arXiv subjects

Parongama Sen

Publications and source records attributed to Parongama Sen.

At least 37 records · Page 2Linked to original sources

arXiv2022

Virtual walks inspired by a mean field kinetic exchange model of opinion dynamics

We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange model of opinion dynamics. The walks are either Markovian or non-Markovian in nature. The opinion dynamics model is characterized by a parameter $p$ which drives an order disorder transition at a critical value $p_c$. The distribution $S(X,t)$ of the displacements $X$ from the origin of the walkers is computed at different times. Below $p_c$, two time scales associated with a crossover behavior in time are detected, which diverge in a power law manner at criticality with different exponent values. $S(X,t)$ also carries the signature of the phase transition as it changes its form at $p_c$. The walks show the features of a biased random walk below $p_c$, and above $p_c$, the walks are like unbiased random walks. The bias vanishes in a power law manner at $p_c$ and the width of the resulting Gaussian function shows a discontinuity. Some of the features of the walks are argued to be comparable to the critical quantities associated with the mean field Ising model, to which class the opinion dynamics model belongs. The results for the Markovian and non-Markovian walks are almost identical which is justified by considering the different fluxes. We compare the present results with some earlier similar studies.

cond-mat.stat-mech↗

arXiv2021

Non-equilibrium dynamics in Ising like models with biased initial condition

We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like models. The stability analysis of the fixed points in the mean field calculation shows the existence of an exponent that depends on the coordination number $z$ in the Ising model. For the generalised voter model, a phase diagram is obtained based on this study. Numerical results for the Ising model for both the mean field case and short ranged models on lattices with different values of $z$ are also obtained. A related study is the behaviour of the exit probability $E(x_0)$, defined as the probability that a configuration ends up with all spins up starting with $x_0$ fraction of up spins. An interesting result is $E(x_0) = x_0$ in the mean field approximation when $z=2$, which is consistent with the conserved magnetisation in the system. For larger values of $z$, $E(x_0)$ shows the usual finite size dependent non linear behaviour both in the mean field model and in Ising model with nearest neighbour interaction on different two dimensional lattices. For such a behaviour, a data collapse of $E(x_0)$ is obtained using $y = \frac{(x_0 - x_c)}{x_c}L^{1/ν}$ as the scaling variable and $f(y)=\frac{1+\tanh(λy)}{2}$ appears as the scaling function. The universality of the exponent and the scaling factor is investigated.

cond-mat.stat-mech↗

arXiv2021

The variational method applied to the harmonic oscillator in presence of a delta function potential

The problem of the harmonic oscillator with a centrally located delta function potential can be exactly solved in one dimension where the eigenfunctions are expressed as superpositions of the Hermite polynomials or as confluent hypergeometric functions in general. The eigenfunctions obtained exactly are difficult to visualise and hence to gain more insight, one can attempt using model wave functions which are explicitly and simply expressed. Here we apply the variational method to verify how close one can approach the exact ground state eigenvalues using such trial wave functions. We obtain the estimates of the ground state energies which are closer to the exact values in comparison to earlier approximate results for both the repulsive and attractive delta potentials.

quant-ph↗

arXiv2021

$A+ A \to \emptyset$ system in one dimension with particle motion determined by nearest neighbour distances: results for parallel updates

A one dimensional $A+A \to \emptyset$ system where the direction of motion of the particles is determined by the position of the nearest neighours is studied. The particles move with a probability $0.5 + \epsi$ towards their nearest neighbours with $-0.5 \leq \epsi \leq 0.5$. This implies a stochastic motion towards the nearest neighbour or away from it for positive and negative values of $\epsi$ respectively, with $\epsi = \pm ~0.5$ the two deterministic limits. The position of the particles are updated in parallel. The macroscopic as well as tagged particle dynamics are studied which show drastic changes from the diffusive case $\epsi=0$. The decay of particle density shows departure from the usual power law behaviour as found in $\epsi =0$, on both sides of $\epsi =0$ and a scaling regime is obtained for $\epsi > 0$. The $\epsi =0.5$ point is characterized by the presence of dimers, which are isolated pairs of particles in adjacent sites that are never annihilated. The persistence probability is also calculated that decays in a stretched exponential manner for $\epsi < 0$ and switches over to power law behaviour for $\epsi \geq 0$, with different exponents for $\epsi =0$ and $\epsi > 0$. For the tagged particle, the probability distribution $Π(x,t)$ that it is at position $x$ at time $t$ shows the existence of a scaling variable $x/t^ν$ where $ν= 0.55 \pm 0.05$ for $\epsi > 0$ and varies with $\epsi$ for $\epsi < 0$. Finally, a comparative analysis for the behaviour of all the relevant quantities for the system using parallel and asynchronous dynamics (studied recently) shows that there are significant differences for $\epsi > 0$ while the results are qualitatively similar for $\epsi < 0$.

