Corpus ID: 237513865

Classifying minimum energy states for interacting particles (II) -- regular simplices

@inproceedings{Davies2021ClassifyingME,
  title={Classifying minimum energy states for interacting particles (II) -- regular simplices},
  author={Cameron Davies and Tongseok Lim and Robert J. McCann},
  year={2021}
}
Consider densities of particles on R which interact pairwise through an attractive-repulsive power-law potential Wα,β(x) = |x|/α − |x|/β in the mildly repulsive regime α ≥ β ≥ 2. For n ≥ 2, we show there exists βn ∈ (2, 4) and a decreasing homeomorphism α∆n of [2, βn] onto [βn, 4] which can be extended (non-homeomorphically) by setting α∆n(β) = β for β > βn such that: distributing the particles uniformly over the vertices of a regular unit diameter n-simplex minimizes the potential energy if… Expand

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References

SHOWING 1-10 OF 37 REFERENCES
Classifying minimum energy states for interacting particles (I) -- spherical shells
Particles interacting through long-range attraction and short-range repulsion given by power-laws have been widely used to model physical and biological systems, and to predict or explain many of theExpand
Isodiametry, Variance, and Regular Simplices from Particle Interactions
Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For aExpand
Stability and clustering of self-similar solutions of aggregation equations
In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K * ρ) in Rd, d ⩾ 2, where K(r) = rγ/γ with γ > 2. It wasExpand
On the strong attraction limit for a class of nonlocal interaction energies
This note concerns the problem of minimizing a certain family of non-local energy functionals over measures on $\mathbb{R}^n$, subject to a mass constraint, in a strong attraction limit. In theseExpand
Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials
In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive–attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses isExpand
Geometry of minimizers for the interaction energy with mildly repulsive potentials
We show that the support of any local minimizer of the interaction energy consists of isolated points whenever the interaction potential is of class $C^2$ and mildly repulsive at the origin;Expand
Geometrical Bounds for Variance and Recentered Moments
We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) concerning measuresExpand
Cardinality estimation of support of the global minimizer for the interaction energy with mildly repulsive potentials
Abstract In this work, we give a cardinality estimation of the support of any global minimizer for the interaction energy with mildly repulsive potentials. For any global minimizer μ , the fact thatExpand
Swarm dynamics and equilibria for a nonlocal aggregation model
We consider the aggregation equation ρt −∇ ·(ρ∇K ∗ ρ) = 0i nR n , where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentialsExpand
PREDICTING 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 onExpand
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