Infinite densities for Lévy walks

Area of Science:

• Physics
• Statistical Mechanics
• Complex Systems

Background:

• Particle motion often mixes diffusive and ballistic behaviors, leading to strong anomalous diffusion.
• Existing diffusion equations and central limit theorems struggle to model these mixed dynamics comprehensively.

Purpose of the Study:

• To resolve the theoretical challenge of describing systems with mixed diffusive-ballistic particle motion.
• To introduce and utilize the concept of infinite density for modeling anomalous diffusion.

Main Methods:

• Employed the Lévy walk model, a widely applicable framework for anomalous diffusion.
• Derived a general expression for non-normalized density based on velocity distribution and diffusion parameters.

Main Results:

• Developed a method using infinite densities to unify the description of diffusive and ballistic scaling in particle motion.
• Established that the derived density is determined by particle velocity, anomalous diffusion exponent (α), and diffusion coefficient (K(α)).

Conclusions:

• Infinite densities provide a robust theoretical framework for understanding complex particle dynamics.
• The study demonstrates how infinite densities can be evaluated from experimental or numerical data for diverse physical processes.