Stochastic resetting on comblike structures

Abstract

We study a diffusion process on a three-dimensional comb under stochastic resetting. We consider three different types of resetting: global resetting from any point in the comb to the initial position, resetting from a finger to the corresponding backbone and resetting from secondary fingers to the main fingers. The transient dynamics along the backbone in all three cases is different due to the different resetting mechanisms, finding a wide range of dynamics for the mean squared displacement. For the particular geometry studied herein, we compute the stationary solution and the mean square displacement and find that the global resetting breaks the transport in the three directions. Regarding the resetting to the backbone, the transport is broken in two directions but it is enhanced in the main axis. Finally, the resetting to the fingers enhances the transport in the backbone and the main fingers but reaches a steady value for the mean squared displacement in the secondary fingers.

Diffusion–Advection Equations on a Comb: Resetting and Random Search

“…To analyze the diffusion dynamics numerically, we use a system of Langevin equations in the presence of drift and stochastic resetting to the initial position [33,44]:…”

“…It was shown that the solution for the PDF approaches a non-equilibrium steady state and, in the long-time limit, its MSD is saturated, x 2 (t) ∼ 1/r (also see the review paper [31] for more details). Moreover, Brownian motion in a two-dimensional comb in the presence of stochastic (Markovian) resetting can be solved analytically [32][33][34]. The marginal PDFs along both the backbone and fingers approach non-equilibrium steady states, and the MSDs are saturated according to the resetting rate: x 2 (t) ∼ 1/ √ r and y 2 (t) ∼ 1/r [32][33][34].…”

“…Moreover, Brownian motion in a two-dimensional comb in the presence of stochastic (Markovian) resetting can be solved analytically [32][33][34]. The marginal PDFs along both the backbone and fingers approach non-equilibrium steady states, and the MSDs are saturated according to the resetting rate: x 2 (t) ∼ 1/ √ r and y 2 (t) ∼ 1/r [32][33][34]. These models have been extended to diffusion processes with non-static resetting [34].…”

First passage in discrete-time absorbing Markov chains under stochastic resetting

“…Resetting can produce a counterintuitive effect: it renders an infinite mean first passage time (MFPT) finite, which can be also minimized at a specific resetting rate. Some extensions have been made in the field, such as temporally or spatially dependent resetting rate [13][14][15], higher dimensions [16,17], complex geometries [18][19][20], noninstantaneous resetting [21][22][23][24], in the presence of external potential [25][26][27], or in the presence of multiple targets [28][29][30], other types of Brownian motion, like runto-tumble particles [31][32][33], active particles [34,35], and so on [36]. These nontrivial findings have triggered enormous recent activities in the field, including statistical physics [37][38][39][40][41][42], stochastic thermodynamics [43][44][45], and single-particle experiments [46,47].…”

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

“…Nonetheless, a prevalent real world observation conforms that selfreproduction is characterized with a stationary distribution that has power law tails, which hinders the practical implementation of the model [8]. A natural way to invoke stationarity is to adapt GBM with a stochastic resetting mechanism which has recently spurred extensive research interests in statistical physics [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], stochastic processes [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] and in single particle experiments [42,43].…”