Isodiametry, Variance, and Regular Simplices from Particle Interactions
Isodiametry, Variance, and Regular Simplices from Particle Interactions
复制标题
粒子相互作用的等径、方差和正则单纯形
DOI：
10.1007/s00205-021-01632-9
复制
发表时间：
2019
影响因子：
2.5
通讯作者：
R. McCann
中科院分区：
文献类型：
--
作者：
Tongseok Lim;R. McCann
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 a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained-apart from translations and rotations-by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant d∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\infty $$\end{document} metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {R}}}^n$$\end{document} whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass 1/(n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(n+1)$$\end{document} to each vertex of a (unit-diameter) regular simplex.
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 a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant d∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\infty $$\end{document} metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {R}}}^n$$\end{document} whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass 1/(n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(n+1)$$\end{document} to each vertex of a (unit-diameter) regular simplex.
影响因子：
3.1
作者：
Carrillo, J. A.;Hittmeir, S.;Yao, Y.
通讯作者：
Yao, Y.
影响因子：
2.5
作者：
Balague, D.;Carrillo, J. A.;Raoul, G.
通讯作者：
Raoul, G.