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

Area of Science:

• Physics
• Applied Mathematics
• Computational Science

Background:

• The narrow escape problem models particle diffusion in domains with small absorbing regions.
• Mean first-passage time (MFPT) is often calculated using Poisson partial differential equations with mixed boundary conditions.

Purpose of the Study:

• To perform direct numerical simulations of Brownian motion in a 3D spherical domain with boundary traps.
• To compute MFPT values and compare them with asymptotic results from the Poisson equation model.
• To investigate phenomena not captured by continuum models, like near-boundary diffusion.

Main Methods:

• Direct numerical simulations of multiple Brownian particles in a 3D spherical domain.
• Averaging Brownian escape times to compute MFPT.
• Comparison of simulation results with analytical solutions of the Poisson equation.

Main Results:

• Close agreement between simulated MFPT values and theoretical predictions was observed.
• Computational validation of the Poisson equation-based continuum model was achieved with 10^4 particle runs.
• Insights into particle dynamics near boundaries, including time spent in boundary layers and diffusion anisotropy, were obtained.

Conclusions:

• Direct Brownian dynamics simulations provide robust validation for continuum models of the narrow escape problem.
• The study confirms the accuracy of the Poisson equation approach for MFPT calculations.
• Simulation methods offer a pathway to explore complex near-boundary dynamics beyond continuum approximations.