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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.