arXiv subjects

Anupam Kundu

Publications and source records attributed to Anupam Kundu.

At least 55 records · Page 3Linked to original sources

arXiv2021

Mean area of the convex hull of a run and tumble particle in two dimensions

We investigate the statistics of the convex hull for a single run-and-tumble particle in two dimensions. Run-and-tumble particle, also known as persistent random walker, has gained significant interest in the recent years due to its biological application in modelling the motion of bacteria. We consider two different statistical ensembles depending on whether (i) the total number of tumbles $n$ or (ii) the total observation time $t$ is kept fixed. Benchmarking the results on perimeter, we study the statistical properties of the area of the convex hull for RTP. Exploiting the connections to extreme value statistics, we obtain exact analytical expressions for the mean area for both ensembles. For fixed-$t$ ensemble, we show that the mean possesses a scaling form in $γt$ (with $γ$ being the tumbling rate) and the corresponding scaling function is exactly computed. Interestingly, we find that it exhibits crossover from $\sim t^3$ scaling at small times $\left( t \ll γ^{-1} \right)$ to $\sim t$ scaling at large times $\left( t \gg γ^{-1} \right)$. On the other hand, for fixed-$n$ ensemble, the mean expectedly grows linearly with $n$ for $n \gg 1$. All our analytical findings are supported with numerical simulations.

cond-mat.stat-mech↗

arXiv2021

Edge fluctuations and third-order phase transition in harmonically confined long-range systems

We study the distribution of the position of the rightmost particle $x_{\max}$ in a $N$-particle Riesz gas in one dimension confined in a harmonic trap. The particles interact via long-range repulsive potential, of the form $r^{-k}$ with $-2 -2$. We also find that these large deviation functions describe a pulled to pushed type phase transition as observed in Dyson's log-gas ($k\to 0$) and $1d$ one component plasma ($k=-1$). Remarkably, we find that the phase transition remains $3^{\rm rd}$ order for the entire regime. Our results demonstrate the striking universality of the $3^{\rm rd}$ order transition even in models that fall outside the paradigm of Coulomb systems and the random matrix theory. We numerically verify our analytical expressions of the large deviation functions via Monte Carlo simulation using an importance sampling algorithm.

cond-mat.stat-mech↗

arXiv2021

Estimation of Constrained Mean-Covariance of Normal Distributions

Estimation of the mean vector and covariance matrix is of central importance in the analysis of multivariate data. In the framework of generalized linear models, usually the variances are certain functions of the means with the normal distribution being an exception. We study some implications of functional relationships between covariance and the mean by focusing on the maximum likelihood and Bayesian estimation of the mean-covariance under the joint constraint $\bmΣ\bmμ = \bmμ$ for a multivariate normal distribution. A novel structured covariance is proposed through reparameterization of the spectral decomposition of $\bmΣ$ involving its eigenvalues and $\bmμ$. This is designed to address the challenging issue of positive-definiteness and to reduce the number of covariance parameters from quadratic to linear function of the dimension. We propose a fast (noniterative) method for approximating the maximum likelihood estimator by maximizing a lower bound for the profile likelihood function, which is concave. We use normal and inverse gamma priors on the mean and eigenvalues, and approximate the maximum aposteriori estimators by both MH within Gibbs sampling and a faster iterative method. A simulation study shows good performance of our estimators.

stat.ME↗

arXiv2021

Harmonically confined long-ranged interacting gas in the presence of a hard wall

In this paper, we compute exactly the average density of a harmonically confined Riesz gas of $N$ particles for large $N$ in the presence of a hard wall. In this Riesz gas, the particles repel each other via a pairwise interaction that behaves as $|x_i - x_j|^{-k}$ for $k>-2$, with $x_i$ denoting the position of the $i^{\rm th}$ particle. This density can be classified into three different regimes of $k$. For $k \geq 1$, where the interactions are effectively short-ranged, the appropriately scaled density has a finite support over $[-l_k(w),w]$ where $w$ is the scaled position of the wall. While the density vanishes at the left edge of the support, it approaches a nonzero constant at the right edge $w$. For $-1<k<1$, where the interactions are weakly long-ranged, we find that the scaled density is again supported over $[-l_k(w),w]$. While it still vanishes at the left edge of the support, it diverges at the right edge $w$ algebraically with an exponent $(k-1)/2$. For $-2<k< -1$, the interactions are strongly long-ranged that leads to a rather exotic density profile with an extended bulk part and a delta-peak at the wall, separated by a hole in between. Exactly at $k=-1$ the hole disappears. For $-2<k< -1$, we find an interesting first-order phase transition when the scaled position of the wall decreases through a critical value $w=w^*(k)$. For $w<w^*(k)$, the density is a pure delta-peak located at the wall. The amplitude of the delta-peak plays the role of an order parameter which jumps to the value $1$ as $w$ is decreased through $w^*(k)$. Our analytical results are in very good agreement with our Monte-Carlo simulations.

