ABSTRACT
Anomalous diffusion is ubiquitous in the natural world. With the rapid development of science and technology, nowadays, it is widely acknowledged that non-Brownian motion, i.e., anomalous diffusion exists in many physical and biology systems. The subdiffusion is widely observed in glass-forming systems, translocation of polymers, solute transport in porous media, etc. While many cellular processes driven by molecular motor show superdiffusion. This chapter firstly introduces several common physical models and the stochastic processes to describe the anomalous and nonergodic diffusion in the natural world. Then we focus on the central part of the monograph: distribution of statistical observables. Then this chapter analyzes the probability density function of the underlying processes for anomalous and nonergodic diffusions, and further derives the macroscopic equations governing the probability density functions of the particle's position and the corresponding functionals. The ergodic property of some common processes are investigated by comparing the ensemble- and time-averaged mean squared displacements. Besides, this chapter extends the studies to more general statistical observables, such as the fractional moments, first passage time and first hitting time, and presents their connections to mathematics and other applications.
