Diffusion Dynamics In Branched Curvilinear Structures

Since many processes in biology,physics,chemistry,geology,and economics are based on diffusion,the study and description of the properties of diffusion in these systems have attracted the interest of many researchers from various disciplines since the beginning of the last century.However,simulation and experiment investigations of diffusion in complex systems have shown a deviation from the normal behavior established in the diffusion theory,called anomalous diffusion,and the principles of the normal diffusion theory are no longer able to describe the emerged phenomenon.In this context,understanding the underlying mechanism has become a goal during the last decades through building multifunctional theoretical models,their assignments are not just to identify the mechanism behind the resulted behavior when conducting an experiment or simulation but also to study the related processes.Random walk in branched structure,comb-like structure as a special case which is composed of the horizontal confined channel(backbone)with lateral branches(fingers),is an example quenched model that is used to understand anomalous diffusion.Particle diffusion in comb-like structures along the x-axis is possible only in the backbone in which fingers play the role of traps,in which the particle executes a random motion inside them until it returns to the backbone by chance and so on.Several modifications of the classical comb,related to real scenes,have been introduced and various behaviors resulting from carrying out distinguished dynamics inside them have been captured.Most of these modifications have concentrated on the shape of fingers and their distribution along the channel,and a little care has been taken on the geometry of the channel(backbone).Examples of branched curvilinear structures are encountered including not limited to intracellular,extracellular,porous environments,and the recent experimental and theoretical studies in these systems have detected matchless geometry-induced phenomena.Therefore,further research on the effects of geometry on the diffusion process is required to understand the mechanism behind these phenomena.In this thesis,we have studied the properties of the diffusion process in branched curvilinear structures.Three branched structures with different geometric properties have been introduced and different scenarios of diffusion dynamics have been taken into account.The structures are Archimedean spiral,generalized elliptical,and spherical comb-like structure.A descriptive analysis study has been conducted using statistical analysis tools;probability density function and mean square displacement which have been achieved through the Fokker-Planck equation and approximated numerical schemes technique and calculated using MATLAB software.