Cotar Codina, Friesecke Gero, Pass Brendan
1Department of Statistical Science, University College London, London, UK.
2Department of Mathematics, Technische Universität München, Munich, Germany.
Calc Var Partial Differ Equ. 2015;54(1):717-742. doi: 10.1007/s00526-014-0803-0. Epub 2014 Dec 17.
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 -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 , 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. with arbitrary fixed inhomogeneity profile ), 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.
我们引入并分析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，对于这类成本的一个自然类别，优化器由独立乘积测度给出，其所有因子均由单体边缘分布给出。这与标准的有限体OT问题形成鲜明对比，在有限体OT问题中优化器通常高度相关，也与具有Gangbo - Swiech成本的无限体OT问题不同。此外，通过采用概率论中可交换过程研究的一种构造，我们证明相应的N体OT问题可以由无限体问题很好地近似。我们的类别包括多电子量子力学中出现的库仑成本。在物理和量子化学文献中，库仑N体OT问题的最优成本作为单体边缘密度的函数以SCE的名称为人所知，并且作为著名的 Hohenberg - Kohn泛函的半经典极限自然出现。我们的结果意味着在非均匀高密度极限（即具有任意固定非均匀性分布的情况下），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两体OT问题。
