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Optimization of the Mean First Passage Time in Near-Disk and Elliptical Domains in 2-D with Small Absorbing Traps [article]

Sarafa A. Iyaniwura, Tony Wong, Colin B. Macdonald, Micheal J. Ward
2020 arXiv   pre-print
The determination of the mean first passage time (MFPT) for a Brownian particle in a bounded 2-D domain containing small absorbing traps is a fundamental problem with biophysical applications.  ...  We develop a hybrid asymptotic-numerical approach to predict optimal configurations of m small stationary circular absorbing traps that minimize the average MFPT in near-disk and elliptical domains.  ...  the average mean first capture time of the particle.  ... 
arXiv:2006.12722v1 fatcat:jkwkhuqy3jh7nbfm5nlsdey2fm

Simulation and Optimization of Mean First Passage Time Problems in 2-D using Numerical Embedded Methods and Perturbation Theory [article]

Sarafa Iyaniwura, Tony Wong, Michael J. Ward, Colin B. Macdonald
2019 arXiv   pre-print
We develop novel numerical methods and perturbation approaches to determine the mean first passage time (MFPT) for a Brownian particle to be captured by either small stationary or mobile traps inside a  ...  bounded 2-D confining domain.  ...  The authors thank Justin Tzou for discussions that lead to the time-relaxation algorithm for moving trap problems.  ... 
arXiv:1911.07842v1 fatcat:p7parohoi5go7kwvbcs4mtrple

Global Optimization of the Mean First Passage Time for Narrow Capture Problems in Elliptic Domains [article]

Jason Gilbert, Alexei Cheviakov
2022 arXiv   pre-print
In a recent paper , an explicit size- and trap location-dependent expansion of the average mean first passage time (MFPT) in a 2D elliptic domain was derived.  ...  Further, an asymptotic formula the for the average MFPT in elliptic domains with N circular traps of arbitrary sizes is derived, and sample optimal configurations involving non-equal traps are computed  ...  Acknowledgements A.C. is thankful to NSERC of Canada for research support through the Discovery grant RGPIN-2019-05570.  ... 
arXiv:2101.08368v2 fatcat:b5kyqyhp6rcr5jebxcqtbccq2y

Mean first passage time for a small rotating trap inside a reflective disk [article]

Justin C. Tzou, Theodore Kolokolnikov
2014 arXiv   pre-print
We compute the mean first passage time (MFPT) for a Brownian particle inside a two-dimensional disk with reflective boundaries and a small interior trap that is rotating at a constant angular velocity.  ...  In the regime 1 ≪ω≪O(ε^-1) the optimal trap path approaches the boundary of the disk. However as ω is further increased to O(ε^-1), the optimal trap path "jumps" closer to the origin.  ...  Kolokolnikov is supported by NSERC discovery and NSERC accelerator grants. We thank Michael Ward for useful discussions and suggestions.  ... 
arXiv:1405.2302v2 fatcat:l3a62xxvjbbk3kynxpfnldcpm4

Mathematical modeling and numerical computation of narrow escape problems

Alexei F. Cheviakov, Ashton S. Reimer, Michael J. Ward
2012 Physical Review E  
The narrow escape problem refers to the problem of calculating the mean first passage time (MFPT) needed for an average Brownian particle to leave a domain with an insulating boundary containing N small  ...  This mean first passage time satisfies the Poisson partial differential equation (PDE) subject to a mixed Dirichlet-Neumann boundary condition on the domain boundary, with the Dirichlet condition corresponding  ...  Acknowledgements The authors are grateful to NSERC for research support through Discovery grants (A.C. and M.J.W.) and a USRA fellowship (A.R.).  ... 
doi:10.1103/physreve.85.021131 pmid:22463176 fatcat:ffdxrz3p2vditilh7viuecse7u

Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points

Daniel Coombs, Ronny Straube, Michael Ward
2009 SIAM Journal on Applied Mathematics  
Here, we attack this problem by calculating asymptotic results for the mean first passage time for a diffusing particle confined to the surface of a sphere, in the presence of N partially absorbing traps  ...  The rate at which the small diffusing molecule becomes captured by one of the traps is determined by asymptotically calculating the principal eigenvalue for the Laplace operator on the sphere with small  ...  In the context of the mean first passage time problem, the constant χ in (3.  ... 
doi:10.1137/080733280 fatcat:wzitzqbktvejddbtzlp44fx3ey

Optimization of First Passage Times by Multiple Cooperating Mobile Traps

A. E. Lindsay, J. C. Tzou, T. Kolokolnikov
2017 Multiscale Modeling & simulation  
We study the mean capture time of an unbiased random walker by multiple absorbing mobile traps in bounded domains of one and two spatial dimensions.  ...  In two dimensions, we consider two small traps rotating with the same angular velocity ω inside a unit disk, and characterize the optimal positions (radii of rotation and relative phase) of the two traps  ...  RGPIN-33798 and NSERC Accelerator Grant No. RG-PAS/461907. We gratefully acknowledge the anonymous reviewers whose feedback greatly improved the presentation of this article.  ... 
doi:10.1137/16m1060169 fatcat:imehoidb6bfwjh4mi4usmcyrpu

