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Ultraslow convergence to ergodicity in transient subdiffusion
Tomoshige Miyaguchi1, Takuma Akimoto
1Department of Applied Physics, Osaka City University, Osaka, Japan. tomo@a-phys.eng.osaka-cu.ac.jp
Summary
We studied continuous time random walks with truncated α-stable trapping times. Results show time-averaged observables follow the Mittag-Leffler distribution, indicating slow convergence to ordinary ergodicity.
Area of Science:
- Physics
- Mathematics
- Statistical Mechanics
Background:
- Continuous time random walks (CTRWs) are fundamental models in statistical physics.
- α-stable distributions and Mittag-Leffler functions are key in describing anomalous diffusion and long-tail waiting times.
Purpose of the Study:
- To investigate the ergodic properties of CTRWs with truncated α-stable trapping times.
- To analyze the behavior of time-averaged observables under these conditions.
Main Methods:
- Mathematical analysis of continuous time random walks.
- Derivation of the probability density function for time-averaged observables.
- Investigation of convergence properties to ergodicity.
Main Results:
- Distributional ergodicity is proven for a class of observables.
- Time-averaged observables are shown to follow the Mittag-Leffler distribution.
- A crossover from distributional to ordinary ergodicity is observed, with slower convergence.
Conclusions:
- Truncated α-stable trapping times lead to long-lasting distributional ergodic behavior.
- The convergence to ordinary ergodicity is significantly slower compared to standard trapping time distributions.
- The Mittag-Leffler distribution characterizes the anomalous ergodic behavior in these systems.
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