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 3D) or a ring (in 2D). Localized, space-filling, codimension zero patterns can occur as well. In this paper, we utilize a dynamical systems approach to predict such behaviors in a given system of particles. More specifically, we develop a non-local linear stability analysis for particles uniformly distributed on a d − 1 sphere. Remarkably, the linear theory accurately characterizes the patterns in the ground states from the instabilities in the pairwise potential. This aspect of the theory then allows us to address the issue of inverse statistical mechanics in selfassembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.

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Abstract:In this paper, we consider distribution solutions to the aggregation equation ρ t + div(ρu) = 0, u = −∇V * ρ in R d , where the density ρ concentrates on a co-dimension one manifold. We show that an evolution equation for the manifold itself completely determines the dynamics of such solutions. We refer to such solutions aggregation sheets. When the equation for the sheet 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. Moreover, we show that if the initial sheet is C 1 then the solution itself remains C 1 as long as it remains Lipschitz. Lastly, we provide conditions on the kernel g(s) = − dV ds that guarantee the solution remains a bi-Lipschitz surface globally in time, and construct explicit solutions that either collapse or blow up in finite time when these conditions fail. BackgroundSystems with a large number of pairwise interacting particles pervade many disciplines, ranging from models of self-assembly processes in physics and chemistry [21][22][23]29] to models for biological swarming [1,8,16,28] to algorithms for the cooperative control of autonomous vehicles [33]. A simple example of these models employs a first order system of ordinary differential equations for the positionsThe interaction kernel g(s) describes the manner in which particles interact with one another, and therefore depends on the particular application for the model. The formal continuum limit of this system then yields the well-known aggregation equation for the density ρ of particles. This equation has received significant attention in recent years, and the majority of the analysis largely falls into two categories. More classical treatments focus on densities ρ that are absolutely continuous with respect to Lebesgue measure, such as those lying in an L p (R d ) space [2][3][4][5][6]9,10,14]. For densities that merely define a Borel measure on R d , such as point masses, ideas from optimal transport have proven fruitful for demonstrating the well-posedness of (2) for some classes of interaction kernels [7,12,13,19,20]. However, several recent studies [15,26,30,31] have found that rings, spheres and more complicated surface-like states naturally occur in the ODE systems (1) and the full PDE models. This suggests that a co-dimension one description of (2) might prove useful for studying such particle distributions (see Fig. 1). In this context, i.e. when the density must have support of co-dimension one, even the most basic well-posedness results do not yet exist. We therefore provide them in this paper.Specifically, we analyze distribution solutions to (2) that have support homeomorphic to the (d − 1) sphere S d−1 ⊂ R d , and so take the formThe map Φ(·, t) :, where ρ Φ (x, t) describes the density of particles along the manifold and dH Φ (x) denotes the surface measure on the manifold. By (3), we mean that ρ acts as a distribution on ψIn the usual manner, we then require thatWell-Posedness Theory for Aggregation Sheetshold for all...