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Multiscale modeling and simulation for anomalous and nonergodic dynamics: From statistics to mathematics
Heng Wang1, Xuhao Li2, Lijing Zhao3
1School of Mathematics and Statistics, State Key Laboratory of Natural Product Chemistry, Lanzhou University, Lanzhou 730000, China.
Fundamental Research
|April 13, 2026
Summary
Anomalous diffusion, where particle movement deviates from standard patterns, is common in nature. This review explores microscopic models, physical mechanisms, and applications of these complex diffusion behaviors in science.
Area of Science:
- Physics
- Chemistry
- Biology
- Mathematics
Background:
- Anomalous and nonergodic diffusion are significant research areas across multiple scientific disciplines.
- The ubiquity of anomalous diffusion was highlighted by the 2004 PRL paper,
- anomalous is normal,
- demonstrating its prevalence in classical particle diffusion on surfaces.
Purpose of the Study:
- To review and build microscopic models (stochastic processes) describing anomalous diffusion phenomena.
- To uncover the underlying physical mechanisms driving these diffusion behaviors.
- To explore potential applications of anomalous diffusion models in various scientific fields.
Main Methods:
- Utilizing frameworks such as continuous time random walks, Langevin equations, and subordinated strong Markov processes.
- Designing statistical observables (e.g., particle position, trajectory functionals, first passage time, escape probability) tailored to specific applications.
- Deriving and mathematically analyzing governing equations for probability density functions, including well-posedness, regularity, and numerical methods.
Main Results:
- Development of microscopic models for anomalous diffusion.
- Mathematical analysis of derived governing equations.
- Identification of potential applications in chemistry and biology.
Conclusions:
- Anomalous diffusion is a widespread phenomenon with diverse microscopic origins.
- Mathematical modeling and analysis are crucial for understanding diffusion mechanisms and predicting behavior.
- Further research into anomalous diffusion holds promise for advancements in chemistry, biology, and other fields.
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