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Langevin equation with fluctuating diffusivity: A two-state model.

Tomoshige Miyaguchi1, Takuma Akimoto2, Eiji Yamamoto2

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Summary
This summary is machine-generated.

This study models fluctuating diffusivity using a Langevin equation. It reveals that while ensemble-averaged mean-square displacement shows subdiffusion, time-averaged measures can reveal aging and long-time correlations, offering insights into anomalous diffusion.

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Area of Science:

  • Statistical Mechanics
  • Soft Matter Physics
  • Physical Chemistry

Background:

  • Single-particle tracking experiments often report anomalous subdiffusion, aging, and scattered diffusion coefficients.
  • The underlying physical origins of these observed anomalous diffusion behaviors remain largely unexplained.

Purpose of the Study:

  • To investigate a theoretical model for anomalous diffusion phenomena using a Langevin equation with fluctuating diffusivity.
  • To explore how a dichotomous stochastic process for diffusivity impacts diffusion measurements.
  • To analyze the behavior of mean-square displacement (MSD) and propagator under different ensemble conditions.

Main Methods:

  • Modeling diffusivity as a dichotomous stochastic process with power-law sojourn times.
  • Analyzing ensemble-averaged and time-averaged mean-square displacement (MSD).
  • Developing a two-state renewal theory for analytical calculations.

Main Results:

  • Non-equilibrium ensembles exhibit transient subdiffusion in ensemble-averaged MSD.
  • Time-averaged MSD shows normal diffusion, but the effective diffusion coefficient displays transient aging.
  • The propagator is non-Gaussian at short times, converging slowly to a Gaussian distribution.
  • For equilibrium ensembles, standard MSDs do not reveal diffusivity fluctuations.
  • Relative standard deviation (RSD) of time-averaged MSD serves as a signature of long-time correlations and relates to the non-Gaussian parameter.

Conclusions:

  • Fluctuating diffusivity can explain anomalous diffusion behaviors observed in experiments.
  • The relative standard deviation (RSD) of the time-averaged MSD is a crucial indicator for detecting diffusivity fluctuations and long-time correlations.
  • The developed two-state renewal theory provides a robust analytical framework for studying such complex diffusion dynamics.