Fractional diffusion equation for an n-dimensional correlated Lévy walk

Area of Science:

• Physics
• Mathematics
• Statistical Mechanics

Background:

• Lévy walks are crucial for modeling superdiffusion, exceeding Brownian motion's speed.
• Previous research lacked a diffusion equation for n-dimensional correlated Lévy walks.

Purpose of the Study:

• Derive a fractional diffusion equation for correlated Lévy walks.
• Analyze superdiffusive dynamics in the large time limit.
• Identify experimentally measurable dynamical mechanisms.

Main Methods:

• Utilized a fractional Klein-Kramers equation as the starting point.
• Employed a moment method combined with a Cattaneo approximation.
• Solved the derived fractional diffusion equation.

Main Results:

• Successfully derived a fractional diffusion equation for short-range auto-correlated Lévy walks.
• The equation is valid for superdiffusive processes in the large time limit.
• Identified distinct dynamical mechanisms contributing to correlated Lévy walk diffusion.

Conclusions:

• The derived equation provides a framework for understanding correlated Lévy walk diffusion.
• The findings offer experimentally verifiable insights into superdiffusive phenomena.
• This work advances the theoretical understanding of anomalous diffusion models.