A Probabilistic Approach to Extreme Statistics of Brownian Escape Times in Dimensions 1, 2, and 3

Lawley, Sean D.; Madrid, Jacob B.

doi:10.1007/s00332-019-09605-9

Cited by 49 publications

(53 citation statements)

References 65 publications

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“…The recent interest in extreme FPTs of diffusion was sparked by the pioneering work in [14], wherein the authors used formal analysis to derive E[T 1,N ] ∼ L 2 /(4D ln N ) for pure diffusion in a class of 2-dimensional domains with small targets. Their work also found that E[T 1,N ] decays like 1/ √ ln N in 3-dimensional domains, which was later corrected for convex domains in [39]. In fact, the correct 3-dimensional result for small targets was first formally derived in [34].…”

Section: Discussionmentioning

confidence: 93%

Universal formula for extreme first passage statistics of diffusion

Lawley

2020

Phys. Rev. E

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The timescales of many physical, chemical, and biological processes are determined by first passage times (FPTs) of diffusion. The overwhelming majority of FPT research studies the time it takes a single diffusive searcher to find a target. However, the more relevant quantity in many systems is the time it takes the fastest searcher to find a target from a large group of searchers. This fastest FPT depends on extremely rare events and has a drastically faster timescale than the FPT of a given single searcher. In this paper, we prove a simple explicit formula for every moment of the fastest FPT. The formula is remarkably universal, as it holds for d-dimensional diffusion processes (i) with general space-dependent diffusivities and force fields, (ii) on any smooth Riemannian manifold, (iii) in the presence of reflecting obstacles, and (iv) with partially absorbing targets. Our results rigorously confirm, generalize, correct, and unify various conjectures and heuristics about the fastest FPT.S l (t) ≈ 1 − e −l 2 /t for t 1.

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Section: Discussionmentioning

confidence: 93%

Universal formula for extreme first passage statistics of diffusion

Lawley

2020

Phys. Rev. E

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“…The impact of such an extreme event on the FRTs and their PDF has been rather extensively discussed within the recent years, and several analytical analyses have been proposed (see, e.g. [84,[91][92][93][94] and references therein). An extension of such analyses for the geometrical setting studied here should be of interest.…”

Section: Discussionmentioning

confidence: 99%

Distribution of first-reaction times with target regions on boundaries of shell-like domains

Metzler

Oshanin

2021

New J. Phys.

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We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted "onion-shell" geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.

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“…Statistics of extreme FPTs of diffusion in one dimensional or spherically symmetric domains were further studied in [32,47,36,48,49,21]. Recently, approximate formulas for the moments of extreme FPTs of diffusion in more general two and three dimensional domains were derived in [50,12,33]. Even more recently, it was proven in significant generality that the mth moment of the kth fastest FPT has the large N behavior,…”

Section: Discussionmentioning

confidence: 99%

“…where T 1,N := T N . While the distribution and statistics of a single FPT, τ 1 , are well understood in a variety of scenarios [26,27,28,29,30], studying the so-called extreme FPTs, T k,N , is notoriously difficult, both analytically and numerically [31,32,12,11,33,34]. The essential difficulty is that extreme FPTs depend on very rare events.…”

Section: Introductionmentioning

confidence: 99%

Distribution of extreme first passage times of diffusion

Lawley

2020

J. Math. Biol.

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Many events in biology are triggered when a diffusing searcher finds a target, which is called a first passage time (FPT). The overwhelming majority of FPT studies have analyzed the time it takes a single searcher to find a target. However, the more relevant timescale in many biological systems is the time it takes the fastest searcher(s) out of many searchers to find a target, which is called an extreme FPT. In this paper, we apply extreme value theory to find a tractable approximation for the full probability distribution of extreme FPTs of diffusion. This approximation can be easily applied in many diverse scenarios, as it depends on only a few properties of the short time behavior of the survival probability of a single FPT. We find this distribution by proving that a careful rescaling of extreme FPTs converges in distribution as the number of searchers grows. This limiting distribution is a type of Gumbel distribution and involves the LambertW function. This analysis yields new explicit formulas for approximations of statistics of extreme FPTs (mean, variance, moments, etc.) which are highly accurate and are accompanied by rigorous error estimates.

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A Probabilistic Approach to Extreme Statistics of Brownian Escape Times in Dimensions 1, 2, and 3

Cited by 49 publications

References 65 publications

Universal formula for extreme first passage statistics of diffusion

Universal formula for extreme first passage statistics of diffusion

Distribution of first-reaction times with target regions on boundaries of shell-like domains

Distribution of extreme first passage times of diffusion

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