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Zero constant formula for first-passage observables in bounded domains.

O Bénichou¹, B Meyer, V Tejedor

¹Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France.

Physical Review Letters

|October 15, 2008

Summary

This study introduces an analytical method to precisely calculate mean first-passage times (MFPTs) for random walks in complex systems. The approach accurately predicts transport processes on fractals and can be extended to other related measures.

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Area of Science:

Physics
Physical Chemistry
Applied Mathematics

Background:

Random walks are fundamental models for transport phenomena in diverse systems.
Calculating mean first-passage times (MFPTs) is crucial for understanding system dynamics but challenging in complex geometries.
Existing methods often lack explicit solutions for intricate transport processes.

Purpose of the Study:

To develop a novel analytical approach for the explicit determination of MFPTs.
To derive precise MFPT expressions for transport on complex media, including fractals.
To provide a generalized framework for first-passage observables.

Main Methods:

Development of an analytical method based on scale-invariance hypothesis.
Application of a large volume expansion for MFPT approximation.
Validation through numerical simulations on fractal and bounded domains.

Main Results:

Explicit expressions for MFPTs derived for the first time for transport on deterministic and random fractals.
The analytical approach demonstrates high accuracy, even for small system sizes.
The method is shown to be generalizable to other first-passage observables.

Conclusions:

The developed analytical approach offers a powerful tool for quantifying transport in complex systems.
This work provides foundational results for understanding diffusion and related processes on fractal structures.
The generalized framework facilitates further studies on first-passage phenomena in various scientific disciplines.