Operator-stable Lévy motions and renormalizing the chaotic dynamics of microbes in anisotropic porous media
Moongyu Park, John Howard Cushman · Journal of Statistical Mechanics Theory and Experiment · 2009
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.