Multiscale modeling and simulation for anomalous and nonergodic dynamics: From statistics to mathematics

In recent decades, anomalous and nonergodic diffusions are topical issues in almost all disciplines. In 2004, the phrase “anomalous is normal” was used in a title of a PRL paper, which reveals that the diffusion of classical particles on a solid surface has rich anomalous behavior controlled by the friction coefficient, meaning that anomalous diffusion phenomena are ubiquitous in the natural world. This review article first builds the microscopic models (stochastic processes) to describe the experimentally observed phenomena of the motion of the physical/abstract particles under the frameworks of continuous time random walks, Langevin picture, subordinated strong Markov process, etc. Beyond directly conducting statistical analyses on these stochastic processes, we target on delving into these microscopic models to uncover their physical mechanisms and digging out their potential applications. According to the application scenarios and the research requirements, we design the approciate statistical observables, e.g., the position of the particles, functional of the trajectories of the particles, probability of the first passage time, escape probability, etc. Then, we derive the governing equations of the probability density functions of the statistical observables. We do the mathematical studies on the equations, including well-posedness and regularity analyses, designing numerical schemes, performing numerical analyses, etc. Finally, we present the applications of the models in chemistry and biology, and propose future prospects in this research field.