Mathematical modeling and numerical computation of narrow escape problems.

Area of Science:

• Statistical Physics
• Mathematical Physics
• Physical Chemistry

Background:

• The narrow escape problem studies the mean first passage time (MFPT) for a Brownian particle to exit a domain through small absorbing windows (traps).
• This problem is modeled using the Poisson partial differential equation with mixed Dirichlet-Neumann boundary conditions.

Purpose of the Study:

• To develop and validate an asymptotic theory for calculating the MFPT in two-dimensional domains and on a unit sphere.
• To investigate the optimization of trap locations for minimizing the MFPT.
• To assess the accuracy of the asymptotic theory under various conditions, including finite trap sizes and proximity.

Main Methods:

• Development of a common asymptotic theory for MFPT in the limit of small total trap size.
• Application of the theory to two-dimensional domains and the unit sphere.
• Comparison of asymptotic results with full numerical simulations.
• Numerical computation of optimal trap configurations on a sphere.

Main Results:

• Asymptotic MFPT formulas were derived, dependent on trap locations, enabling global optimization.
• The asymptotic theory demonstrated high accuracy (within 1%) compared to numerical simulations, even for moderately small total trap sizes and closely spaced traps.
• Optimal trap configurations on a unit sphere were determined to minimize average MFPT, considering identical and varied trap sizes.

Conclusions:

• The developed asymptotic theory provides a robust and accurate method for calculating MFPT in narrow escape problems.
• The theory's accuracy extends beyond asymptotically small trap sizes, offering practical benefits.
• Insights into trap configuration optimization can be gained, relevant for designing systems with controlled escape dynamics.