The mathematics of urban morphology. With a foreword by Michael Batty
This monograph gives a comprehensive and contemporary overview of many recent advances on random walks over comb-like structures. Comb models were introduced to understand anomalous transport in percolation. Revolving around the topic of comb-like structure, the authors present a random walk description of the transport in these simple but intriguing geometries covering a wide range of topics, including simple random walks, super-diffusion, sub-diffusion, Lévy process, and fractional Brownian motion, to name a few. This monograph is written in an explanatory and concise manner with plenty of concrete examples rather than a formal definition-theorem-proof textbook form. Nevertheless, necessary elements and preliminaries are presented to the reader and the results covered here reflect the state-of-the-art of the research on comb structures. The monograph is split into four parts. In Part I, methods and techniques that are useful in the latter development are introduced. These include continuous time random walk, elements of fractional calculus, some special functions such as Mittag-Leffler function, Markov process and stochastic differential equations. Part II is devoted to various topics about random walks on comb models. Mesoscopic description of random walks on combs is first presented followed by the Langevin formulation and inhomogeneous advection and super-diffusion. The last section in Part II introduces ultra-slow diffusion in three-dimensional cylindrical comb. Part III presents reaction-transport processes in combs. Two realms of front propagation in combs are discussed: the spiny dendrites and actin polymerization in a comb micrograph. Both are presented in a gradual manner with sufficient introductions and detailed examples. In Part IV, the authors describe varied interesting extensions of the comb model. They include Lévy processes on a generalized fractal comb, a quantum comb model, and path integral description of combs. The culmination of the monograph is the bibliography and index.
scientific article; zbMATH DE number 7029509