Renormalizing chaotic dynamics in fractal porous media with application to microbe motility

Abstract

Motivated by the need to understand the movement of microbes in natural porous systems and the evolution of their genetic information, a renormalization procedure for motile particles in media with fractal functionality between upper and lower cutoffs is developed and applied to Lévy particles. On the micro scale, particle trajectories are the solution to an integrated stochastic ordinary differential equation (SODE) with Markov, stationary, ergodic drift subject to Lévy diffusion. The Lévy diffusion allows for self‐motile particles. On the meso scale, the trajectory is the solution to an integrated SODE with Lévy drift and diffusion arising from the microscale asymptotics. Lévy drift is associated with the fractal character of the Lagrangian velocity. On the macro scale, the process is driven by the asymptotics of the mesoscale drift without additional diffusion. Renormalized dispersion equations are presented on the meso and macro scales.

On upscaling operator-stable Lévy motions in fractal porous media

“…In subsequent analysis we model the meso scale drift in two ways. The first parallels that of [10,11] with the Lagrangian velocity a-stable Lévy. The second and novel approach is to model the Lagrangian mesoscale acceleration as a-stable Lévy.…”

“…[2]. Randomly swimming microbes can often be modeled as a-stable Lévy motions [11]. Using the concept of directional preference put forth in [8], swimming in a preferential direction based on an energy gradient can be modeled using an operator-stable Lévy motion.…”

Scaling the fractional advective–dispersive equation for numerical evaluation of microbial dynamics in confined geometries with sticky boundaries

“…Hence for dilute concentrations of microbes swimming through a given fluid at standard conditions in an experimental flow cell, a probability distribution can be applied to describe the motility pattern. It has been observed that Levy motion defined by a-stable probability distributions of increments best describes the runs and tumbles of microbial motion [6,18,19]. The mixing measure within the definition of a Levy process may be employed to account for microbe food (energy) sources by skewing the movement along a gradient.…”

“…2 and 3). These interesting properties of the a-stable variables have found widespread use in a variety of fields, such as in the study of microbial dynamics [18,19], transport in porous media [29][30][31], anomalous dispersion [32], and super diffusion in turbulence [33][34][35].…”

Two-scale renormalization-group classification of diffusive processes

“…Processes with infinite MSD, such as α-stable Lévy motion, have been used to study nonlocal transport [18,19], movement patterns of marine predators [20], and microbial motion [21], to name a few examples. It can be shown that α-stable Lévy motion is 1/αself-similar [3] and therefore 1/α-diffusive on both the long and short time scales.…”