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ISBN 9783843905374

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978-3-8439-0537-4, Reihe Mathematik

Stephan Martin
Applied Kinetic PDEs: Collective behavior models and Hamiltonian energy dynamics

145 Seiten, Dissertation Technische Universität Kaiserslautern (2012), Hardcover, A5

Zusammenfassung / Abstract

In recent years, mathematics has seen a reviving interest in kinetic theory and its equations. Not only has there been great recognition of advancements in theoretical analysis, but new applications of kinetic models have come up and today attract the attention of researchers both in applied and pure mathematics. This thesis explores the application of kinetic models in two selected modern applications. In both problems, one is interested in investigating particular stationary states of the underlying partial differential equation, which describes the evolution of a distribution function. First, we draw our attention to swarming models in mathematical biology. The complex behavior of animal groups leads to the emergence of coherent patterns of motion, frequently observed in nature for example in swarms of fish or flock of birds. After a review on recent advances on the subject, we provide a new class of interaction potentials which reassemble the commonly accepted set of behavior rules, but also allow the explicit computation of stationary states, even in higher dimensions. Our considerations will be based on characteristic pattern equations derived from a suitable hydrodynamic closure of the continuous model equations. Numerical studies validate the agreement of our results with particle simulations. The second part is devoted to kinetic equilibria in Hamiltonian systems. We are interested in the deviation of the stationary energy equilibrium of linear Fokker-Planck equations in presence of a second deterministic and periodic force, whose frequency is not identical to the Hamiltonian motion. Stochastic averaging is a classical method to derive a self-consistent asymptotic equation for the evolution of Hamiltonian energy distributions, but is unable to capture the induced energy deviations. We develop a method of higher-order stochastic averaging based on a formal energy projection and a Chapman-Enskog type splitting. The construction is given for a general setting and computed explicitly for the special case of a linear oscillator. Additionally, we apply higher-order averaging to a model of fiber lay-down processes and obtain convincing comparisons of the resulting coefficients with Monte-Carlo simulations of the original systems.