Abstract
A particle moves with Brownian motion in a unit disc with reflection from the boundaries except for a portion (called a “window” or “gate”) in which it is absorbed. The main problems are to determine the first hitting time and spatial distribution. A closed formula for the mean first hitting time is discovered and proven for a gate of any size. Also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed. Numerical simulations of the stochastic process with finite step size and a sufficient number of sample paths are compared with the exact solution to the Brownian motion (the limit of zero step size), providing an empirical formula for the difference. Histograms of first hitting times are also generated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, X., Friedman, A.: Asymptotic Analysis for the Narrow Escape Problem, Preprint.
Cheviakov, A.F., Ward, M.J., Straube, R.: An asymptotic analysis of the mean first passage time for narrow escape problems. Part II: the sphere. SIAM Multiscale Model. Simul. (to appear)
Grigoriev I.V., Makhnovskii Y.A., Berezhkovskii A.M., Zitserman V.Y.: Kinetics of escape through a small hole. J. Chem. Phys. 116, 9574–9577 (2002)
Holcman D., Schuss Z.: Escape through a small opening: receptor trafficking in a synaptic membrane. J. Stat. Phys. 117, 975–1014 (2004)
Holcman D., Schuss Z.: Diffusion escape through a cluster of small absorbing windows. J. Phys. A: Math. Theory 41, 155001 (2008)
Pillay, S., Ward, M.J., Peirce, A., Kolokolnikov, T.: An asymptotic analysis of the mean first passage time for narrow escape problems. Part I: two dimensional domains. SIAM Multiscale Model. Simul. (to appear)
Redner, S.: A Guide to First Passage Time Processes. Cambridge University Press, 2001
Schuss Z.: Theory and Applications of Stochastic Differential Equations: An Analytical Approach. Springer, New York (2010)
Singer A., Schuss Z., Holcman D.: Narrow escape Part II: the circular disk. J. Stat. Phys. 122, 465–489 (2006)
Zwanzig Z.: A rate process with an entropy barrier. J. Chem. Phys. 94, 6147–6152 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Dafermos
Rights and permissions
About this article
Cite this article
Caginalp, C., Chen, X. Analytical and Numerical Results for an Escape Problem. Arch Rational Mech Anal 203, 329–342 (2012). https://doi.org/10.1007/s00205-011-0455-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-011-0455-6