Infinite-body optimal transport with Coulomb cost

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• 作者：Codina Cotar ; Gero Friesecke ; Brendan Pass
• 关键词：N ; representability ; Density functional theory ; Hohenberg ; Kohn functional ; N ; body optimal transport ; Infinite ; body optimal transport ; Coulomb cost ; Exchange ; correlation functional ; de Finetti’s Theorem ; Finite exchangeability ; N ; extendability ; 49S05 ; 65K99 ; 81V55 ; 82B05 ; 82C70 ; 92E99 ; 35Q40
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：54
• 期：1
• 页码：717-742
• 全文大小：660 KB
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• 作者单位：Codina Cotar (1)
  Gero Friesecke (2)
  Brendan Pass (3)
  
  1. Department of Statistical Science, University College London, London, UK
  2. Department of Mathematics, Technische Universit?t München, Munich, Germany
  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

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 \(N\)-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. \(N\rightarrow \infty \) with arbitrary fixed inhomogeneity profile \(\rho {/}N\)), 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.