Diffusion equations for a Markovian jumping process

Area of Science:

• Statistical Physics
• Stochastic Processes
• Anomalous Diffusion

Background:

• Markovian jumping processes are fundamental in modeling systems with random transitions.
• Understanding diffusion dynamics, especially anomalous diffusion, is crucial in various scientific fields.
• Position-dependent frequencies and power-law behavior introduce complexity to standard diffusion models.

Purpose of the Study:

• To investigate the diffusion limit of a Markovian jumping process with position-dependent frequency and power-law form.
• To derive and analyze the Fokker-Planck and fractional equations governing normal, sub-, and superdiffusion.
• To define and calculate a fractional diffusion coefficient for systems with divergent variance.

Main Methods:

• Derivation of the Fokker-Planck equation for small steps.
• Construction and solution of a fractional equation for Lévy distributed step sizes.
• Numerical solution of the master equation.
• Calculation of fractional moments.

Main Results:

• Demonstration of normal diffusion, subdiffusion, and superdiffusion phenomena.
• Solution of a fractional equation with a variable coefficient in the diffusion limit.
• Definition and calculation of a fractional diffusion coefficient.
• Observation of deviations from Lévy stable distribution for large wave numbers in numerical simulations.

Conclusions:

• The study successfully characterizes different diffusion regimes (normal, sub-, super-) within a generalized Markovian jumping process.
• Fractional equations provide a suitable framework for analyzing anomalous diffusion, particularly with Lévy-type step sizes.
• The proposed fractional diffusion coefficient offers a robust measure for systems exhibiting divergent variances.
• Numerical results highlight the limitations of stable Lévy distributions in describing complex stochastic processes at higher wave numbers.