cond-mat.stat-mech↗

arXiv2020

Block size dependence of coarse graining in discrete opinion dynamics model: Application to the US presidential elections

The electoral college of voting system for the US presidential election is analogous to a coarse graining procedure commonly used to study phase transitions in physical systems. In a recent paper, opinion dynamics models manifesting a phase transition, were shown to be able to explain the cases when a candidate winning more number of popular votes could still lose the general election on the basis of the electoral college system. We explore the dependence of such possibilities on various factors like the number of states and total population (i.e., system sizes) and get an interesting scaling behavior. In comparison with the real data, it is shown that the probability of the minority win, calculated within the model assumptions, is indeed near the highest possible value. In addition, we also implement a two step coarse graining procedure, relevant for both opinion dynamics and information theory.

physics.soc-ph↗

arXiv2020

$A+ A \to \emptyset$ reaction for particles with a dynamic bias to move away from their nearest neighbour in one dimension

We consider the dynamics of particles undergoing the reaction $A+A \to \emptyset$ in one dimension with a dynamic bias. Here the particles move towards their nearest neighbour with probability $0.5+ε$ where $-0.5 \leq ε< 0$. $ε_c = -0.5$ is the deterministic limit where the nearest neighbour interaction is strictly repulsive. We show that the negative bias changes drastically the behaviour of the fraction of surviving particles $ρ(t)$ and persistence probability $P(t)$ with time $t$. $ρ(t)$ decays as $a/ (\log t)^b$ where $b$ increases with $ε- ε_c$. $P(t)$ shows a stretched exponential decay with non-universal decay parameters. The probability $Π(x,t)$ that a tagged particle is at position $x$ from its origin is found to be Gaussian for all $ε<0$; the associated scaling variable is $x/t^α$ where $α$ approaches the known limiting value $1/4$ as $ε\to ε_c$, in a power law manner. Some additional features of the dynamics by tagging the particles are also studied. The results are compared to the case of positive bias, a well studied problem.

cond-mat.stat-mech↗

arXiv2020

Long route to consensus: Two stage coarsening in a binary choice voting model

Formation of consensus, in binary yes/no type of voting, is a well defined process. However, even in presence of clear incentives, the dynamics involved can be incredibly complex. Specifically, formations of large groups of similarly opinionated individuals could create a condition of `support-bubbles' or spontaneous polarization that renders consensus virtually unattainable (e.g., the question of the UK exiting the EU). There have been earlier attempts in capturing the dynamics of consensus formation in societies through simple $Z_2$-symmetric models hoping to capture the essential dynamics of average behaviorof a large number of individuals in a statistical sense. However, in absence of external noise, they tend to reach a frozen state with fragmented and polarized states i.e., two or more groups of similarly opinionated groups with frozen dynamics. Here we show in a kinetic exchange opinion model (KEM) considered on $L \times L$ square lattices, that while such frozen states could be avoided, an exponentially slow approach to consensus is manifested. Specifically, the system could either reach consensus in a time that scales as $L^2$ or a long lived metastable state (termed a domain-wall state) for which formation of consensus takes a time scaling as $L^{3.6}$. The latter behavior is comparable to some voter-like models with intermediate states studied previously. The late-time anomaly in the time scale is reflected in the persistence probability of the model. Finally, the interval of zero-crossing of the average opinion i.e., the time interval over which the average opinion does not change sign is shown to follow a scale free distribution, which is compared with that seen in the opinion surveys regarding Brexit and associated issues in the last 40 years. The issue of minority spreading is also addressed by calculating the exit probability.

physics.soc-ph↗

arXiv2020

Covid-19 spread: Reproduction of data and prediction using a SIR model on Euclidean network

We study the datafor the cumulative as well as daily number of cases in the Covid-19 outbreak in China. The cumulative data can be fit to an empirical form obtained from a Susceptible-Infected-Removed (SIR) model studied on an Euclidean network previously. Plotting the number of cases against the distance from the epicenter for both China and Italy, we find an approximate power law variation with an exponent $\sim 1.85$ showing strongly that the spatial dependence plays a key role, a factor included in the model. We report here that the SIR model on the Eucledean network can reproduce with a high accuracy the data for China for given parameter values, and can also predict when the epidemic, at least locally, can be expected to be over.

