Power law diffusion coefficient and anomalous diffusion: analysis of solutions and first passage time

Area of Science:

• Physics
• Physical Chemistry
• Mathematical Physics

Background:

• Diffusion processes are fundamental in various scientific fields.
• Understanding anomalous diffusion requires analyzing non-constant diffusion coefficients.
• Quartic potentials introduce complex dynamics in physical systems.

Purpose of the Study:

• To investigate one-dimensional diffusion equations with non-constant diffusion coefficients.
• To compare the behaviors of diffusion coefficients placed inside vs. between derivatives.
• To analyze the mean first passage time for different diffusion equation configurations.

Main Methods:

• Solving one-dimensional diffusion equations with D(x) proportional to |x|^(-theta).
• Employing a quartic potential to introduce system complexity.
• Analyzing and comparing results from two distinct mathematical approaches.
• Calculating and evaluating the mean first passage time.

Main Results:

• Diffusion equations with non-constant coefficients exhibit diverse behaviors.
• The position of the diffusion coefficient (inside vs. between derivatives) alters system dynamics.
• Significant differences in mean first passage times were observed based on coefficient placement.
• The choice of D(x) and potential influences the diffusion regimes.

Conclusions:

• The placement of non-constant diffusion coefficients critically affects diffusion dynamics.
• Mean first passage time analysis reveals distinct system responses.
• This research provides insights into anomalous diffusion modeling.