cond-mat.stat-mech↗

arXiv2021

Dynamical regimes of finite temperature discrete nonlinear Schrödinger chain

We show that the one dimensional discrete nonlinear Schrödinger chain (DNLS) at finite temperature has three different dynamical regimes (ultra-low, low and high temperature regimes). This has been established via (i) one point macroscopic thermodynamic observables (temperature $T$ , energy density $ε$ and the relationship between them), (ii) emergence and disappearance of an additional almost conserved quantity (total phase difference) and (iii) classical out-of-time-ordered correlators (OTOC) and related quantities (butterfly speed and Lyapunov exponents). The crossover temperatures $T_{\textit{l-ul}}$ (between low and ultra-low temperature regimes) and $T_{\textit{h-l}}$ (between high and low temperature regimes) extracted from these three different approaches are consistent with each other. The analysis presented here is an important step forward towards the understanding of DNLS which is ubiquitous in many fields and has a non-separable Hamiltonian form. Our work also shows that the different methods used here can serve as important tools to identify dynamical regimes in other interacting many body systems.

cond-mat.stat-mech↗

arXiv2021

Dynamics of a randomly kicked particle

Levy walk (LW) process has been used as a simple model for describing anomalous diffusion in which the mean squared displacement of the walker grows non-linearly with time in contrast to the diffusive motion described by simple random walks or Brownian motion. In this paper we study a simple extension of the LW model in one dimension by introducing correlation among the velocities of the walker in different (flight) steps. Such correlation is absent in the LW model. The correlations are introduced by making the velocity at a step dependent on the velocity at the previous step in addition to the usual random noise ('kick') that the particle gets at random time intervals from the surrounding medium as in the LW model. Consequently the dynamics of the position becomes non-Markovian. We study the statistical properties of velocity and position of the walker at time t, both analytically and numerically. We show how different choices of the distribution of the random time intervals and the degree of correlation, controlled by a parameter r, affect the late time behaviour of these quantities.

cond-mat.stat-mech↗

arXiv2021

Stochastic resetting with stochastic returns using external trap

In the past few years, stochastic resetting has become a subject of immense interest. Most of the theoretical studies so far focused on instantaneous resetting which is, however, a major impediment to practical realization or experimental verification in the field. This is because in the real world, taking a particle from one place to another requires finite time and thus a generalization of the existing theory to incorporate non-instantaneous resetting is very much in need. In this paper, we propose a method of resetting which involves non-instantaneous returns facilitated by an external confining trap potential $U(x)$ centered at the resetting location. We consider a Brownian particle that starts its random motion from the origin. Upon resetting, the trap is switched on and the particle starts experiencing a force towards the center of the trap which drives it to return to the origin. The return phase ends when the particle makes a first passage to this center. We develop a general framework to study such a set up. Importantly, we observe that the system reaches a non-equilibrium steady state which we analyze in full details for two choices of $U(x)$, namely, (i) linear and (ii) harmonic. Finally, we perform numerical simulations and find an excellent agreement with the theory. The general formalism developed here can be applied to more realistic return protocols opening up a panorama of possibilities for further theoretical and experimental applications.

cond-mat.stat-mech↗

arXiv2021

Resetting with stochastic return through linear confining potential

We consider motion of an overdamped Brownian particle subject to stochastic resetting in one dimension. In contrast to the usual setting where the particle is instantaneously reset to a preferred location (say, the origin), here we consider a finite time resetting process facilitated by an external linear potential $V(x)=λ|x|~ (λ>0)$. When resetting occurs, the trap is switched on and the particle experiences a force $-\partial_x V(x)$ which helps the particle to return to the resetting location. The trap is switched off as soon as the particle makes a first passage to the origin. Subsequently, the particle resumes its free diffusion motion and the process keeps repeating. In this set-up, the system attains a non-equilibrium steady state. We study the relaxation to this steady state by analytically computing the position distribution of the particle at all time and then analysing this distribution using the spectral properties of the corresponding Fokker-Planck operator. As seen for the instantaneous resetting problem, we observe a `cone spreading' relaxation with travelling fronts such that there is an inner core region around the resetting point that reaches the steady state, while the region outside the core still grows ballistically with time. In addition to the unusual relaxation phenomena, we compute the large deviation functions associated to the corresponding probability density and find that the large deviation functions describe a dynamical transition similar to what is seen previously in case of instantaneous resetting. Notably, our method, based on spectral properties, complements the existing renewal formalism and reveals the intricate mathematical structure responsible for such relaxation phenomena. We verify our analytical results against extensive numerical simulations.

