Burger M (2018)
Publication Status: Published
Publication Type: Journal article
Publication year: 2018
Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD
Book Volume: 28
Pages Range: 409-451
Journal Issue: 3
URI: https://arxiv.org/pdf/1610.08108.pdf
DOI: 10.1142/S0218202518500112
We consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. An example of this class of models is the so-called Kucken-Champod model describing the formation of fingerprint patterns. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In contrast to isotropic interaction models the anisotropic forces in our class of models cannot be derived from a potential. The underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the variation of this parameter, describing the transition between the isotropic and the anisotropic model, analytically and numerically. We analyze the equilibria of the corresponding mean-field partial differential equation and investigate pattern formation numerically in two dimensions by studying the dependence of the parameters in the model on the resulting patterns.
APA:
Burger, M. (2018). Pattern formation of a nonlocal, anisotropic interaction model. Mathematical Models & Methods in Applied Sciences, 28(3), 409-451. https://dx.doi.org/10.1142/S0218202518500112
MLA:
Burger, Martin. "Pattern formation of a nonlocal, anisotropic interaction model." Mathematical Models & Methods in Applied Sciences 28.3 (2018): 409-451.
BibTeX: Download