Pattern formation of a nonlocal, anisotropic interaction model

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 Kuecken-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. We study the stationary states for the transition between the isotropic and the anisotropic model analytically and numerically. Based on these theoretical and numerical results we adapt the forces in the Kuecken-Champod model in such a way that we can model fingerprint patterns (and more general any desired pattern) as stationary solutions. This is joint work with M. Burger, B. Duering, P. Markowich and C.-B. Schoenlieb.