cond-mat.stat-mech↗

arXiv2021

Local time of an Ornstein-Uhlenbeck particle

In this paper, we study the local time spent by an Ornstein-Uhlenbeck particle at some location till time t. Using the Feynman-Kac formalism, the computation of the moment generating function of the local time can be mapped to the problem of finding the eigenvalues and eigenfunctions of a quantum particle. We employ quantum perturbation theory to compute the eigenvalues and eigenfunctions in powers of the argument of the moment generating function which particularly help to directly compute the cumulants and correlations among local times spent at different locations. In particular, we obtain explicit expressions of the mean, variance, and covariance of the local times in the presence and in the absence of an absorbing boundary, conditioned on survival. In the absence of absorbing boundaries, we also study large deviations of the local time and compute exact asymptotic forms of the associated large deviation functions explicitly. In the second part of the paper, we extend our study of the statistics of local time of the Ornstein-Uhlenbeck particle to the case not conditioned on survival. In this case, one expects the distribution of the local time to reach a stationary distribution in the large time limit. Computations of such stationary distributions are known in the literature as the problem of first passage functionals. In this paper, we study the approach to this stationary state with time by providing a general formulation for evaluating the moment generating function. From this moment generating function, we compute the cumulants of the local time exhibiting the approach to the stationary values explicitly for a free particle and a Ornstein-Uhlenbeck particle. Our analytical results are verified and supported by numerical simulations.

cond-mat.stat-mech↗

arXiv2021

COVID-19: Analytic results for a modified SEIR model and comparison of different intervention strategies

The Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model is one of the standard models of disease spreading. Here we analyse an extended SEIR model that accounts for asymptomatic carriers, believed to play an important role in COVID-19 transmission. For this model we derive a number of analytic results for important quantities such as the peak number of infections, the time taken to reach the peak and the size of the final affected population. We also propose an accurate way of specifying initial conditions for the numerics (from insufficient data) using the fact that the early time exponential growth is well-described by the dominant eigenvector of the linearized equations. Secondly we explore the effect of different intervention strategies such as social distancing (SD) and testing-quarantining (TQ). The two intervention strategies (SD and TQ) try to reduce the disease reproductive number, $R_0$, to a target value $R^{\rm target}_0 < 1$, but in distinct ways, which we implement in our model equations. We find that for the same $R^{\rm target}_0 < 1$, TQ is more efficient in controlling the pandemic than SD. However, for TQ to be effective, it has to be based on contact tracing and our study quantifies the required ratio of tests-per-day to the number of new cases-per-day. Our analysis shows that the largest eigenvalue of the linearised dynamics provides a simple understanding of the disease progression, both pre- and post- intervention, and explains observed data for many countries. We apply our results to the COVID data for India to obtain heuristic projections for the course of the pandemic, and note that the predictions strongly depend on the assumed fraction of asymptomatic carriers.

q-bio.PE↗

arXiv2020

MLE of Jointly Constrained Mean-Covariance of Multivariate Normal Distributions

Estimating the unconstrained mean and covariance matrix is a popular topic in statistics. However, estimation of the parameters of $N_p(μ,Σ)$ under joint constraints such as $Σμ= μ$ has not received much attention. It can be viewed as a multivariate counterpart of the classical estimation problem in the $N(θ,θ^2)$ distribution. In addition to the usual inference challenges under such non-linear constraints among the parameters (curved exponential family), one has to deal with the basic requirements of symmetry and positive definiteness when estimating a covariance matrix. We derive the non-linear likelihood equations for the constrained maximum likelihood estimator of $(μ,Σ)$ and solve them using iterative methods. Generally, the MLE of covariance matrices computed using iterative methods do not satisfy the constraints. We propose a novel algorithm to modify such (infeasible) estimators or any other (reasonable) estimator. The key step is to re-align the mean vector along the eigenvectors of the covariance matrix using the idea of regression. In using the Lagrangian function for constrained MLE (Aitchison et al. 1958), the Lagrange multiplier entangles with the parameters of interest and presents another computational challenge. We handle this by either iterative or explicit calculation of the Lagrange multiplier. The existence and nature of location of the constrained MLE are explored within a data-dependent convex set using recent results from random matrix theory. A simulation study illustrates our methodology and shows that the modified estimators perform better than the initial estimators from the iterative methods.

