arXiv subjects

Arnab Pal

Publications and source records attributed to Arnab Pal.

At least 55 records · Page 3Linked to original sources

arXiv2023

Microscopic Theory of Adsorption Kinetics

Adsorption is the accumulation of a solute at an interface that is formed between a solution and an additional gas, liquid, or solid phase. The macroscopic theory of adsorption dates back more than a century and is now well-established. Yet, despite recent advancements, a detailed and self-contained theory of single-particle adsorption is still lacking. Here, we bridge this gap by developing a microscopic theory of adsorption kinetics, from which the macroscopic properties follow directly. One of our central achievements is the derivation of the microsopic version of the seminal Ward-Tordai relation which connects the surface and subsurface adsorbate concentrations via a universal equation that holds for arbitrary adsorption dynamics. Furthermore, we present a microscopic interpretation of the Ward-Tordai relation which, in turn, allows us to generalize it to arbitrary dimension, geometry and initial conditions. The power of our approach is showcased on a set of hitherto unsolved adsorption problems to which we present exact analytical solutions. The framework developed herein sheds fresh light on the fundamentals of adsorption kinetics, which opens new research avenues in surface science with applications to artificial and biological sensing and to the design of nano-scale devices.

cond-mat.stat-mech↗

arXiv2022

The fate of the American dream: A first passage under resetting approach to income dynamics

Detailed knowledge of individual income dynamics is crucial for investigating the existence of the American dream: Are we able to improve our income status during our working life? This key question simply boils down to observing individual status and how it moves between two thresholds: the current income and the desired income. Yet, our knowledge of these temporal properties of income remains limited since we rely on estimates coming from transition matrices which simplify income dynamics by aggregating the individual changes into quantiles and thus overlooking significant microscopic variations. Here, we bridge this gap by employing First Passage Time concepts in a baseline stochastic process with resetting used for modeling income dynamics and developing a framework that is able to crucially disaggregate the temporal properties of income to the level of an individual worker. We find analytically and illustrate numerically that our framework is orthogonal to the transition matrix approach and leads to improved and more granular estimates. Moreover, to facilitate empirical applications of the framework, we introduce a publicly available statistical methodology, and showcase the application using the USA income dynamics data. These results help to improve our understanding on the temporal properties of income in real economies and provide a set of tools for designing policy interventions.

econ.GN↗

arXiv2022

Fick-Jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting

Transport of particles through channels is of paramount importance in physics, chemistry and surface science due to its broad real world applications. Much insights can be gained by observing the transition paths of a particle through a channel and collecting statistics on the lifetimes in the channel or the escape probabilities from the channel. In this paper, we consider the diffusive transport through a narrow conical channel of a Brownian particle subject to intermittent dynamics, namely, stochastic resetting. As such, resetting brings the particle back to a desired location from where it resumes its diffusive phase. To this end, we extend the Fick-Jacobs theory of channel-facilitated diffusive transport to resetting-induced transport. Exact expressions for the conditional mean first passage times, escape probabilities and the total average lifetime in the channel are obtained, and their behaviour as a function of the resetting rate are highlighted. It is shown that resetting can expedite the transport through the channel -- rigorous constraints for such conditions are then illustrated. Furthermore, we observe that a carefully chosen resetting rate can render the average lifetime of the particle inside the channel minimal. Interestingly, the optimal rate undergoes continuous and discontinuous transitions as some relevant system parameters are varied. The validity of our one-dimensional analysis and the corresponding theoretical predictions are supported by three-dimensional Brownian dynamics simulations. We thus believe that resetting can be useful to facilitate particle transport across biological membranes -- a phenomena that can spearhead further theoretical and experimental studies.

cond-mat.stat-mech↗

arXiv2022

Chiral run-and-tumble walker: transport and optimizing search

We study the statistical properties of a non-Markovian chiral run-and-tumble particle (CRTP) in two dimensions in continuous space and time. In our model, the possible orientations of the particle correspond to the four cardinal directions. The particle can reorient by turning left, right or reversing its direction of motion at different rates. We show how chirality manifests itself in the transport properties like the spatial moments of the marginal position distribution and the first-passage properties of a CRTP. Interestingly, we find that the chirality leads to enhanced diffusion and a \textit{looping} tendency in the trajectory space. Furthermore, our results show that chirality plays a pivotal role in the improvement of the search strategy -- notably, there exists an optimal bias in tumbling that minimizes the mean search time. This key observation can play a crucial role in determining how living systems efficiently search under non-equilibrium conditions.

cond-mat.stat-mech↗

arXiv2022

Resetting mediated navigation of active Brownian searcher in a homogeneous topography

