mechanical response of naturally abundant amorphous solids such as gels, jammed grains, and biological tissues are not described by the conventional paradigm of broken symmetry that defines crystalline elasticity. In contrast, the response of such athermal solids are governed by local conditions of mechanical equilibrium, i.e., force and torque balance of its constituents. Here we show that these constraints have the mathematical structure of a generalized electromagnetism, where the electrostatic limit successfully captures the anisotropic elasticity of amorphous solids. The emergence of elasticity from local mechanical constraints offers a new paradigm for systems with no broken symmetry, analogous to emergent gauge theories of quantum spin liquids. Specifically, our $U(1)$ rank-2 symmetric tensor gauge theory of elasticity translates to the electromagnetism of fractonic phases of matter with the stress mapped to electric displacement and forces to vector charges. We corroborate our theoretical results with numerical simulations of soft frictionless disks in both two and three dimensions, and experiments on frictional disks in two dimensions. We also present experimental evidence indicating that force chains in granular media are sub-dimensional excitations of amorphous elasticity similar to fractons.

cond-mat.dis-nn↗

arXiv2020

Athermal fluctuations in disordered crystals

We analyze the fluctuations in particle positions and inter-particle forces in disordered jammed crystals in the limit of weak disorder. We demonstrate that such athermal systems are fundamentally different from their thermal counterparts, characterized by constrained fluctuations of forces perpendicular to the lattice directions. We develop a disorder perturbation expansion in polydispersity about the crystalline state, which we use to derive exact results to linear order. We show that constrained fluctuations result as a consequence of local force balance conditions, and are characterized by non-Gaussian distributions which we derive exactly. We analytically predict several properties of such systems, including the scaling of the average coordination with polydispersity and packing fraction, which we verify with numerical simulations using soft disks with one-sided harmonic interactions.

cond-mat.soft↗

arXiv2020

Arrested States in Persistent Active Matter: Gelation without Attraction

We explore phase separation and kinetic arrest in a model active colloidal system consisting of self-propelled, hard-core particles with nonconvex shapes. The passive limit of the model, namely cross-shaped particles on a square lattice, exhibits a first-order transition from a fluid phase to a solid phase with increasing density. Quenches into the two-phase coexistence region exhibit an aging regime. The nonconvex shape of the particles eases jamming in the passive system and leads to strong inhibition of rotations of the active particles. Using numerical simulations and analytical modeling, we quantify the nonequilibrium phase behavior as a function of density and activity. If we view activity as the analog of attraction strength, the phase diagram exhibits strong similarities to that of attractive colloids, exhibiting both aging, glassy states and gel-like arrested states. The two types of dynamically arrested states, glasses and gels, are distinguished by the appearance of density heterogenities in the latter. In the infinitely persistent limit, we show that a coarse-grained model based on the asymmetric exclusion process quantitatively predicts the density profiles of the gel states. The predictions remain qualitatively valid for finite rotation rates. Using these results, we classify the activity-driven phases and identify the boundaries separating them.

cond-mat.soft↗

arXiv2020

Timescale divergence at the shear jamming transition

We find that in simulations of quasi-statically sheared frictional disks, the shear jamming transition can be characterized by an abrupt jump in the number of force bearing contacts between particles. This mechanical coordination number increases discontinuously from $Z = 0$ to $Z \gtrsim d +1$ at a critical shear value $γ_c$, as opposed to a smooth increase in the number of geometric contacts. This is accompanied by a diverging timescale $τ^*$ that characterizes the time required by the system to attain force balance when subjected to a perturbation. As the global shear $γ$ approaches the critical value $γ_c$ from below, one observes the divergence of the time taken to relax to a state where all the inter-particle contacts have uniformly zero force. Above $γ_{c}$, the system settles into a state characterized by finite forces between particles, with the timescale also increasing as $γ\to γ_{c}^{+}$. By using two different protocols to generate force balanced configurations, we show that this timescale divergence is a robust feature that accompanies the shear jamming transition.

cond-mat.soft↗

arXiv2018

Motion of active tracer in a lattice gas with cross-shaped particles

We analyze the dynamics of an active tracer particle embedded in a thermal lattice gas. All particles are subject to exclusion up to third nearest neighbors on the square lattice, which leads to slow dynamics at high densities. For the case with no rotational diffusion of the tracer, we derive an analytical expression for the resulting drift velocity v of the tracer in terms of non-equilibrium density correlations involving the tracer particle and its neighbors, which we verify using numerical simulations. We show that the properties of the passive system alone do not adequately describe even this simple system of a single non-rotating active tracer. For large activity and low density, we develop an approximation for v. For the case where the tracer undergoes rotational diffusion independent of its neighbors, we relate its diffusion coefficient to the thermal diffusion coefficient and v. Finally we study dynamics where the rotation of the tracer is limited by the presence of neighboring particles. We find that the effect of this rotational locking may be quantitatively described in terms of a reduction of the rotation rate.

cond-mat.soft↗

arXiv2018

Microscopic origin of frictional rheology in dense suspensions: correlations in force space

We develop a statistical framework for the rheology of dense, non-Brownian suspensions, based on correlations in a space representing forces, which is dual to position space. Working with the ensemble of steady state configurations obtained from simulations of suspensions in two dimensions, we find that the anisotropy of the pair correlation function in force space changes with confining shear stress ($σ_{xy}$) and packing fraction ($ϕ$). Using these microscopic correlations, we build a statistical theory for the macroscopic friction coefficient: the anisotropy of the stress tensor, $μ= σ_{xy}/P$. We find that $μ$ decreases (i) as $ϕ$ is increased and (ii) as $σ_{xy}$ is increased. Using a new constitutive relation between $μ$ and viscosity for dense suspensions that generalizes the rate-independent one, we show that our theory predicts a Discontinuous Shear Thickening (DST) flow diagram that is in good agreement with numerical simulations, and the qualitative features of $μ$ that lead to the generic flow diagram of a DST fluid observed in experiments.

cond-mat.soft↗