Diffusion in a Crowded, Rearranging Environment.

Area of Science:

• Physics
• Statistical Mechanics
• Soft Matter

Background:

• Brownian motion in rearranging environments exhibits Fickian mean squared displacement (MSD) but non-Gaussian displacement probability distributions.
• This deviation from Gaussian behavior is attributed to time-dependent, or diffusing, diffusivity.
• Analytical solutions for diffusing diffusivity models exist only for short times, necessitating simulations for longer timescales.

Purpose of the Study:

• To develop analytical solutions for diffusing diffusivity models applicable across all timescales.
• To investigate the connection between diffusing diffusivity and Brownian motion with a sink.
• To characterize the transition of displacement probability distributions from short-time to long-time regimes.

Main Methods:

• Equivalence established between a diffusing diffusivity model and Brownian motion in the presence of a sink.
• Development of a new class of diffusing diffusivity models amenable to analytical solutions.
• Derivation of analytical expressions for MSD and displacement probability distributions.

Main Results:

• The mean squared displacement ⟨(x(T))(2)⟩ is shown to be proportional to time (T) for all times, confirming Fickian behavior.
• The displacement probability distribution function transitions from exponential at short times to Gaussian at long times and large displacements.
• Analytical solutions are provided for both MSD and probability distributions.

Conclusions:

• The study provides a unified analytical framework for understanding Brownian motion with diffusing diffusivity.
• The observed crossover in displacement distributions offers a new perspective on anomalous diffusion phenomena.
• The findings are relevant for systems with time-varying diffusion coefficients, such as in complex fluids or biological environments.