Universal Formula for the Mean First Passage Time in Planar Domains

Area of Science:

• Mathematical Physics
• Statistical Mechanics
• Physical Chemistry

Background:

• Mean First Passage Time (MFPT) is crucial for understanding diffusion processes.
• Existing models often simplify complex geometries and diffusion coefficients.
• Accurate MFPT calculations are vital for chemical kinetics and reaction rates.

Purpose of the Study:

• To derive a general, exact formula for MFPT in planar domains.
• To investigate the role of the escape region's properties on MFPT.
• To provide a new perspective on diffusion escape problems.

Main Methods:

• Solving a mixed Dirichlet-Neumann boundary value problem.
• Utilizing conformal mapping to transform the domain to a unit disk.
• Exact analytical solution and back-transformation to the original domain.

Main Results:

• A general exact formula for MFPT with arbitrary space-dependent diffusion coefficients.
• An explicit, universal leading logarithmic term for MFPT.
• Identification of harmonic measure as the key parameter, not perimeter.
• Altered MFPT scaling near the escape region.

Conclusions:

• The study offers a novel framework for analyzing diffusion escape problems.
• Findings challenge conventional scaling laws and highlight the importance of harmonic measure.
• The results have implications for understanding kinetics in complex, multiscale systems.