Brownian motion with alternately fluctuating diffusivity: Stretched-exponential and power-law relaxation

Area of Science:

• Physics
• Statistical Mechanics
• Physical Chemistry

Background:

• Brownian motion is fundamental to understanding particle dynamics in complex systems.
• Fluctuating diffusivity can arise in various physical and biological environments.
• Characterizing relaxation dynamics in such systems is crucial for many applications.

Purpose of the Study:

• To develop a theoretical framework for analyzing Brownian motion with alternating diffusivity.
• To investigate the influence of sojourn-time distributions on relaxation functions.
• To connect theoretical predictions with observable quantities like correlation functions.

Main Methods:

• Development of an alternating renewal process theory.
• Analysis of a relaxation function dependent on integrated diffusivity.
• Relating the relaxation function to position correlation and self-intermediate scattering functions.
• Comparison of theoretical results with numerical simulations.

Main Results:

• Short-time dynamics exhibit exponential or stretched-exponential relaxation, dependent on sojourn-time power-law indices.
• Long-time dynamics show a power-law decay with an exponential cutoff.
• The influence of initial ensemble conditions (equilibrium vs. nonequilibrium) on relaxation is clarified.

Conclusions:

• The study provides a comprehensive theoretical model for fluctuating diffusivity systems.
• Distinct relaxation behaviors are predicted based on the statistical properties of state switching.
• The findings offer insights into particle dynamics in complex environments with time-dependent properties.