Comparison of two models of tethered motion

Abstract

We consider a random walker whose motion is tethered around a focal point. We use two models that exhibit the same spatial dependence in the steady state but widely different dynamics. In one case, the walker is subject to a deterministic bias towards the focal point, while in the other case, it resets its position to the focal point at random times. The deterministic tendency of the biased walker makes the forays away from the focal point more unlikely when compared to the random nature of the returns of the resetting walker. This difference has consequences on the spatio-temporal dynamics at intermediate times. To show the differences in the two models, we analyze their probability distribution and their dynamics in presence and absence of partially or fully absorbing traps. We derive analytically various quantities: (i) mean first-passage times to one target, where we recover results obtained earlier by a different technique, (ii) splitting probabilities to either of two targets as well as survival probabilities when one or either target is partially absorbing. The interplay between confinement, diffusion and absorbing traps produces interesting non-monotonic effects in various quantities, all potentially accessible in experiments. The formalism developed here may have a diverse range of applications, from study of animals roaming within home ranges and of electronic excitations moving in organic crystals to developing efficient search algorithms for locating targets in a crowded environment.

Anderson-like localization transition of random walks with resetting

“…Owing to the fact that resetting events break detailed balance, the probability density of the particle position develops a non-equilibrium steady state (NESS), instead of following the standard Gaussian distribution. The NESSs of diffusive systems with resetting have been characterized in arbitrary spatial dimension [3], for multiplicative processes [4], continuous time random walks [5], or in presence of absorbing boundaries and partially absorbing traps [6]. Furthermore, resetting is interesting for applications to random search problems [1,2], or problems that require the completion of a random computing task [7], since the mean time needed to reach a target state for the first time by a process with resetting is finite and may be minimized with respect to the resetting rate.…”

Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?

“…In part due to their relevance for random searches, there has been in recent years a marked interest for processes with resetting or restart. Processes under resetting are related to a variety of phenomena such as animal foraging [14], genome conversion [15], extinctions in population dynamics [16], problem solving in computer science [17], data search [18,19,20] or catalytic reactions [21,22,23] (see also [24] for a review). Bringing a given process back to its starting point at regular or random time intervals profoundly affects several important static and dynamic observables.…”

Controlling particle currents with evaporation and resetting from an interval

“…Stochastic processes whereby incremental changes are interspersed with sudden and large changes occurring at unpredictable times are common in nature [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Examples range from epidemics (e.g., to financial markets (e.g., the 2008 crisis) and biology (e.g., the catastrophic events of sudden shrinkage during the polymerization of a microtubule [1,3,4], the flashing ratchet mechanism of molecular motors [2], etc.).…”

“…A paradigmatic model for these phenomena is provided by a Brownian particle diffusing and also resetting instantaneously its position to a fixed value at exponentially distributed random time intervals [5]. The concept of stochastic resetting has been invoked in many different fields such as first-passage properties [6,15], continuous-time random walks [16], foraging [7,[17][18][19], reaction-diffusion models [8], fluctuating interfaces [9,20], exclusion processes [21], phase transitions [10], large deviations [22], RNA transcription [11,23], quantum dynamics [24], cellular sensing [12], population dynamics [25], stochastic thermodynamics [26], and active matter [27]; see Ref. [28] for a recent review.…”