Infinite-body optimal transport with Coulomb cost.

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 → ∞ with arbitrary fixed inhomogeneity profile ρ / 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.

N-density representability and the optimal transport limit of the Hohenberg-Kohn functional.

We derive and analyze a hierarchy of approximations to the strongly correlated limit of the Hohenberg-Kohn functional. These "density representability approximations" are obtained by first noting that in the strongly correlated limit, N-representability of the pair density reduces to the requirement that the pair density must come from a symmetric N-point density. One then relaxes this requirement to the existence of a representing symmetric k-point density with k < N. The approximate energy can be computed by simulating a fictitious k-electron system. We investigate the approximations by deriving analytically exact results for a 2-site model problem, and by incorporating them into a self-consistent Kohn-Sham calculation for small atoms. We find that the low order representability conditions already capture the main part of the correlations.

Should females prefer to mate with low-quality males?

We analyse a model of mate choice when males differ in reproductive quality and provide care for their offspring. Females choose males on the basis of the success they will obtain from breeding with them and a male chooses his care time on the basis of his quality so as to maximise his long-term rate of reproductive success. We use this model to establish whether high-quality males should devote a longer period of care to their broods than low-quality males and whether females obtain greater reproductive success from mating with higher quality males. We give sufficient conditions for optimal care times to decrease with increasing male quality. When care times decrease, this does not necessarily mean that high-quality males are less valuable to the female because quality may more than compensate for the lack of care. We give a necessary and sufficient condition for high-quality males to be less valuable mates, and hence for females to prefer low-quality males. Females can prefer low-quality males if offspring produced and cared for by high-quality males do well even if care is short, and do not significantly benefit from additional care, while offspring produced and cared for by low-quality males do well only if they receive a long period of care.