physics.soc-ph↗

arXiv2020

Space-time dependence of corona virus (COVID-19) outbreak

We analyse the data for the global corona virus (COVID-19) outbreak using the results of a previously studied Susceptible-Infected-Removed (SIR) model of epidemic spreading on Euclidean networks. We also directly study the correlation of the distance from the epicenter and the number of cases. An inverse square law is seen to exist approximately. The studies are made for China and the rest of the world separately.

physics.soc-ph↗

arXiv2020

Tagged particle dynamics in one dimensional $A+ A \to kA$ models with the particles biased to diffuse towards their nearest neighbour

Dynamical features of tagged particles are studied in a one dimensional $A+A \rightarrow kA$ system for $k=0$ and 1, where the particles $A$ have a bias $ε$ $(0 \leq ε\leq 0.5)$ to hop one step in the direction of their nearest neighboring particle. $ε=0$ represents purely diffusive motion and $ε=0.5$ represents purely deterministic motion of the particles. We show that for any $ε$, there is a time scale $t^*$ which demarcates the dynamics of the particles. Below $t^*$, the dynamics are governed by the annihilation of the particles, and the particle motions are highly correlated, while for $t \gg t^*$, the particles move as independent biased walkers. $t^*$ diverges as $(ε_c-ε)^{-γ}$, where $γ=1$ and $ε_c =0.5$. $ε_c$ is a critical point of the dynamics. At $ε_c$, the probability $S(t)$, that a walker changes direction of its path at time $t$, decays as $S(t) \sim t^{-1}$ and the distribution $D(τ)$ of the time interval $τ$ between consecutive changes in the direction of a typical walker decays with a power law as $D(τ) \sim τ^{-2}$.

cond-mat.stat-mech↗

arXiv2019

Scaling and crossover behaviour in a truncated long range quantum walk

We consider a discrete time quantum walker in one dimension, where at each step, the step length $\ell$ is chosen from a distribution $P(\ell) \propto \ell^{-δ-1}$ with $\ell \leq \ell_{max}$. We evaluate the probability $f(x,t)$ that the walker is at position $x$ at time $t$ and its first two moments. As expected, the disorder effectively localizes the walk even for large values of $δ$. Asymptotically, $\langle x^2 \rangle \propto t^{3/2}$ and $\langle x \rangle \propto t^{1/2}$ independent of $δ$ and $\ell$, both finite. The scaled distribution $f(x,t)t^{1/2}$ plotted versus $x/t^{1/2}$ shows a data collapse for $x/t < α(δ,\ell_{max}) \sim \mathcal O(1) $ indicating the existence of a universal scaling function. The scaling function is shown to have a crossover behaviour at $δ= δ^* \approx 4.0$ beyond which the results are independent of $\ell_{max}$. We also calculate the von Neumann entropy of entanglement which gives a larger asymptotic value compared to the quantum walk with unique step length even for large $δ$, with negligible dependence on the initial condition.

quant-ph↗

arXiv2019

Persistent quantum walks: dynamic phases and diverging timescales

A discrete time quantum walk is considered in which the step lengths are chosen to be either $1$ or $2$ with the additional feature that the walker is persistent with a probability $p$. This implies that with probability $p$, the walker repeats the step length taken in the previous step and is otherwise antipersistent. We estimate the probability $P(x,t)$ that the walker is at $x$ at time $t$ and the first two moments. Asymptotically, $\langle x^2 \rangle = t^ν$ for all $p$. For the extreme limits $p=0$ and $1$, the walk is known to show ballistic behaviour, i.e., $ν= 2$. As $p$ is varied from zero to 1, the system is found in four different phases characterised by the value of $ν$: $ν=2$ at $p=0$, $1 \leq ν\leq 3/2$ for $0 < p < p_c$, $ν= 3/2$ for $ p_c < p <1$ and $ν= 2$ again at $p=1$. $p_c$ is found to be very close to $1/3$ numerically. Close to $p=0,1$, the scaling behaviour shows a crossover in time. Associated with this crossover, two diverging timescales varying as $1/p$ and $1/(1-p)$ close to $p=0$ and $p=1$ respectively are detected. Using a different scheme in which the antipersistence behaviour is suppressed, one gets $ν= 3/2$ for the entire region $0 < p< 1$. Further, a measure of the entropy of entanglement is studied for both the schemes.

quant-ph↗

arXiv2019

Active-absorbing phase transition and small world behaviour in Ising model on finite addition type networks in two dimensions

We consider the ordering dynamics of the Ising model on a square lattice where an additional fixed number of bonds connect any two sites chosen randomly. The total number of shortcuts added is controlled by two parameters $p$ and $α$. The structural properties of the network are investigated which show that that the small world behaviour is obtained along the line $α=\frac{\ln (N/2p)}{\ln N}$, which separates regions with ultra small world like behaviour and short ranged lattice like behaviour. We obtain a rich phase diagram in the $p-α$ plane showing the existence of different types of active and absorbing states to which Ising model evolves and their boundaries.

