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- Title
PREDICTING PATTERN FORMATION IN PARTICLE INTERACTIONS.
- Authors
VON BRECHT, JAMES H.; UMINSKY, DAVID; KOLOKOLNIKOV, THEODORE; BERTOZZI, ANDREA L.; Bellomo, N.; Brezzi, F.
- Abstract
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, co-dimension 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 nonlocal 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 self-assembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.
- Subjects
PARTICLES; MANIFOLDS (Mathematics); STATISTICAL mechanics; DIMENSIONS; PREDICTION theory; SYSTEM analysis; MATHEMATICAL analysis
- Publication
Mathematical Models & Methods in Applied Sciences, 2012, Vol 22, p1140002-1
- ISSN
0218-2025
- Publication type
Article
- DOI
10.1142/S0218202511400021