Optimal mean first-passage time of a Brownian searcher with resetting in one and two dimensions: Experiments, theory and numerical tests

Faisant, Felix; Besga, Benjamin; Petrosyan, Artyom; Ciliberto, Sergio; Majumdar, Satya N.

doi:10.48550/arxiv.2106.09113

Cited by 8 publications

(12 citation statements)

References 44 publications

(99 reference statements)

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“…Developments on the experimental realisation of stochastic resetting (using optical techniques) have been reported in [30][31][32], with extension to two dimensions and investigation of periodic resetting protocols. An experimental realisation of our resetting prescription for a random walker on a ring could be provided by a one-armed mechanical clock, suspended vertically.…”

Section: Discussionmentioning

confidence: 99%

Winding number of a Brownian particle on a ring under stochastic resetting

Grange¹

2022

J. Phys. A: Math. Theor.

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We consider a random walker on a ring, subjected to resetting at Poisson-distributed times to the initial position (the walker takes the shortest path along the ring to the initial position at resetting times). In the case of a Brownian random walker the mean first-completion time of a turn is expressed in closed form as a function of the resetting rate. The value is shorter than in the ordinary process if the resetting rate is low enough. Moreover, the mean first-completion time of a turn can be minimised in the resetting rate. At large time the distribution of winding numbers does not reach a steady state, which is in contrast with the non-compact case of a Brownian particle under resetting on the real line. The mean total number of turns and the variance of the net number of turns grow linearly with time, with a proportionality constant equal to the inverse of the mean first-completion time of a turn.

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Section: Discussionmentioning

confidence: 99%

Winding number of a Brownian particle on a ring under stochastic resetting

Grange¹

2022

J. Phys. A: Math. Theor.

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“…In order to confirm our prediction, we visualize the temporal evolution of the observables x 2 (t) and C ∆ (t) for a fixed choice of parameters. See 43) and ( 44) into (42), we obtain the following expression for the TAMSD in the long time limit…”

Section: Ergodicity Of Sample Mean Under Stochastic Resettingmentioning

confidence: 99%

“…This volume of work spans different fields starting from non-equilibrium transport properties [6][7][8][9][10][11][12][13][14][15][16], first-passage [17][18][19][20][21][22][23][24][25][26][27][28] and search theory [29][30][31][32], stochastic thermodynamics [33][34][35][36], extreme value statistics [37,38] and stochastic resonance [39]. Along with the rapid progress in the theoretical front, resetting has also become an active playground for several experimental groups which have led to new understanding [40][41][42]. We refer to [43] by Evans et al which provides an overview of the current state of the problem.…”

Section: Introductionmentioning

confidence: 99%

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Stojkoski,

Sandev,

Kocarev

et al. 2021

Preprint

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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-squareddisplacement (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.

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“…The phenomena has been catalyzed even further since over the last decade, resetting has found overreaching applications in statistical physics [43][44][45][46][47][48][49][50][51][52], computer science [53,54], ecology [55][56][57], complex systems [58][59][60] operation research [61][62][63] and economics [64][65][66]. Recently, the field has also seen advancements in experiments [67][68][69]. A paradigm model in the field is the Brownian motion with stochastic resetting for which many interesting results exist [47,[70][71][72][73][74][75][76].…”

Section: Introductionmentioning

confidence: 99%

First-passage Brownian functionals with stochastic resetting

Singh,

Pal

2022

Preprint

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We study the statistical properties of first-passage time functionals of a one dimensional Brownian motion in the presence of stochastic resetting. A firstpassage functional is defined aswhere 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:, and (iii) functionals of the form A n = ´tf 0 dτ [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 ∼ exp (−A n /a n ) 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.

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Optimal mean first-passage time of a Brownian searcher with resetting in one and two dimensions: Experiments, theory and numerical tests

Cited by 8 publications

References 44 publications

Winding number of a Brownian particle on a ring under stochastic resetting

Winding number of a Brownian particle on a ring under stochastic resetting

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

First-passage Brownian functionals with stochastic resetting

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