Ergodicity testing for anomalous diffusion: small sample statistics.

Area of Science:

• Statistical physics
• Complex systems analysis
• Non-equilibrium thermodynamics

Background:

• Experimental trajectory analysis often relies on time averages, which are equivalent to ensemble averages only if the system is ergodic.
• Ergodicity is a fundamental assumption in statistical mechanics, linking microscopic dynamics to macroscopic observables.
• Anomalous diffusion processes challenge traditional assumptions, necessitating methods to assess their ergodic behavior.

Purpose of the Study:

• To develop and implement computational tools for assessing ergodic properties in diffusion models.
• To investigate the ergodicity of fractional Brownian motion and subordinated Ornstein-Uhlenbeck processes.
• To establish a universal methodology for analyzing ergodicity in experimental trajectory data, even with limited samples.

Main Methods:

• Implementation of statistical tools for ergodicity analysis.
• Application of rigorous statistical methods including mean square displacement (MSD) and confidence intervals.
• Utilization of the dynamical functional test to probe system dynamics.

Main Results:

• Fractional Brownian motion was identified as an ergodic process.
• The subordinated Ornstein-Uhlenbeck process was found to be non-ergodic.
• Demonstrated the effectiveness of statistical methods in distinguishing ergodic from non-ergodic diffusion.

Conclusions:

• The developed methodology provides a robust framework for evaluating ergodicity in complex systems.
• The findings highlight the distinct ergodic behaviors of different anomalous diffusion models.
• The approach is versatile, applicable to diverse experimental datasets, including those with few recorded trajectories.