First-passage times to anisotropic partially reactive targets

Area of Science:

• Mathematical Physics
• Physical Chemistry
• Biophysics

Background:

• Restricted diffusion in bounded domains is crucial for understanding reaction dynamics.
• Partially reactive targets in complex geometries present significant analytical challenges.

Purpose of the Study:

• To propose an explicit approximation for the principal eigenvalue of the Laplace operator with mixed boundary conditions.
• To determine the mean first-reaction time, survival probability decay, and reaction rates.
• To analyze the influence of target shape and anisotropy on diffusion-controlled reactions.

Main Methods:

• Developing an explicit approximation involving harmonic capacity, surface area, domain volume, diffusion coefficient, and reactivity.
• Validating the approximation using the finite-elements method.
• Computing harmonic capacities for prolate and oblate spheroids in various dimensions.

Main Results:

• The approximation accurately predicts reaction dynamics, including mean first-reaction time and overall reaction rate.
• A key lengthscale of the target was identified, determining its trapping capacity.
• Target anisotropy significantly affects the principal eigenvalue, with calculations performed for different spheroid shapes.

Conclusions:

• The proposed approximation offers a powerful tool for studying restricted diffusion and reaction kinetics in complex systems.
• Understanding target shape effects is vital for predicting reaction outcomes in chemical physics and biophysics.
• The findings provide insights into diffusion-controlled processes relevant to biological and chemical systems.