Infinite-body optimal transport with Coulomb cost
Infinite-body optimal transport with Coulomb cost
复制标题
具有库仑成本的无限体最优输运
DOI：
10.1007/s00526-014-0803-0
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发表时间：
2013
影响因子：
2.1
通讯作者：
Brendan Pass
中科院分区：
文献类型：
--
作者：
Codina Cotar;G. Friesecke;Brendan Pass
我们引入并分析了具有对势形式成本函数的对称无限体最优传输（OT）问题。我们表明，对于此类成本的自然类别，优化器由独立乘积度量给出，其所有因素均由单体边际给出。这与标准有限体 OT 问题（其中优化器通常高度相关）以及具有 Gangbo-Swiech 成本的无限体 OT 问题形成鲜明对比。此外，通过采用概率论中可交换过程研究的构造，我们证明了相应的 N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}-body OT 问题很好地近似为无限体问题。我们的课程属于多电子量子力学中出现的库仑成本。库仑 N 体 OT 问题的最优成本作为单体边际密度的函数，在物理学和量子化学文献中以 SCE 泛函的名称为人所知，并且自然地作为著名的 Hohenberg-Kohn 泛函的半经典极限而出现。我们的结果表明，在非均匀高密度限制下（即 N→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\rightarrow \infty $$\end{document} 具有任意固定不均匀性轮廓 ρ/N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho {/}N$$\end{document})，SCE 泛函收敛到平均场泛函。我们还将无限体和 N 体 OT 问题重新表述为具有可表示性约束的二体 OT 问题。
We introduce and analyze symmetric infinite-body optimal transport (OT) problems with cost function of pair potential form. We show that for a natural class of such costs, the optimizer is given by the independent product measure all of whose factors are given by the one-body marginal. This is in striking contrast to standard finite-body OT problems, in which the optimizers are typically highly correlated, as well as to infinite-body OT problems with Gangbo–Swiech cost. Moreover, by adapting a construction from the study of exchangeable processes in probability theory, we prove that the corresponding N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}-body OT problem is well approximated by the infinite-body problem. To our class belongs the Coulomb cost which arises in many-electron quantum mechanics. The optimal cost of the Coulombic N-body OT problem as a function of the one-body marginal density is known in the physics and quantum chemistry literature under the name SCE functional, and arises naturally as the semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results imply that in the inhomogeneous high-density limit (i.e. N→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\rightarrow \infty $$\end{document} with arbitrary fixed inhomogeneity profile ρ/N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho {/}N$$\end{document}), the SCE functional converges to the mean field functional. We also present reformulations of the infinite-body and N-body OT problems as two-body OT problems with representability constraints.
Hohenberg-Kohn 泛函的 N 密度表示性和最佳输运极限
DOI：
10.1063/1.4821351
发表时间：
2013
期刊：
The Journal of chemical physics
影响因子：
--
作者：
Gero Friesecke;Christian B. Mendl;Brendan Pass;Codina Cotar;Claudia Klüppelberg
通讯作者：
Claudia Klüppelberg