Analytical expressions for the first passage time distribution and hit distribution in two and three dimensions

Clarkson, Alexander, Lam, Chi-Hang and Deng, Hai-Yao ORCID: https://orcid.org/0000-0001-6065-483X 2024. Analytical expressions for the first passage time distribution and hit distribution in two and three dimensions. American Journal of Physics 92 (4) , 299–307. 10.1119/5.0121165

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Abstract

The distribution of the time elapsed before a random variable reaches a threshold value for the first time, called the first passage time (FPT) distribution, is a fundamental characteristic of stochastic processes. Here, by solving the standard macroscopic diffusion equation, we derive analytical expressions for the FPT distribution of a diffusing particle hitting a spherical object in two dimensions (2D) and three dimensions (3D) in the course of unrestricted diffusion in open space. In addition, we calculate, analytically, the angular dependence of the FPT, known as the hit distribution. The analytical results are also compared to simulations of the motions of a random walker on a discrete lattice. This topic could be of wide pedagogical interest because the FPT is important not only in physics but also in chemistry, biology, medicine, agriculture, engineering, and finance. Additionally, the central equations often appear in physics and engineering with only trivial variations, making the solution techniques widely applicable.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Physics and Astronomy
Publisher:	American Association of Physics Teachers
ISSN:	0002-9505
Date of First Compliant Deposit:	21 March 2024
Date of Acceptance:	12 January 2024
Last Modified:	26 Mar 2024 10:00
URI:	https://orca.cardiff.ac.uk/id/eprint/167431

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