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Anomalous diffusion driven by the redistribution of internal stresses
J Cleland1,2, M A K Williams1,2,3
1School of Fundamental Sciences, Massey University, Palmerston North 4442, New Zealand.
Physical Review. E
|August 20, 2021
Summary
This study introduces a mathematical model for anomalous diffusion driven by internal stresses, not thermal noise. The model uses a continuous time random walk framework, showing a shift from subdiffusive to diffusive behavior and eventual Gaussianity in soft-matter systems.
Area of Science:
- Mathematical Physics
- Soft Matter Physics
- Statistical Mechanics
Background:
- Anomalous diffusion is often attributed to thermal fluctuations.
- Internal stresses in soft-matter systems can also drive complex dynamics.
- A unified mathematical framework for stress-driven diffusion is lacking.
Purpose of the Study:
- To develop a mathematical description of anomalous diffusion driven by internal stresses.
- To model the dynamics of internal stresses using a continuous time random walk (CTRW) framework.
- To analyze the resulting diffusion equation and its solutions.
Main Methods:
- Utilized a continuous time random walk (CTRW) framework.
- Described waiting times between displacements using the generalized Gamma distribution.
- Identified and solved the associated generalized diffusion equation.
- Employed Fox H functions for analytical solutions.
Main Results:
- The probability density function transitions from non-Gaussian to Gaussian at longer timescales.
- The second moment of the diffusion exhibits transient behavior, shifting between subdiffusive and diffusive characteristics.
- A generalized diffusion equation was derived and solved analytically.
Conclusions:
- The developed mathematical framework accurately describes anomalous diffusion driven by internal stresses.
- The model predicts a transition from non-Gaussian to Gaussian behavior and from subdiffusive to diffusive dynamics.
- This approach offers potential applications for understanding phenomena like quaking in soft-matter systems.
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