Mean first passage time for a particle diffusing on a disk with two absorbing traps at the boundary.

Area of Science:

• Physics
• Statistical Mechanics
• Mathematical Physics

Background:

• Brownian motion describes random particle movement.
• Understanding particle diffusion on surfaces with boundaries is crucial in various scientific fields.
• Previous studies often relied on simulations or approximations for complex boundary conditions.

Purpose of the Study:

• To analytically determine the survival probability and mean first passage time of a Brownian particle on a disk with a reflective boundary and two absorbing arcs.
• To develop a method for calculating effective trapping rates on complex boundaries.
• To provide a simplified analytical expression for specific limiting cases.

Main Methods:

• Application of boundary homogenization framework to calculate effective trapping rates.
• Utilizing conformal mapping to simplify the geometric configuration of the problem.
• Derivation of analytical expressions for mean first passage time.

Main Results:

• An analytical solution for the mean first passage time was derived.
• The method of boundary homogenization was successfully applied to estimate effective trapping rates.
• Simplified expressions were obtained for cases with small absorbing arcs or gaps.

Conclusions:

• The derived analytical expressions for mean first passage time are accurate and compare well with Brownian particle simulations and existing literature.
• The study provides a robust analytical framework for analyzing diffusion on surfaces with complex boundaries.
• The conformal mapping technique offers an efficient way to solve such problems.