1 - Fractional Calculus and Anomalous Transport

Summary

We initially introduce the standard diffusion model solving the PDF of the Brownian motion/process, satisfying the normal scaling property. This happens through a new definition of the process increments, where they are no longer drawn from a normal distribution, leading to α-stable Lévy flights at the microscopic level and correspondingly an anomalous diffusion model with a fractional Laplacian at the macroscopic scale. Next, we show how the Riemann–Liouville fractional derivatives emerge in another anomalous diffusion model corresponding to the asymmetric α-stable Lévy flights at small scales. Subsequently, we introduce the notion of subdiffusion stochastic processes, in which the Caputo time-fractional derivative appears in the anomalous subdiffusion fractional model. We combine the previous two cases, and construct continuous-time random walks, where a space-time fractional diffusion model will solve the evolution of the probability density function of the stochastic process. Next, we motivate and introduce many other types of fractional derivatives that will code more complexity and variability at micro-to-macroscopic scales, including fractional material derivatives, time-variable diffusivity for the fractional Brownian motion, tempered/variable-order/distributed-order/vector fractional calculus, etc.