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Fractional diffusion-reaction stochastic simulations.

Basil S Bayati¹

¹Intellectual Ventures Laboratory, 1600 132nd Ave. NE, Bellevue, Washington 98004, USA. bbayati@intven.com

The Journal of Chemical Physics

|March 22, 2013

Summary

This study introduces a new simulation method for fractional diffusion in Markov processes. The novel approach demonstrates that fractional diffusion can accelerate wave propagation compared to classical diffusion models.

Area of Science:

Computational Physics
Mathematical Modeling
Stochastic Processes

Background:

Classical diffusion models often fail to capture complex physical phenomena.
Fractional diffusion offers a more generalized framework for describing anomalous diffusion processes.
Accurate simulation methods are crucial for understanding systems exhibiting fractional diffusion.

Purpose of the Study:

To develop and validate a novel computational method for simulating discrete state space, continuous time Markov processes under fractional diffusion.
To investigate the impact of fractional diffusion on the dynamics of phenomena like the Fisher-KPP wavefront.
To establish the necessity of fractional diffusion methods for accurate simulations in various physical processes.

Main Methods:

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Implementation of a Lie-Trotter operator splitting technique to decouple diffusion and reaction terms in the master equation.

Development of a diffusion kernel based on the discretized solution of the fractional diffusion equation.

Validation of the algorithm using simulations of the Fisher-KPP wavefront propagation.

Main Results:

The proposed method successfully simulates Markov processes with fractional diffusion.
Simulations of the Fisher-KPP wavefront show that wave speed is dependent on the order of the fractional derivative.
Lower orders of the fractional derivative lead to faster wave propagation compared to classical diffusion.

Conclusions:

Fractional diffusion modeling is essential for accurately simulating physical processes that deviate from classical diffusion behavior.
The developed simulation method provides a valuable tool for studying systems governed by fractional diffusion.
The findings highlight the significant influence of fractional dynamics on wave propagation phenomena.