“Phenomenology and stochastic modelling of dispersive processes: the Continuous Time Random Walk framework”

 

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Abstract

 

In several environments mass is transported through space and get macroscopically dispersed. Different mechanisms underlying the mass transport lead to different dispersion dynamics and different macroscopic properties (like long tailing of breakthrough curves or late arrival times distributions). Various stochastic models have been proposed to represent these macroscopic dispersive phenomena, with very different underlying mechanisms, such as mobile-immobile mass exchange, long-range correlated spatial motions, or heavy-tailed trapping time distributions. These different models may provide equally good fits to data, such as first passage time distributions. Yet, their implications can be dramatic when transport-driven processes are considered, such as chemical reactions or biofilm growth. A key challenge is to relate these upscaled dispersive models to the micro-scale properties. In this lecture it will be presented the framework of Continuous Time Random Walk (CTRW) model, where a random walker jumps instantaneously from one site to another, following a waiting period on a site whose duration “t" is drawn according to a PDF of waiting times "ψ(t)”. While it was originally proposed to describe physical diffusion processes in semiconductors, its use has recently been extended to other situations where anomalous diffusion arises. Recent developments of the theory allow the incorporation of reactive processes within the CTRW framework. 

 

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“Phenomenology and stochastic modelling of dispersive processes: the Continuous Time Random Walk framework”

 

================

 

Abstract

 

In several environments mass is transported through space and get macroscopically dispersed. Different mechanisms underlying the mass transport lead to different dispersion dynamics and different macroscopic properties (like long tailing of breakthrough curves or late arrival times distributions). Various stochastic models have been proposed to represent these macroscopic dispersive phenomena, with very different underlying mechanisms, such as mobile-immobile mass exchange, long-range correlated spatial motions, or heavy-tailed trapping time distributions. These different models may provide equally good fits to data, such as first passage time distributions. Yet, their implications can be dramatic when transport-driven processes are considered, such as chemical reactions or biofilm growth. A key challenge is to relate these upscaled dispersive models to the micro-scale properties. In this lecture it will be presented the framework of Continuous Time Random Walk (CTRW) model, where a random walker jumps instantaneously from one site to another, following a waiting period on a site whose duration “t" is drawn according to a PDF of waiting times "ψ(t)”. While it was originally proposed to describe physical diffusion processes in semiconductors, its use has recently been extended to other situations where anomalous diffusion arises. Recent developments of the theory allow the incorporation of reactive processes within the CTRW framework. 

 

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