stat.ME↗

arXiv2020

Local time for run and tumble particle

We investigate the local time $(T_{loc})$ statistics for a run and tumble particle in an one dimensional inhomogeneous medium. The inhomogeneity is introduced by considering the position dependent rate of the form $R(x) = γ\frac{|x|^α}{l^α}$ with $α\geq 0$. For $α=0$, we derive the probability distribution of $T_{loc}$ exactly which is expressed as a series of $δ$-functions in which the coefficients can be interpreted as the probability of multiple revisits of the particle to the origin starting from the origin. For general $α$, we show that the typical fluctuations of $T_{loc}$ scale with time as $T_{loc} \sim t^{\frac{1+α}{2+α}}$ for large $t$ and their probability distribution possesses a scaling behaviour described by a scaling function which we have computed analytically. In the second part, we study the statistics of $T_{loc}$ till the RTP makes a first passage to $x=M~(>0)$. In this case also, we show that the probability distribution can be expressed as a series sum of $δ$-functions for all values of $α~(\geq 0)$ with coefficients appearing from appropriate exit problems. All our analytical findings are supported with the numerical simulations.

cond-mat.stat-mech↗

arXiv2020

Particles confined in arbitrary potentials with a class of finite-ranged interactions

In this paper, we develop a large-$N$ field theory for a system of $N$ classical particles in one dimension at thermal equilibrium. The particles are confined by an arbitrary external potential, $V_\text{ex} (x)$, and repel each other via a class of pairwise interaction potentials $V_\text{int}(r)$ (where $r$ is distance between a pair of particles) such that $ V_\text{int} \sim |r|^{-k}$ when $r \to 0$. We consider the case where every particle is interacting with $d$ (finite range parameter) number of particles to its left and right. Due to the intricate interplay between external confinement, pairwise repulsion and entropy, the density exhibits markedly distinct behavior in three regimes $k>0$, $k \to 0$ and $k<0$. From this field theory, we compute analytically the average density profile for large $N$ in these regimes. We show that the contribution from interaction dominates the collective behaviour for $k > 0$ and the entropy contribution dominates for $k<0$, and both contributes equivalently in the $k\to 0$ limit (finite range log-gas). Given the fact that these family of systems are of broad relevance, our analytical findings are of paramount importance. These results are in excellent agreement with brute-force Monte-Carlo simulations.

cond-mat.stat-mech↗

arXiv2020

Transport, correlations, and chaos in a classical disordered anharmonic chain

We explore transport properties in a disordered nonlinear chain of classical harmonic oscillators and thereby identify a regime exhibiting behavior analogous to that seen in quantum many-body-localized systems. Through extensive numerical simulations of this system connected at its ends to heat baths at different temperatures, we computed the heat current and the temperature profile in the nonequilibrium steady state as a function of system size $N$, disorder strength $Δ$, and temperature $T$. The conductivity $κ_N$, obtained for finite length ($N$) systems, saturates to a value $κ_\infty >0$ in the large $N$ limit, for all values of disorder strength $Δ$ and temperature $T>0$. We show evidence that for any $Δ>0$ the conductivity goes to zero faster than any power of $T$ in the $(T/Δ) \to 0$ limit, and find that the form $κ_\infty \sim e^{-B |\ln(C Δ/T)|^3}$ fits our data. This form has earlier been suggested by a theory based on the dynamics of multi-oscillator chaotic islands. The finite-size effect can be $κ_N < κ_{\infty}$ due to boundary resistance when the bulk conductivity is high (the weak disorder case), or $κ_N > κ_{\infty}$ due to direct bath-to-bath coupling through bulk localized modes when the bulk is weakly conducting (the strong disorder case). We also present results on equilibrium dynamical correlation functions and on the role of chaos on transport properties. Finally, we explore the differences in the growth and propagation of chaos in the weak and strong chaos regimes by studying the classical version of the Out-of-Time-Ordered-Commutator.

cond-mat.stat-mech↗

arXiv2020

Universal scaling in active single-file dynamics

We study the single-file dynamics of three classes of active particles: run-and-tumble particles, active Brownian particles and active Ornstein-Uhlenbeck particles. At high activity values, the particles, interacting via purely repulsive and short-ranged forces, aggregate into several motile and dynamical clusters of comparable size, and do not display bulk phase-segregation. In this dynamical steady-state, we find that the cluster size distribution of these aggregates is a scaled function of the density and activity parameters across the three models of active particles with the same scaling function. The velocity distribution of these motile clusters is non-Gaussian. We show that the effective dynamics of these clusters can explain the observed emergent scaling of the mean-squared displacement of tagged particles for all the three models with identical scaling exponents and functions. Concomitant with the clustering seen at high activities, we observe that the static density correlation function displays rich structures, including multiple peaks that are reminiscent of particle clustering induced by effective attractive interactions, while the dynamical variant shows non-diffusive scaling. Our study reveals a universal scaling behavior in the single-file dynamics of interacting active particles.

