Journal
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
Volume 22, Issue -, Pages -Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218202511400021
Keywords
Nonlocal differential equations; pattern formation; particle self-assembly
Categories
Funding
- UC
- NSF [DMS-0902792, EFRI-1024765, DMS-0907931]
- ONR [N000141010641]
- NSERC [47050]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0907931] Funding Source: National Science Foundation
- Directorate For Engineering [1024765] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [902792] Funding Source: National Science Foundation
- Emerging Frontiers & Multidisciplinary Activities [1024765] Funding Source: National Science Foundation
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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.
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