Designing navigation strategies for search time optimization remains of interest in various interdisciplinary branches in science. In here, we focus on microscopic self-propelled searchers namely active Brownian walkers in noisy and confined environment which are mediated by one such autonomous strategy namely resetting. As such, resetting stops the motion and compels the walkers to restart from the initial configuration intermittently according to an external timer that does not require control by the walkers. In particular, the resetting coordinates are either quenched (fixed) or annealed (fluctuating) over the entire topography. Although the strategy relies upon simple rules, it shows a significant ramification on the search time statistics in contrast to the original search. We show that the resetting driven protocols mitigate the performance of these active searchers based, robustly, on the inherent search time fluctuations. Notably, for the annealed condition, resetting is always found to expedite the search process. These features, as well as their applicability to more general optimization problems starting from queuing systems, computer science to living systems, make resetting based strategies universally promising.

cond-mat.soft↗

arXiv2022

First-passage Brownian functionals with stochastic resetting

We study the statistical properties of first-passage time functionals of a one dimensional Brownian motion in the presence of stochastic resetting. A first-passage functional is defined as $V=\int_0^{t_f} Z[x(τ)]$ where $t_f$ is the first-passage time of a reset Brownian process $x(τ)$, i.e., the first time the process crosses zero. In here, the particle is reset to $x_R>0$ at a constant rate $r$ starting from $x_0>0$ and we focus on the following functionals: (i) local time $T_{loc} = \int _0^{t_f}d τ~ δ(x-x_R)$, (ii) residence time $T_{res} = \int _0^{t_f} d τ~θ(x-x_R)$, and (iii) functionals of the form $A_n = \int _{0}^{t_f} d τ[x(τ)]^n $ with $n >-2$. For first two functionals, we analytically derive the exact expressions for the moments and distributions. Interestingly, the residence time moments reach minima at some optimal resetting rates. A similar phenomena is also observed for the moments of the functional $A_n$. Finally, we show that the distribution of $A_n$ for large $A_n$ decays exponentially as $\sim \text{exp}\left( -A_n/a_n\right)$ for all values of $n$ and the corresponding decay length $a_n$ is also estimated. In particular, exact distribution for the first passage time under resetting (which corresponds to the $n=0$ case) is derived and shown to be exponential at large time limit in accordance with the generic observation. This behavioural drift from the underlying process can be understood as a ramification due to the resetting mechanism which curtails the undesired long Brownian first passage trajectories and leads to an accelerated completion. We confirm our results to high precision by numerical simulations.

cond-mat.stat-mech↗

arXiv2021

The Pólya and Sisyphus lattice random walks with resetting -- a first passage under restart approach

We revisit the simple lattice random walk (Pólya walk) and the Sisyphus random walk in $\mathbb{Z}$, in the presence of random restarts. We use a relatively direct approach namely First passage under restart for discrete space and time which was recently developed by us in PRE 103, 052129 (2021) and rederive the first passage properties of these walks under the memoryless geometric restart mechanism. Subsequently, we show how our method could be generalized to arbitrary first passage process subject to more complex restart mechanisms such as sharp, Poisson and Zeta distribution where the latter is heavy tailed. We emphasize that the method is very useful to treat such problems both analytically and numerically.

cond-mat.stat-mech↗

arXiv2021

Mitigating long queues and waiting times with service resetting

What determines the average length of a queue which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations induced delays are in high demand as queue structures appear in various natural and man-made systems. Here we demonstrate that a simple service resetting mechanism can reverse the deleterious effects of large fluctuations in service times, thus turning a marked drawback into a favourable advantage. This happens when stochastic fluctuations are intrinsic to the server, and we show that the added feature of service resetting can then dramatically cut down average queue lengths and waiting times. While the analysis presented herein is based on the M/G/1 queueing model where service is general but arrivals are assumed to be Markovian, Kingman's formula asserts that the benefits of service resetting will carry over to queues with general arrivals. We thus expect results coming from this work to find widespread application to queueing systems ranging from telecommunications, via computing, and all the way to molecular queues that emerge in enzymatic and metabolic cycles of living organisms.

cond-mat.stat-mech↗

arXiv2021

Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity

We explore the role of non-ergodicity in the relationship between income inequality, the extent of concentration in the income distribution, and mobility, the feasibility of an individual to change their position in the income distribution. For this purpose, we explore the properties of an established model for income growth that includes "resetting" as a stabilising force which ensures stationary dynamics. We find that the dynamics of inequality is regime-dependent and may range from a strictly non-ergodic state where this phenomenon has an increasing trend, up to a stable regime where inequality is steady and the system efficiently mimics ergodic behaviour. Mobility measures, conversely, are always stable over time, but the stationary value is dependent on the regime, suggesting that economies become less mobile in non-ergodic regimes. By fitting the model to empirical data for the dynamics of income share of the top earners in the United States, we provide evidence that the income dynamics in this country is consistently in a regime in which non-ergodicity characterises inequality and immobility dynamics. Our results can serve as a simple rationale for the observed real world income dynamics and as such aid in addressing non-ergodicity in various empirical settings across the globe.

econ.GN↗

arXiv2021

Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process

We study the effects of stochastic resetting on geometric Brownian motion (GBM), a canonical stochastic multiplicative process for non-stationary and non-ergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains non-ergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the non-ergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.

