Normal and Anomalous Diffusion in Heterogeneous and Bounded Domains

: Lattice Random Walks and Brownian Local Time Approaches

Abstract

Diffusion is a ubiquitous transport mechanism across countless natural and synthetic systems. In many of these systems the spatial domain is not homogeneous and often contains spatial heterogeneities and boundaries. Due to recent advances in the resolution of tracking technologies, there is an ongoing demand to develop sophisticated theoretical techniques to model diffusive
processes. In this thesis we contribute to such demand by constructing analytical frameworks to study diffusive motion in the presence of reactive and inert heterogeneities and boundaries. Throughout, we take two complementary approaches to tackle this problem, namely lattice random walks and Brownian local time, where the latter characterizes the time spent at a given location. Firstly, we study reactive heterogeneities and boundaries using the defect technique and find different first-passage related quantities. Using lattice random walks, we elucidate the connection to the local time and use Feynman-Kac theory to study its statistics. Subsequently, we develop a technique to study defects that are reflecting, and discover the role local time plays in such scenarios. A large part of the thesis is also devoted to study diffusion in the presence of permeable barriers. For that we generalize the diffusion equation to account for the presence of permeable barriers, and use it to study first-passage statistics and various generalizations. In addition, we develop a statistical representation of a permeable barrier through subordination techniques using the local time. From this we study crossing statistics and derive renewal type
equations to describe the dynamics through permeable barriers. As a particular application we study extreme value statistics and the Arcsine laws of Brownian motion in the presence of a permeable barrier. Finally, we consider the extension to when the transport mechanism is subdiffusive, and we study reactive and permeable barriers using continuous time random walks and generalized Feynman-Kac techniques.

Date of Award	25 Mar 2024
Original language	English
Awarding Institution	University of Bristol
Supervisor	Luca Giuggioli (Supervisor) & Mario Di Bernardo (Supervisor)