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Pattern Formation in Particle Interactions


Large systems of particles interacting pairwise in $d$-dimensions give rise to extraordinarily rich patterns. These patterns generally occur in two types. On one hand, the particles may concentrate on a co-dimension one manifold such as a sphere in three dimensions or a ring in two dimensions. Localized, space-filling, co-dimension zero patterns can occur as well. This work develops an understanding of such patterns by exploring how the prediction and design of patterns relates to the stability and well-posedness properties of the underlying mathematical equations.

At the outset we use dynamical systems theory to predict the types behaviors a given system of particles will exhibit. Specifically, we develop a non-local linear stability analysis for particles that distribute uniformly on a $d-1$ sphere. Remarkably, this linear theory accurately characterizes the patterns in the ground states from the instabilities in the pairwise potential. We then leverage this aspect of the theory to address the issue of inverse statistical mechanics in self-assembly, i.e. the construction of a potential that will produce a desired pattern. As the linear theory indicates that potentials with a small number of spherical harmonic instabilities may produce very complex patterns, we naturally arrive at the linearized inverse statistical mechanics question: given a finite set of unstable modes, can we construct a potential that possesses precisely these linear instabilities? An affirmative answer would allow for the design of potentials with arbitrarily intricate spherical symmetries in the ground state. We solve the linearized inverse problem in full, and present a wide variety of designed ground states.

To conclude we begin the task of transferring aspects of the linear theory apply to the fully nonlinear problem. In particular, we address the well-posedness of distribution solutions to the aggregation equation $\rho_{t} + \mathrm{div}(\rho \mathbf{u} ) = 0, \; \mathbf{u} = -\nabla V * \rho$ in $\Rd$ where the density $\rho$ concentrates on a co-dimension one manifold. When the equation for such a solution is linearly well-posed, we show that the fully non-linear evolution is also well-posed locally in time for the class of bi-Lipschitz surfaces. In this aspect at least, the linear and non-linear theories therefore coincide.

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