Correction to: Asymptotic Formulas for Extreme Statistics of Escape Times in 1, 2 and 3-Dimensions
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Correction to: Asymptotic Formulas for Extreme Statistics of Escape Times in 1, 2 and 3-Dimensions K. Basnayake1 · Z. Schuss2 · D. Holcman1 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Correction to: Journal of Nonlinear Science (2019) 29:461–499 https://doi.org/10.1007/s00332-018-9493-7 In section 5.1 entitled “The shortest NEP from a bounded domain in R2,3 ”” for dimension 3, we found an error in Eq. 97 of page 20. This leads to the conclusion that also in ¯ (1) three dimensions, √ the mean time for the fastest τ depends on the reciprocal of log N and not on 1/ log N , as written prviously. Here are the corrected computations: τ
(3dim)
∞
=
[Pr {t1 > t}] N dt ⎧ ⎫ ⎛ 1 2 ⎞⎬ ∞ ⎨ 2 δ − a − √ ⎠ dt 4t ≈ ex p Nlog ⎝1 − √ t 2 e ⎩ ⎭ δ π 0 ⎧ ⎫ 1 2 ∞ ⎨ a 2 − − √δ ⎬ 4t ex p −N √ t 2 e ≈ dt ⎩ ⎭ δ π 0 δ2 ∞ 1 − u1 du ≈ ex p −N √ e 4 0 u 0
where N = variable,
2N a 2 √ . π δ2
(97)
(98)
Using the method developed in section 3.1 with the change of
Zeev Schuss passed away in July 2018. The original article can be found online at https://doi.org/10.1007/s00332-018-9493-7.
B
D. Holcman [email protected]
1
Applied Mathematics and Computational Biology, Ecole Normale Supérieure, 46 rue d’Ulm, 75005 Paris, France
2
Department of Applied Mathematics, Tel-Aviv University, 69978 Tel Aviv, Israel
123
Journal of Nonlinear Science Fig. 3 b MFPT of the fastest particles versus the number of particles N . The asymptotic solution (green curve) log(Nα )+β fitted the stochastic simulations, obtained with 2000 runs
1 1 1 w = w(t) = √ e−1/t , w (t) = √ e− t t t
1 1 − + 2 . 2t 4t
(99)
We have with w = 4w(log(w))2 τ¯
(3dim)
∞ ≈δ
2
exp(−N w) du. 4w(log(w))2
0
We then follow the computation of section 3.1 “Escape from a Ray”. When the diffusion coefficient is D, we obtain the formula τ¯ 3dim ≈
δ2 4Dlog 2N
.
a2
(100)
π 1/2 δ 2
When there are p windows, whose distances from the initial position of the Brownian particles are dk = dist(P0 , Pk ), formula (100) changes to τ¯ 3 ≈
δ2 4Dlog 2N
a2
,
(101)
π 1/2 δ 2
where δ 2 = min(d12 , ...d 2p ). The asymptotic formula (100) is compared with results of Brownian simulations and shows very good agreement (Fig. 3). Fig.3b of the main manuscript has to be replaced by Fig. 3, which fits the dependency of τ¯ (3dim) with respect to the total number of particles N . Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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