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Role of infinite invariant measure in deterministic subdiffusion
Takuma Akimoto1, Tomoshige Miyaguchi
1Department of Mechanical Engineering, Keio University, Yokohama 223-8522, Japan. akimoto@z8.keio.jp
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
Summary
This study reveals that the diffusion coefficient in deterministic subdiffusion is governed by infinite ergodic theory. The Mittag-Leffler distribution characterizes the diffusion coefficient as a random variable, offering new insights into transport processes.
Area of Science:
- Statistical physics
- Ergodic theory
- Transport phenomena
Background:
- Subdiffusion describes anomalous transport where particles spread slower than Brownian motion.
- Understanding the statistical properties of transport coefficients is crucial for modeling complex systems.
- Infinite ergodic theory provides a framework for analyzing systems with long-term memory and non-ergodic behavior.
Purpose of the Study:
- To investigate the statistical properties of transport coefficients in deterministic subdiffusion.
- To connect these properties to the concepts of infinite invariant measure and reduced maps.
- To characterize the behavior of the diffusion coefficient and its distribution under specific conditions.
Main Methods:
- Application of infinite ergodic theory to deterministic subdiffusion models.
- Analysis of the reduced map and its infinite invariant measure.
- Investigation of the time-averaged mean square displacement for small time differences.
- Characterization of the limit distribution of the diffusion coefficient.
Main Results:
- The averaged diffusion coefficient is determined by the infinite invariant measure of the reduced map.
- For short time differences, the mean square displacement shows linear dependence.
- The diffusion coefficient is shown to be a random variable.
- The limit distribution of the diffusion coefficient follows the Mittag-Leffler distribution.
Conclusions:
- Infinite ergodic theory offers a powerful framework for understanding subdiffusion.
- The Mittag-Leffler distribution is a universal law governing the diffusion coefficient in this context.
- The findings provide a deeper statistical understanding of anomalous transport phenomena.
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