Relaxed many-body optimal transport and related asymptotics

Abstract

Abstract Optimization problems on probability measures in ℝ d {\mathbb{R}^{d}} are considered where the cost functional involves multi-marginal optimal transport. In a model of N interacting particles, for example in Density Functional Theory, the interaction cost is repulsive and described by a two-point function c ⁢ ( x , y ) = ℓ ⁢ ( | x - y | ) {c(x,y)=\ell(\lvert x-y\rvert)} where ℓ : ℝ + → [ 0 , ∞ ] {\ell:\mathbb{R}_{+}\to[0,\infty]} is decreasing to zero at infinity. Due to a possible loss of mass at infinity, non-existence may occur and relaxing the initial problem over sub-probabilities becomes necessary. In this paper, we characterize the relaxed functional generalizing the results of [4] and present a duality method which allows to compute the Γ-limit as N → ∞ {N\to\infty} under very general assumptions on the cost ℓ ⁢ ( r ) {\ell(r)} . We show that this limit coincides with the convex hull of the so-called direct energy. Then we study the limit optimization problem when a continuous external potential is applied. Conditions are given with explicit examples under which minimizers are probabilities or have a mass < 1 {<1} . In a last part, we study the case of a small range interaction ℓ N ⁢ ( r ) = ℓ ⁢ ( r / ε ) {\ell_{N}(r)=\ell(r/\varepsilon)} ( ε ≪ 1 {\varepsilon\ll 1} ) and we show how the duality approach can also be used to determine the limit energy as ε → 0 {\varepsilon\to 0} of a very large number N ε {N_{\varepsilon}} of particles.