Operator-stable Lévy motions and renormalizing the chaotic dynamics of microbes in anisotropic porous media

Self-motile particles, such as microbes, that can preferentially orient their microscale movement are embedded in a multiscale anisotropic porous medium. Such an environment is consistent with most natural geological systems. The preferential directional orientation and position of the self-motile particles at the microscale is accounted for with an operator-stable Lévy process. The anisotropy of the porous medium on the mesoscale is accounted for with an operator-stable velocity with either finite or infinite second moments. Upscaling is accomplished with generalized central limit theorems which can be shown to be equivalent to a renormalized group approach. If the mesoscale drift is operator-stable Lévy (fractal), then the macroscale Fokker–Planck equation has time-dependent diffusion tensor and fractional derivatives which are directionally dependent.