Subordinated diffusion and continuous time random walk asymptotics.

Area of Science:

• Physics
• Mathematical Physics
• Computational Physics

Background:

• Anomalous transport phenomena are typically modeled using continuous time random walks (CTRWs) or fractional Fokker-Planck equations (FFPEs).
• Existing literature establishes asymptotic relationships between scaled CTRWs and fractional diffusion processes.
• Accurate approximation of anomalous diffusion processes remains a key challenge in theoretical and computational physics.

Purpose of the Study:

• To investigate and compare two distinct numerical methods for approximating anomalous diffusion processes.
• To establish a correspondence between CTRWs and time and space fractional diffusion equations.
• To evaluate the numerical performance and accuracy of these methods against known analytical solutions.

Main Methods:

• Monte Carlo simulation of uncoupled CTRW with Lévy α-stable spatial jumps and Mittag-Leffler waiting times.
• Discretization of a subordinated Langevin equation where physical time is a random variable.
• Numerical testing and verification using the Green function of a space-time fractional diffusion equation.

Main Results:

• A trade-off exists between solution precision and computational cost for both methods.
• The subordinated Langevin equation method yields higher accuracy in approximating anomalous diffusion.
• CTRW with Mittag-Leffler waiting times efficiently approximates FFPE solutions and converges to the subordinated process's probability density function in the long-time limit.

Conclusions:

• Both CTRW and subordinated Langevin equation methods provide valuable frameworks for studying anomalous transport.
• The choice of method depends on the desired balance between accuracy and computational efficiency.
• Further research can explore the nuances of these models for complex transport systems.