Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions.

Area of Science:

• Physics
• Biophysics
• Statistical Mechanics

Background:

• Fractional Brownian motion (fBm) models anomalous diffusion with a constant Hurst exponent.
• Single-particle tracking in cells shows complex diffusion beyond standard fBm.
• Existing models fail to capture trajectory-to-trajectory variations in diffusion behavior.

Purpose of the Study:

• To develop a mathematical framework for fractional Brownian motion with a randomly varying Hurst exponent.
• To analyze the diffusion and correlation properties of this generalized process.
• To explain complex anomalous diffusion observed in biological systems.

Main Methods:

• Developed a general mathematical framework for analytical, numerical, and statistical analysis.
• Derived explicit formulas for probability density function, mean-squared displacement, and autocovariance.
• Investigated three Hurst exponent distributions: two-point, uniform, and beta.

Main Results:

• Demonstrated accelerating diffusion and persistence transitions.
• Provided analytical and numerical evidence for these phenomena.
• Characterized the impact of random Hurst exponent variations on diffusion properties.

Conclusions:

• Fractional Brownian motion with a random Hurst exponent provides a more realistic model for anomalous diffusion in biological systems.
• The framework allows for detailed analysis of complex diffusion dynamics.
• The findings explain previously unexplained diffusion behaviors in single-particle tracking experiments.