Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous...

Area of Science:

• Physics
• Physical Chemistry
• Biophysics

Background:

• Anomalous diffusion is often modeled using scaled Brownian motion (SBM), a Gaussian process.
• SBM features a power-law time-dependent diffusion coefficient, with mean squared displacement ⟨x²(t)⟩ ≃ 2t⁽ᵃ⁻¹⁾ for 0 < α < 2.
• In unbounded systems, SBM's probability density function aligns with fractional Brownian motion.

Purpose of the Study:

• To critically evaluate the applicability and limitations of scaled Brownian motion (SBM) in describing anomalous diffusion.
• To investigate the ergodicity and stationarity of SBM under both free and confined conditions.
• To compare SBM dynamics with fractional Brownian motion and continuous-time random walks, particularly in confined environments.

Main Methods:

• Theoretical analysis of scaled Brownian motion (SBM).
• Investigation of SBM's ergodicity and stationarity properties.
• Comparative analysis of SBM with fractional Brownian motion and continuous-time random walks.

Main Results:

• Free scaled Brownian motion (SBM) is weakly non-ergodic but shows limited time-averaged mean squared displacement scatter.
• Under confinement, SBM dynamics significantly diverge from fractional Brownian motion and continuous-time random walks.
• Scaled Brownian motion (SBM) exhibits high non-stationarity, rendering it unsuitable for thermalized stationary systems.

Conclusions:

• Scaled Brownian motion (SBM) is not a universally applicable model for anomalous diffusion, especially in confined systems.
• The non-stationary nature of SBM limits its use in describing particles within thermalized, stationary environments.
• Findings impact single particle tracking models, particularly for intracellular transport and optical tweezer experiments.