Calculation of the mean first passage time tested on simple two-dimensional models

Area of Science:

• Statistical Physics
• Physical Chemistry
• Computational Physics

Background:

• Particle diffusion in confined geometries is crucial for understanding molecular transport.
• Analyzing diffusion in two-dimensional (2D) systems presents unique challenges compared to one-dimensional (1D) systems.
• The Fick-Jacobs equation is a common tool for simplifying diffusion problems in complex spaces.

Purpose of the Study:

• To calculate the mean first passage time for a diffusing particle in a 2D channel.
• To evaluate the accuracy of the 1D Fick-Jacobs equation and its modified versions for this 2D diffusion problem.
• To compare the results from the simplified 1D models with the full 2D diffusion dynamics.

Main Methods:

• Modeling particle diffusion in a 2D container representing two passing circular particles.
• Deriving 1D Fick-Jacobs equations by mapping the 2D diffusion onto a longitudinal coordinate.
• Calculating mean first passage time and comparing it with the hopping time (inverse of the lowest eigenvalue of the 2D problem).

Main Results:

• The 1D Fick-Jacobs equation and its modification provide reliable estimations for mean first passage time in the 2D channel.
• These simplified models show good agreement with results derived from the full 2D diffusion equation.
• The accuracy of the mapped equations contrasts with the predictions of basic transition state theory.

Conclusions:

• The Fick-Jacobs equation, even in its simplified 1D form, is a valuable tool for analyzing diffusion in complex 2D geometries.
• This approach offers a computationally efficient alternative to solving full 2D diffusion equations for first passage time calculations.
• The study highlights limitations of the simplest transition state theory in accurately describing diffusion dynamics in such systems.