Diffusion in an expanding medium: Fokker-Planck equation, Green's function, and first-passage properties

Area of Science:

• Statistical physics
• Complex systems

Background:

• The Fokker-Planck equation describes diffusion processes.
• Understanding diffusion in dynamic, expanding environments is crucial for various scientific fields.

Purpose of the Study:

• To derive the Fokker-Planck equation for diffusion in an expanding medium from first principles.
• To analyze the behavior of diffusion, its moments, and first-passage properties in such systems.

Main Methods:

• Classical, mesoscopic derivation starting from a generalized Chapman-Kolmogorov equation.
• Analytical and numerical investigation of the Green's function (propagator) and its moments.
• Analysis of first-passage time properties in expanding hyperspherical geometries.

Main Results:

• An analytical expression for the Green's function was obtained.
• The system's behavior is governed by Brownian conformal time (τ).
• Crossover effects in diffusion mixing and first-passage properties were identified for power-law expansion, particularly around γ=1/2.

Conclusions:

• Brownian conformal time is a key determinant of diffusion behavior in expanding media.
• The nature of medium expansion significantly influences diffusion dynamics and mixing.
• A stationary probability distribution emerges in contracting media with exponential scale factors.