Reinhard Illner

Modeling with dynamical systems and kinetic equations

Pages 199-243
Received: 18 January 2007
Mathematics Subject Classification (2000): 82D25 - 92C37 - 60K20 - 82D99

Abstract This article contains both a survey of and some novelties about mathematical modeling problems which emerged within recent years in physical chemistry, microbiology, and multi-lane traffic flow. Specifically, we first present a generalization of the Kolmogorov-Avrami model for crystallization dynamics for cases where the crystallization is incomplete and the classical model fails; second, the concept and an application of transcriptional-translational oscillators operating inside a living cell. The feasibility and significance of such oscillators is an important topic in molecular biology, and, as will be shown, their interactions may be the cause underlying phenomena like circadian rhythms. The basic equations are stochastic differential equations including Markovian random variables, and their associated Kolmogorov master equations are a linear kinetic system of PDEs. Third, we engage in a discussion of traffic flow models for the multi-lane scenario, with emphasis on Fokker-Planck type systems and their properties. We summarize and discuss results from a series of papers in which such models were introduced as alternatives to other, rather different kinetic models, and we conduct a comparison. There are discussions on the relationship between kinetic and macroscopic models, on fundamental diagrams, and on modeling ingredients which may lead to more than one equilibrium solution, a scenario known as a bifurcated (or multi-valued) fundamental diagram. We go though an extensive presentation of the properties and degeneracies of the new models, and we briefly introduce relative entropy and discuss its applications.