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Published on: May 1, 2018
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Subdiffusion with particle immobilization process described by a differential equation with Riemann-Liouville-type
1Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland.
Physical Review. E
|August 16, 2023
Summary
This study introduces a new equation for subdiffusion with particle immobilization using the continuous time random walk model. The derived fractional time derivative equation predicts an exponential distribution in the long-term stationary state.
Area of Science:
- Physics
- Mathematical Modeling
- Physical Chemistry
Background:
- Subdiffusion describes anomalous particle movement, deviating from standard Brownian motion.
- Particle immobilization can significantly alter transport dynamics in various systems.
- Continuous Time Random Walk (CTRW) models are widely used to describe anomalous diffusion.
Purpose of the Study:
- To derive a mathematical equation for subdiffusion incorporating particle immobilization.
- To develop a method for analyzing the time-domain behavior of the immobilization kernel.
- To investigate the long-time behavior and stationary state of the subdiffusion-immobilization process.
Main Methods:
- Derivation of a fractional time derivative equation based on the CTRM model.
- Utilizing Laplace transform to define the kernel controlling immobilization.
- Developing a method for inverse Laplace transform to obtain the time-domain kernel.
Main Results:
- An equation featuring a fractional time derivative of Riemann-Liouville type was successfully derived.
- A novel method for calculating the inverse Laplace transform of the immobilization kernel was proposed.
- The subdiffusion-immobilization process was shown to reach a stationary state with an exponential probability density function in the long-time limit.
Conclusions:
- The derived fractional equation accurately models subdiffusion with immobilization.
- The proposed inverse Laplace transform method provides insights into the immobilization dynamics.
- The long-term stationary state is characterized by an exponential particle distribution, indicating a predictable final state.
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