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Generalized diffusion: a microscopic approach.

James F Lutsko¹, Jean Pierre Boon

¹Physics Department, Université Libre de Bruxelles, Code Postal 231, 1050-Bruxelles, Belgium. jlutsko@ulb.ac.be

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 23, 2008

Summary

This summary is machine-generated.

This study derives a generalized Fokker-Planck equation for nonlinear jump probabilities in random walks. Anomalous diffusion is possible with power-law nonlinearities, reducing the equation to the porous media equation, confirmed by simulations.

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Area of Science:

Statistical Physics
Nonlinear Dynamics
Computational Physics

Background:

The classical Fokker-Planck equation describes random walker dynamics.
Nonlinear jump probabilities can lead to complex diffusion behaviors.

Purpose of the Study:

Derive a generalized Fokker-Planck equation for nonlinear jump probabilities.
Investigate conditions for anomalous diffusion and scaling solutions.
Analyze the role of drift and external fields.

Main Methods:

Theoretical derivation of a generalized Fokker-Planck equation.
Analysis of nonlinear jump probability functions.
Comparison with the porous media equation.
Monte Carlo simulations for validation.

Main Results:

A generalized Fokker-Planck equation accounting for nonlinear jump probabilities was derived.
Anomalous diffusion and scaling solutions emerge under power-law nonlinearities.
The generalized equation reduces to the porous media equation in specific cases.
Simulations confirmed theoretical predictions.

Conclusions:

Nonlinear jump probabilities significantly alter random walk dynamics.
Power-law nonlinearities are crucial for anomalous diffusion described by the porous media equation.
The derived generalized Fokker-Planck equation provides a framework for studying complex diffusion phenomena.