Diffusion in complex system: is the anomalous usual?

Starting from the works of Fourier, Einstein, Langevin (among others) up to the elegant approach of Montroll, diffusive processes have fascinated researchers from different fields of science. It is a ubiquitous process in nature and has deep connections to general results of probability theory, such as the Central Limit Theorem. This connection provides the most common fingerprints of the usual diffusion: the linear spreading of system (variance is linear in time) and the Gaussian propagator (no matter the initial condition of the system, for long enough times it will become a Gaussian).

However, the emergence of these features is conditional on some assumptions that may not always hold. When this happens, people usually report on anomalous diffusion. The first work related to anomalous diffusion is contributed by Richardson [Proc. R. Soc. Lond. A 110, 709–737 (1926)]. However, his work did not raise much attention, and it took over 20 years and the presentation of experimental evidence for it to become popular in the scientific community. More recent studies have applied the concepts of diffusion to situations that are very far from its traditional domain. For instance, biological, economical, social and other complex systems in general have been studied using this framework. Naturally, living systems including human beings are quite different from inanimate matter and, consequently, the assumptions leading to usual diffusion often do not hold. Long-range correlations such as those present in financial market volatility, the composed swimming motion of small organisms such as marine bacteria, and the strange patterns of vehicle flows are a just few examples where the Central Limit Theorem does not apply.

Thus, I believe that the answer to my question given by the title is: Yes, complex systems are the ideal workplace for discovering anomalous diffusion.

— Haroldo Ribeiro