q-fin.RM↗

arXiv2021

Thermodynamic uncertainty relations for many-body systems with fast jump rates and large occupancies

A universal large $\mathcal{N}$ theory of nonequilibrium fluctuations emerges in the limit of fast jump rates and large occupancies. We use this theory to derive a set of coarse grained thermodynamic uncertainty relations (TUR) -- one of them being an activity bound. Importantly, the activity serves as a tighter bound for the entropy production in 1D systems. These results are particularly useful in the many-body regime, where typically a coarse grained approach is required to handle the large microscopic state space.

cond-mat.stat-mech↗

arXiv2021

The inspection paradox in stochastic resetting

The remaining travel time of a plane shortens with every minute that passes from its departure, and a flame diminishes a candle with every second it burns. Such everyday occurrences bias us to think that processes which have already begun will end before those which have just started. Yet, the inspection paradox teaches us that the converse can also happen when randomness is at play. The paradox comes from probability theory, where it is often illustrated by measuring how long passengers wait upon arriving at a bus stop at a random time. Interestingly, such passengers may on average wait longer than the mean time between bus arrivals -- a counter-intuitive result, since one expects to wait less when coming some time after the previous bus departed. In this \textit{viewpoint}, we review the inspection paradox and its origins. The insight gained is then used to explain why, in some situations, stochastic resetting expedites the completion of random processes. Importantly, this is done with elementary mathematical tools which help develop a probabilistic intuition for stochastic resetting and how it works. This viewpoint can thus be used as an accessible introduction to the subject.

cond-mat.stat-mech↗

arXiv2021

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution in the level of a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift-diffusion processes are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. The current study provides an important baseline that unifies two different approaches, used ubiquitously in economics and physics, for studying the ergodic properties in diffusion with resetting. We believe that our results can be verified in single particle experimental set-ups and thus have strong implications in the field of resetting.

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

Thermodynamic uncertainty relation for first-passage times on Markov chains

We derive a thermodynamic uncertainty relation (TUR) for first-passage times (FPTs) on continuous time Markov chains. The TUR utilizes the entropy production coming from bidirectional transitions, and the net flux coming from unidirectional transitions, to provide a lower bound on FPT fluctuations. As every bidirectional transition can also be seen as a pair of separate unidirectional ones, our approach typically yields an ensemble of TURs. The tightest bound on FPT fluctuations can then be obtained from this ensemble by a simple and physically motivated optimization procedure. The results presented herein are valid for arbitrary initial conditions, out-of-equilibrium dynamics, and are therefore well suited to describe the inherently irreversible first-passage event. They can thus be readily applied to a myriad of first-passage problems that arise across a wide range of disciplines.

cond-mat.stat-mech↗

arXiv2021

Extremal statistics for stochastic resetting systems

While averages and typical fluctuations often play a major role to understand the behavior of a non-equilibrium system, this nonetheless is not always true. Rare events and large fluctuations are also pivotal when a thorough analysis of the system is being done. In this context, the statistics of extreme fluctuations in contrast to the average plays an important role, as has been discussed in fields ranging from statistical and mathematical physics to climate, finance and ecology. Herein, we study Extreme Value Statistics (EVS) of stochastic resetting systems which have recently gained lot of interests due to its ubiquitous and enriching applications in physics, chemistry, queuing theory, search processes and computer science. We present a detailed analysis for the finite and large time statistics of extremals (maximum and arg-maximum i.e., the time when the maximum is reached) of the spatial displacement in such system. In particular, we derive an exact renewal formula that relates the joint distribution of maximum and arg-maximum of the reset process to the statistical measures of the underlying process. Benchmarking our results for the maximum of a reset-trajectory that pertain to the Gumbel class for large sample size, we show that the arg-maximum density attains to a uniform distribution regardless of the underlying process at a large observation time. This emerges as a manifestation of the renewal property of the resetting mechanism. The results are augmented with a wide spectrum of Markov and non-Markov stochastic processes under resetting namely simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process and random acceleration process in one dimension. Rigorous results are presented for the first two set-ups while the latter two are supported with heuristic and numerical analysis.

cond-mat.stat-mech↗

arXiv2021

First passage under restart for discrete space and time: application to one dimensional confined lattice random walks

First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to outperform the completion of many first passage processes which otherwise would take longer time to finish. However, most of the studies so far assumed continuous time underlying first passage time processes and moreover considered continuous time resetting restricting out restart processes broken up into synchronized time steps. To bridge this gap, in this paper, we study discrete space and time first passage processes under discrete time resetting in a general set-up. We sketch out the steps to compute the moments and the probability density function which is often intractable in the continuous time restarted process. A criterion that dictates when restart remains beneficial is then derived. We apply our results to a symmetric and a biased random walker (RW) in one dimensional lattice confined within two absorbing boundaries. Numerical simulations are found to be in excellent agreement with the theoretical results. Our method can be useful to understand the effect of restart on the spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions.

cond-mat.stat-mech↗