Mean First Passage Time for a Small Rotating Trap inside a Reflective Disk

J. C. Tzou, T. Kolokolnikov
2015 Multiscale Modeling & simulation  
We compute the mean first passage time (MFPT) for a Brownian particle inside a two-dimensional disk with reflective boundaries and a small interior trap that is rotating at a constant angular velocity.  ...  In Figure 1 (a), we illustrate a scenario in which a trap is located at x = 1/2 on a domain with reflecting boundaries at x = 0 and x = 1.  ...  The authors thank Michael Ward for useful discussions and suggestions.  ... 
doi:10.1137/140968604 fatcat:rftgu25prfhcziiukz6dudklr4

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains

S. Pillay, M. J. Ward, A. Peirce, T. Kolokolnikov
2010 Multiscale Modeling & simulation  
The mean first passage time (MFPT) is calculated for a Brownian particle in a bounded two-dimensional domain that contains N small nonoverlapping absorbing windows on its boundary.  ...  In the asymptotic limit where the absorbing patches have small measure, the method of matched asymptotic expansions is used to calculate the MFPT in an arbitrary two-dimensional domain with a smooth boundary  ...  Cyril Muratov of NJIT for exhibiting the connection between our results and the dilute fraction limit of homogenization theory discussed in section 3.  ... 
doi:10.1137/090752511 fatcat:at2rm5tbmrhnrbznopcxl3clam

Asymptotic Methods For PDE Problems In Fluid Mechanics and Related Systems With Strong Localized Perturbations In Two-Dimensional Domains [chapter]

Michael J. Ward, Mary-Catherine Kropinski
2010 Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances  
bodies, eigenvalue problems in perforated domains, the calculation of the mean first passage time for Brownian motion in a domain with small traps, etc.  ...  Denoting the trajectory of the Brownian particle by X(t), the mean first passage time (MFPT) v(x) is defined as the expectation value of the time τ taken for the Brownian particle to become absorbed somewhere  ...  a cell-signaling problem in mathematical biology (cf. [52] ), the narrow escape problem from a sphere that has small absorbing windows on its boundary, and the mean first passage time in a three-dimensional  ... 
doi:10.1007/978-3-7091-0408-8_2 fatcat:gbxmxmkozvgt5dzmkhl2mcvvu4

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

Alexei F. Cheviakov, Michael J. Ward, Ronny Straube
2010 Multiscale Modeling & simulation  
The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in R 3 that contains N small nonoverlapping absorbing windows, or traps, on its boundary.  ...  Denoting the trajectory of the Brownian particle by X(t), the mean first passage time (MFPT) v(x) is defined as the expectation value of the time τ taken for the Brownian particle to become absorbed somewhere  ...  We would like to thank Sheldon Richards and Professor Raymond Spiteri of the Department of Computer Science at the University of Saskatchewan for providing Table 4 .2.  ... 
doi:10.1137/100782620 fatcat:vxrmr2xtvffu3attqafeqmeoya

Trapping of Planar Brownian Motion: Full First Passage Time Distributions by Kinetic Monte-Carlo, Asymptotic and Boundary Integral Methods [article]

Jake Cherry, Alan E. Lindsay, Adrian Navarro Hernandez, Bryan Quaife
2021 arXiv   pre-print
In contrast to problems posed in finite domains, simple moments of the distribution, such as the mean (MFPT) and variance, are not defined and it is necessary to obtain the full arrival statistics.  ...  To validate the results of this theory, and to obtain the arrival time statistics in very general configurations of absorbers, we introduce an efficient Kinetic Monte Carlo (KMC) method that describes  ...  Preceding works on the theory of first arrival times to small absorbing sites have largely focussed on determining moments such as the mean first passage time (MFPT) [11, 15, 18-20, 33, 34] and in some  ... 
arXiv:2112.06842v1 fatcat:ykesoefvsvby7ob35naopbercm

Time scale of diffusion in molecular and cellular biology

D Holcman, Z Schuss
2014 Journal of Physics A: Mathematical and Theoretical  
The narrow escape time NET in diffusion theory is the mean first passage time (MFPT) of a Brownian trajectory to a small absorbing part of an otherwise reflecting boundary of a bounded domain.  ...  The coarse-grained time scale of cellular function is determined from molecular diffusion by the mean first passage time of molecular Brownian motion to a small targets or through narrow passages.  ... 
doi:10.1088/1751-8113/47/17/173001 fatcat:2jbp55cpvnh6xhpy6rak755eza

Geometrical structure of Laplacian eigenfunctions [article]

Denis S. Grebenkov, Binh-Thanh Nguyen
2013 arXiv   pre-print
We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition.  ...  The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.  ...  The authors thank H. Obuse and A. Barnett for providing their images of eigenfunctions and allowing us to reproduce them. We acknowledge fruitful discussions with Y. G. Sinai and A. L. Delitsyn.  ... 
arXiv:1206.1278v2 fatcat:pmeiikacdrhjxjqjpoh44tvaii

Surveying Diffusion in Complex Geometries. An Essay [article]

Denis Grebenkov
2009 arXiv   pre-print
We look into the role of geometrical complexity at different levels, from boundary microroughness to hierarchical structure and connectivity of the whole diffusion-confining domain.  ...  Here, the geometrical complexity, entangled with the stochastic character of diffusive motion, results in numerous fascinating and sometimes unexpected effects like diffusion screening or localization.  ...  Plapp for careful proofreading of the manuscript and numerous advices.  ... 
arXiv:0909.1588v1 fatcat:2wdehn7m75f3fjfodvfbvn77cu
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