Distribution of time-averaged observables for weak ergodicity breaking

Area of Science:

• Statistical Mechanics
• Complex Systems
• Stochastic Processes

Background:

• Subdiffusive continuous time random walks (CTRWs) are crucial for modeling anomalous transport in various complex systems.
• Understanding the statistical properties of time-averaged observables in these systems is essential but challenging.
• Existing frameworks often struggle with nonergodic behavior inherent in anomalous diffusion.

Purpose of the Study:

• To develop a general formula for the distribution of time-averaged observables in subdiffusive CTRWS.
• To establish a statistical mechanical framework for anomalous subdiffusion.
• To analyze the behavior of time-averaged particle position and its fluctuations.

Main Methods:

• Derivation of a general formula for time-averaged observables.
• Application of Lévy's generalized central limit theorem for anomalous diffusion.
• Analysis of Gaussian random walks coupled to a thermal bath.
• Calculation of the distribution of the time-averaged position (X) for unbiased and biased particles.

Main Results:

• A general formula for time-averaged observables in subdiffusive CTRWS was established.
• Ergodicity and Boltzmann statistics were recovered for Gaussian random walks.
• A weakly nonergodic statistical mechanical framework was constructed for anomalous subdiffusion.
• Large fluctuations of the time-averaged position (X) compared to the ensemble average () were demonstrated.

Conclusions:

• The developed framework provides a unified approach to time-averaged observables in subdiffusive systems.
• The study highlights the nonergodic nature of anomalous diffusion and its impact on statistical properties.
• The findings offer insights into particle transport and fluctuations in complex systems.