Survival probability and order statistics of diffusion on disordered media

Area of Science:

• Physics
• Mathematics
• Materials Science

Background:

• Understanding random walker behavior is crucial in disordered media.
• First passage time statistics are key to diffusion processes.
• Order statistics of random walkers in complex environments present analytical challenges.

Purpose of the Study:

• To investigate the first passage time for multiple random walkers in disordered media.
• To develop and validate an analytical method for predicting these times.
• To compare the efficacy of chemical versus Euclidean distance metrics.

Main Methods:

• Assumed survival probability decay for disordered media mirrors that of Euclidean and fractal lattices.
• Utilized simulations on a 2D incipient percolation aggregate.
• Expressed arbitrary moments of first passage time using asymptotic series.

Main Results:

• Derived an asymptotic series for first passage time moments, applicable to disordered, Euclidean, and fractal lattices.
• Simulations on a 2D percolation aggregate showed good agreement with the derived series.
• The model performed better when using chemical distance compared to Euclidean distance.

Conclusions:

• The analytical approach provides a robust framework for understanding random walker dynamics in disordered systems.
• Chemical distance is a more accurate metric than Euclidean distance for these types of diffusion problems.
• The findings have implications for transport phenomena in complex materials.