cond-mat.stat-mech↗

arXiv2020

Generalised `Arcsine' laws for run-and-tumble particle in one dimension

The 'Arcsine' laws of Brownian particles in one dimension describe distributions of three quantities: the time $t_m$ to reach maximum position, the time $t_r$ spent on the positive side and the time $t_\ell$ of the last visit to the origin. Interestingly, the cumulative distribution of all the three quantities are same and given by Arcsine function. In this paper, we study distribution of these three times $t_m,~t_r$ and $t_\ell$ in the context of single run-and-tumble particle in one dimension, which is a simple non-Markovian process. We compute exact distributions of these three quantities for arbitrary time and find that all three distributions have delta function part and a non-delta function part. Interestingly, we find that the distributions of $t_m$ and $t_r$ are identical (reminiscent of the Brownian particle case) when the initial velocities of the particle are chosen with equal probability. On the other hand, for $t_\ell$, only the non-delta function part is same with the other two. In addition, we find explicit expressions of the joint distributions of the maximum displacement and the time at which this maxima occurs. We verify all our analytical results through numerical simulations.

cond-mat.stat-mech↗

arXiv2020

Run-and-Tumble particle in inhomogeneous media in one dimension

We investigate the run and tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise $σ(t)$ drives the particle which changes between $\pm 1$ values with some rates. Denoting the rate of flip from $1$ to $-1$ as $R_1$ and the converse rate as $R_2$, we consider the position and direction dependent rates of the form $R_1(x)=\left(\frac{\mid x \mid}{l}\right) ^α\left[γ_1~θ(x)+γ_2 ~θ(-x)\right]$ and $R_2(x)=\left(\frac{\mid x \mid}{l}\right) ^α\left[γ_2~θ(x)+γ_1 ~θ(-x)\right]$ with $α\geq 0$. For $γ_1 >γ_2$, we find that the particle exhibits a steady-state probability distriution even in an infinite line whose exact form depends on $α$. For $α=0$ and $1$, we solve the master equations exactly for arbitrary $γ_1$ and $γ_2$ at large $t$. From our explicit expression for time-dependent probability distribution $P(x,t)$ we find that it exponentially relaxes to the steady-state distribution for $γ_1 > γ_2$. On the other hand, for $γ_1<γ_2$, the large $t$ behaviour of $P(x,t)$ is drastically different than $γ_1=γ_2$ case where the distribution decays as $t^{-\frac{1}{2}}$. Contrary to the latter, detailed balance is not obeyed by the particle even at large $t$ in the former case. For general $α$, we argue that the approach to the steady state in $γ_1>γ_2$ case is exponential which we numerically demonstrate....

cond-mat.stat-mech↗

arXiv2020

Spatio-temporal spread of perturbations in a driven dissipative Duffing chain: an OTOC approach

Out-of-time-ordered correlators (OTOC) have been extensively used as a major tool for exploring quantum chaos and also recently, there has been a classical analogue. Studies have been limited to closed systems. In this work, we probe an open classical many-body system, more specifically, a spatially extended driven dissipative chain of coupled Duffing oscillators using the classical OTOC to investigate the spread and growth (decay) of an initially localized perturbation in the chain. Correspondingly, we find three distinct types of dynamical behavior, namely the sustained chaos, transient chaos and non-chaotic region, as clearly exhibited by different geometrical shapes in the heat map of OTOC. To quantify such differences, we look at instantaneous speed (IS), finite time Lyapunov exponents (FTLE) and velocity dependent Lyapunov exponents (VDLE) extracted from OTOC. Introduction of these quantities turn out to be instrumental in diagnosing and demarcating different regimes of dynamical behavior. To gain control over open nonlinear systems, it is important to look at the variation of these quantities with respect to parameters. As we tune drive, dissipation and coupling, FTLE and IS exhibit transition between sustained chaos and non-chaotic regimeswith intermediate transient chaos regimes and highly intermittent sustained chaos points. In the limit of zero nonlinearity, we present exact analytical results for the driven dissipative harmonic system and we find that our analytical results can very well describe the non-chaotic regime as well as the late time behavior in the transient regime of the Duffing chain. We believe, this analysis is an important step forward towards understanding nonlinear dynamics, chaos and spatio-temporal spread of perturbations in many-particle open systems.

cond-mat.stat-mech↗