Brownian dynamics simulations for the narrow escape problem in the unit sphere

• Published in: Physical review. E 2021-12, Vol.104 (6-1), p.064113-064113, Article 064113
• Main Authors: Srivastava, Vaibhava, Cheviakov, Alexei
• Format: Article
• Language: English
• ISSN: 2470-0045; 2470-0053
• DOI: 10.1103/PhysRevE.104.064113

The narrow escape problem is a first-passage problem that concerns the calculation of the time needed for a Brownian particle to leave a domain with localized absorbing boundary traps, such that the measure of these traps is asymptotically small compared to the domain size. A common model for the mean first-passage time (MFPT) as a function of particle's starting location in a given domain with constant diffusivity is given by a Poisson partial differential equation subject to mixed Dirichlet-Neumann boundary conditions. The primary objective of this work is to perform direct numerical simulations of multiple particles undergoing Brownian motion in a three-dimensional spherical domain with boundary traps, compute MFPT values by averaging Brownian escape times, and compare these with explicit asymptotic results obtained previously by approximate solution of the Poisson problem. A close agreement of MFPT values is observed already at 10^{4} particle runs from a single starting point, providing a computational validation of the Poisson equation-based continuum model. Direct Brownian dynamics simulations are also used to study additional features of particle dynamics in narrow escape problems that cannot be captured in a continuum approach, such as average times spent by particles in a thin layer near the domain boundary, and effects of isotropic vs anisotropic near-boundary diffusion.