Scaled Brownian motion as a mean-field model for continuous-time random walks

Area of Science:

• Statistical Physics
• Stochastic Processes
• Physical Chemistry

Background:

• Scaled Brownian motion (sBm) models diffusion with a time-dependent coefficient, D(t)=αD0tα-1.
• This model is frequently applied to experimental subdiffusion data where the underlying mechanism is unclear.
• Subdiffusion, characterized by anomalous diffusion exponents (α<1), is prevalent in various complex systems.

Purpose of the Study:

• To elucidate the relationship between scaled Brownian motion and subdiffusive continuous-time random walks.
• To analyze the properties of sBm, particularly its nonstationary and nonergodic characteristics.
• To investigate the ergodicity breaking phenomenon in sBm and its dependence on trajectory length.

Main Methods:

• Theoretical analysis of the scaled Brownian motion process.
• Comparison of sBm with subdiffusive continuous-time random walk models.
• Mathematical derivation of the heterogeneity parameter for sBm.

Main Results:

• Scaled Brownian motion is identified as a close relative of subdiffusive continuous-time random walks.
• sBm accurately describes the rescaled mean position of independent, diffusing particles.
• The heterogeneity parameter of sBm approaches zero for long trajectories, suggesting ergodicity is restored over time.

Conclusions:

• Scaled Brownian motion shares nonstationary and nonergodic properties with subdiffusive continuous-time random walks.
• Despite its nonergodicity, sBm exhibits limited variation between different realizations for extended time scales.
• The findings provide a deeper understanding of subdiffusion modeling and its implications in complex systems.