Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions

Abstract

We present a framework for systems in which diffusion-advection transport of a tracer substance in a mobile zone is interrupted by trapping in an immobile zone. Our model unifies different model approaches based on distributed-order diffusion equations, exciton diffusion rate models, and random walk models for multi-rate mobile-immobile mass transport. We study various forms for the trapping time dynamics and their effects on the tracer mass in the mobile zone. Moreover we find the associated breakthrough curves, the tracer density at a fixed point in space as function of time, as well as the mobile and immobile concentration profiles and the respective moments of the transport. Specifically we derive explicit forms for the anomalous transport dynamics and an asymptotic power-law decay of the mobile mass for a Mittag-Leffler trapping time distribution. In our analysis we point out that even for exponential trapping time densities transient anomalous transport is observed. Our results have direct applications in geophysical contexts but also in biological, soft matter, and solid state systems.

arXiv comment34 pages, 14 figures

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• Negative diffusion of excitons in quasi-two-dimensional systems
• Subdiffusion equation with Caputo fractional derivative with respect to another function in modelling diffusion in a complex system consisting of matrix and channels
• Hidden Markov modeling of single particle diffusion with stochastic tethering
• Simulation of the continuous-time random walk using subordination schemes
• Two-states Brownian particle in a Harmonic Potential
• Sharp decay estimates and numerical analysis for weakly coupled systems of two subdiffusion equations