Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

Kang, Kyungkeun; Kim, Hwa Kil; Lim, Tongseok; Seo, Geuntaek

doi:10.1007/s00526-020-01882-7

Cited by 8 publications

(3 citation statements)

References 27 publications

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“…There is a partially alternate proof of Lemma 7, which proceeds by verifying the claimed formulas using Fourier transforms in the spirit of our proof of Lemma 4. More precisely, one verifies (16) for −1 < α < 0, (15) for 0 < α < 1 and ( 14) for 1 < α < 2. (In these cases the Fourier transform of the convolution kernels is well-defined without the need of analytic continuation.)…”

Section: Proof Of Theoremmentioning

confidence: 64%

Minimizers for a one-dimensional interaction energy

Frank

2022

Nonlinear Analysis

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Section: Proof Of Theoremmentioning

confidence: 64%

Minimizers for a one-dimensional interaction energy

Frank

2022

Nonlinear Analysis

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“…Proof. For α > β > 0, [30,Lemma 1] shows the diameter of support of all (d 2 -local) minimizers for E W α,β is bounded by the positive zero, say z α,β , of the function w α,β (r) = r α /α − r β /β. It is easily seen that z α,β increases as α ց β to the limit e [5].…”

Section: Resultsmentioning

confidence: 99%

Classifying minimum energy states for interacting particles: Spherical Shells

Davies¹,

Lim²,

McCann³

2021

Preprint

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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 the patterns they display. Apart from rare values of the attractive and repulsive exponents (α, β), the energy minimizing configurations of particles are not explicitly known, although simulations and local stability considerations have led to conjectures with strong evidence over a much wider region of parameters. For a segment β = 2 < α < 4 on the mildly repulsive frontier we employ strict convexity to conclude that the energy is uniquely minimized (up to translation) by a spherical shell. In a companion work, we show that in the mildly repulsive range α > β ≥ 2, a unimodal threshold 2 < α ∆ n (β) ≤ max{β, 4} exists such that equidistribution of particles over a unit diameter regular n-simplex minimizes the energy if and only if α ≥ α ∆ n (β) (and minimizes uniquely up to rigid motions if strict inequality holds). At the point (α, β) = (2, 4) separating these regimes, we show the minimizers all lie on a sphere and are precisely characterized by sharing all first and second moments with the spherical shell. Although the minimizers need not be asymptotically stable, our approach establishes d α -Lyapunov nonlinear stability of the associated (d 2 -gradient) aggregation dynamics near the minimizer in both of these adjacent regimeswithout reference to linearization. The L α -Kantorovich-Rubinstein distance d α which quantifies stability is chosen to match the attraction exponent.

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“…The important observation now is that (15) holds at least for 0 < α < 2. (We restrict ourselves here to α > 0 so that the integral on the left side converges absolutely.)…”

Section: Proof Of Theoremmentioning

confidence: 99%

Minimizers for a one-dimensional interaction energy

Frank¹

2021

Preprint

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We solve explicitly a certain minimization problem for probability measures in one dimension involving an interaction energy that arises in the modelling of aggregation phenomena. We show that in a certain regime minimizers are absolutely continuous with an unbounded density, thereby settling a question that was left open in previous works.Recently, Davies, Lim and McCann [10, Theorem 2.2] have shown that for α ≥ 3 the minimizers for E α are precisely of the form µ = 2 −1 (δ a−1/2 + δ a+1/2 ) for some a ∈ R. Earlier, Kang, Kim, Lim and Seo [15, Theorem 2] had shown that in the case 2 < α < 3, for any m ∈ (0, 1) and a ∈ R the measure mδ a−1/2 + (1 − m)δ a+1/2 is a saddle point for E α (with respect to the ∞-Wasserstein metric) and therefore, in

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Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

Cited by 8 publications

References 27 publications

Minimizers for a one-dimensional interaction energy

Minimizers for a one-dimensional interaction energy

Classifying minimum energy states for interacting particles: Spherical Shells

Minimizers for a one-dimensional interaction energy

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