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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.02322 |
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| _version_ | 1866913296850878464 |
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| author | Davies, Cameron |
| author_facet | Davies, Cameron |
| contents | We study a two parameter family of energy minimization problems for interaction energies $\mathcal{E}_{α,β}$ with attractive-repulsive potential $W_{α,β}$. We develop a concavity principle, which allows us to provide a lower bound on $\mathcal{E}_{α,β}$ if there exist $β_0<β<β_1$ with minimizers of $\mathcal{E}_{α,β_0}$ and $\mathcal{E}_{α,β_1}$ known. In addition to this, we also derive new conclusions about the limiting behaviour of $\mathcal{E}_{α,β}$ for $β\approx 2.$ Finally, we describe a method to show that, for certain values of $(α,β),$ $\mathcal{E}_{α,β}$ cannot be minimized by the uniform distribution over a top-dimensional regular unit simplex. Our results are made possible by two key factors -- recent progress in identifying minimizers of $\mathcal{E}_{α,β}$ for a range of $α$ and $β$, and an analysis of $\inf\mathcal{E}_{α,β}$ as a function on parameter space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02322 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds and Limiting Minimizers for a Family of Interaction Energies Davies, Cameron Optimization and Control Mathematical Physics Analysis of PDEs Primary 49Q10. Secondary 31B10, 35Q70, 37L30, 70F45, 90C20 We study a two parameter family of energy minimization problems for interaction energies $\mathcal{E}_{α,β}$ with attractive-repulsive potential $W_{α,β}$. We develop a concavity principle, which allows us to provide a lower bound on $\mathcal{E}_{α,β}$ if there exist $β_0<β<β_1$ with minimizers of $\mathcal{E}_{α,β_0}$ and $\mathcal{E}_{α,β_1}$ known. In addition to this, we also derive new conclusions about the limiting behaviour of $\mathcal{E}_{α,β}$ for $β\approx 2.$ Finally, we describe a method to show that, for certain values of $(α,β),$ $\mathcal{E}_{α,β}$ cannot be minimized by the uniform distribution over a top-dimensional regular unit simplex. Our results are made possible by two key factors -- recent progress in identifying minimizers of $\mathcal{E}_{α,β}$ for a range of $α$ and $β$, and an analysis of $\inf\mathcal{E}_{α,β}$ as a function on parameter space. |
| title | Bounds and Limiting Minimizers for a Family of Interaction Energies |
| topic | Optimization and Control Mathematical Physics Analysis of PDEs Primary 49Q10. Secondary 31B10, 35Q70, 37L30, 70F45, 90C20 |
| url | https://arxiv.org/abs/2404.02322 |