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Anomalous diffusion and the first passage time problem

Rangarajan1, Ding

  • 1Department of Mathematics and Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012, India.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|November 23, 2000
PubMed
Summary
This summary is machine-generated.

This study analyzes first passage time distributions in anomalous diffusion using a fractional Fokker-Planck equation. We provide new analytical expressions for various drift and barrier conditions, generalizing Brownian motion results.

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Area of Science:

  • Physics
  • Physical Chemistry
  • Statistical Mechanics

Background:

  • Anomalous diffusion describes systems where particle movement deviates from standard Brownian motion.
  • First passage time (FPT) distributions are crucial for understanding transport phenomena in complex systems.
  • Levy type diffusion is a key model for anomalous transport, characterized by long-tailed waiting time distributions.

Purpose of the Study:

  • To derive explicit and analytical expressions for the first passage time distribution in Levy type anomalous diffusion.
  • To extend existing theories of diffusion to more complex scenarios, including non-zero drift and multiple barriers.
  • To provide a unified framework that encompasses ordinary diffusion (Brownian motion) as a special case.

Main Methods:

  • Application of the recently formulated fractional Fokker-Planck equation.
  • Derivation of explicit FPT distribution expressions using Fox or H functions for zero drift.
  • Obtaining analytical Laplace transform expressions for the FPT distribution in non-zero drift cases.
  • Expressing the FPT distribution as a power series for systems with two absorbing barriers.

Main Results:

  • An explicit expression for the FPT distribution in terms of Fox or H functions for zero drift Levy diffusion.
  • An analytical expression for the Laplace transform of the FPT distribution for non-zero drift Levy diffusion.
  • A power series representation of the FPT distribution for Levy diffusion with two absorbing barriers.
  • Demonstration that standard Brownian motion results are recovered as special cases.

Conclusions:

  • The fractional Fokker-Planck equation provides a powerful tool for analyzing FPT distributions in anomalous diffusion.
  • The derived results offer a more general and comprehensive understanding of transport phenomena in complex systems.
  • This work bridges the gap between anomalous diffusion models and the well-established theory of Brownian motion.