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Continuous-time random walk: crossover from anomalous regime to normal regime
1Departamento de Física, Universidade Estadual de Maringá, Av Colombo 5790, 87020-900 Maringá, PR, Brazil.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 28, 2010
Summary
This study introduces a continuous time random walk model with finite waiting times and jump lengths. The model exhibits normal diffusion long-term but can describe anomalous subdiffusive, normal, and superdiffusive behaviors at shorter times.
Area of Science:
- Statistical Physics
- Stochastic Processes
- Complex Systems
Background:
- Continuous time random walks (CTRWs) are fundamental models for anomalous diffusion.
- Understanding diffusion across different timescales requires flexible waiting time and jump length distributions.
- Existing models often focus on specific anomalous regimes or long-time behavior.
Purpose of the Study:
- To develop a CTRW model with finite characteristic waiting time and jump length variance.
- To investigate diffusion behaviors across all timescales, from short to long.
- To analyze the system's probability distribution and its exact solutions.
Main Methods:
- Utilized approximate probability distributions for jump length and waiting time, combining power-law and exponential functions.
- Analyzed the model's diffusive behavior by considering the derived waiting time probability distribution.
- Investigated the exact solution for the system's probability distribution.
Main Results:
- The model demonstrates normal diffusive behavior in the long-time limit due to finite characteristic waiting time and jump length variance.
- Anomalous diffusion, including subdiffusion, normal diffusion, and superdiffusion, is accurately described at short and intermediate times.
- Exact solutions for the system's probability distribution were derived.
Conclusions:
- The proposed CTRW model offers a unified framework for describing diffusion across various timescales.
- The model's flexibility allows for capturing both anomalous and normal diffusive regimes.
- This work provides a comprehensive understanding of diffusion dynamics in systems with finite waiting time and jump length characteristics.
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