Non-Markovian Lévy diffusion in nonhomogeneous media

Area of Science:

• Physics
• Applied Mathematics
• Statistical Mechanics

Background:

• Investigating anomalous diffusion phenomena is crucial for understanding complex systems.
• Fractional calculus and memory kernels offer advanced tools for modeling non-standard diffusion processes.
• The diffusion equation with position-dependent coefficients presents significant analytical challenges.

Purpose of the Study:

• To analyze a diffusion equation featuring a position-dependent, power-law diffusion coefficient.
• To incorporate the Riesz-Weyl fractional operator and memory kernels into the diffusion model.
• To explore the resulting particle distributions and diffusion properties under different kernel conditions.

Main Methods:

• Solving the diffusion equation in the diffusion limit for small wave numbers.
• Analyzing the mathematical properties of the Riesz-Weyl fractional operator.
• Detailed examination of two specific memory kernels: exponential and power-law.
• Introduction of a renormalized fractional moment for comparative analysis.

Main Results:

• The study demonstrates that the resulting distributions conform to the Lévy process for all considered kernels.
• The exponential kernel case simplifies to the well-known telegrapher's equation.
• The power-law kernel case also yields Lévy process distributions, highlighting universality.
• The renormalized fractional moment effectively quantifies and compares diffusion characteristics.

Conclusions:

• The diffusion equation with a position-dependent, power-law diffusion coefficient and fractional operators inherently leads to Lévy process behavior.
• The choice of memory kernel influences the specific form of the diffusion but preserves the Lévy nature of the distributions.
• The introduced renormalized fractional moment provides a robust method for characterizing anomalous diffusion systems.