cond-mat.stat-mech↗

arXiv2019

Effect of bias in a reaction diffusion system in two dimensions

We consider a single species reaction diffusion system on a two dimensional lattice where the particles $A$ are biased to move towards their nearest neighbours and annihilate as they meet; $A + A \to \emptyset$. Allowing the bias to take both negative and positive values parametrically, any nonzero bias is seen to drastically affect the behaviour of the system compared to the unbiased (simple diffusive) case. For positive bias, a finite number of dimers, which are isolated pairs of particles occurring as nearest neighbours, exist while for negative bias, a finite density of particles survive. Both the quantities vanish in a power law manner close to the diffusive limit with different exponents. In addition, a discontinuity is observed at the fully positive bias limit. The persistence behaviour is also analysed for the system.

cond-mat.stat-mech↗

arXiv2018

Unusual scaling in a discrete quantum walk with random long range steps

A discrete time quantum walker is considered in one dimension, where at each step, the translation can be more than one unit length chosen randomly. In the simplest case, the probability that the distance travelled is $\ell$ is taken as $P(\ell) = αδ(\ell-1) + (1-α) δ(\ell-2^n)$ with $n \geq 1$. Even the $n=1$ case shows a drastic change in the scaling behaviour for any $α\neq 0,1$. Specifically, $\langle x^2\rangle \propto t^{3/2}$ for $0 < α< 1$, implying the walk is slower compared to the usual quantum walk. This scaling behaviour, which is neither conventional quantum nor classical, can be justified using a simple form for the probability density. The decoherence effect is characterized by two parameters which vanish in a power law manner close to $α=0$ and $1$ with an exponent $\approx 0.5$. It is also shown that randomness is the essential ingredient for the decoherence effect.

quant-ph↗

arXiv2018

Effect of localized loading on failure threshold of fiber bundles

We investigate the global failure threshold of an interconnected set of elements, when a finite fraction of the elements initially share an externally applied load. The study is done under the framework of random fiber bundle model, where the fibers are linear elastic objects attached between two plates. The failure threshold of the system varies non-monotonically with the fraction of the system on which the load is applied initially, provided the load sharing mechanism following a local failure is sufficiently wide. In this case, there exists a finite value for the initial loading fraction, for which the damage on the system will be maximum, or in other words the global failure threshold will be minimum for a finite value of the initial loading fraction. This particular value of initial loading fraction, however, goes to zero when the load sharing is sufficiently local. Such crossover behavior, seen for both one and two dimensional versions of the model, can give very useful information about stability of interconnected systems with random failure thresholds.

cond-mat.stat-mech↗

arXiv2018

Virtual walks in spin space: a study in a family of two-parameter models

We investigate the dynamics of classical spins mapped as walkers in a virtual "spin" space using a generalised two-parameter family of spin models characterized by parameters $y$ and $z$ [M. J. de Oliveira, J. F. F. Mendes and M. A. Santos, J. Phys. A Math. Gen. \textbf{26}, 2317 (1993)]. The behavior of $S(x,t)$, the probability that the walker is at position $x$ at time $t$ is studied in detail. In general $S(x,t) \sim t^{-α}f(x/t^α)$ with $α\simeq 1$ or $0.5$ at large times depending on the parameters. In particular, $S(x,t)$ for the point $y=1, z=0.5$ corresponding to the voter model shows a crossover in time; associated with this crossover, two timescales can be defined which vary with the system size $L$ as $L^2\log L$. We also show that as the voter model point is approached from the disordered regions along different directions, the width of the Gaussian distribution $S(x,t)$ diverges in a power law manner with different exponents. For the majority voter case, the results indicate that the the virtual walk can detect the phase transition perhaps more efficiently compared to other non-equilibrium methods.

cond-mat.stat-mech↗

arXiv2017

Critical noise can make the minority candidate win: The US Presidential Election cases

A national voting population, when segmented into groups like, for example, different states, can yield a counter-intuitive scenario where the winner may not necessarily get the most number of total votes. A recent example is the 2016 presidential election in the US. We model the situation by using interacting opinion dynamics models and look at the effect of coarse graining near the critical points where the spatial fluctuations are high. We establish that the sole effect of coarse graining, which mimics the `winner takes all' electoral college system in the US, can give rise to finite probabilities of such events of minority candidate winning even in the large size limit near the critical point. The overall probabilities of victory of the minority candidate can be predicted from the models which indicates that one may expect more instances of minority candidate winning in the future.

physics.soc-ph↗