When optimal Coulomb transport must split mass

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Family 373 · Partial differential equations
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Suppose three repelling particles each have the same prescribed spatial probability density. How should their positions be correlated to minimize their average Coulomb repulsion?

One economical description chooses the first particle’s position $x$, then places the others deterministically at $T_2(x)$ and $T_3(x)$. If these maps preserve the prescribed density, all three particles have the correct distribution. This is the Monge, or co-motion, ansatz.

Must some optimal arrangement always have this deterministic form when the density is smooth?

Family 373 claims a counterexample. It constructs a smooth compactly supported density on $\mathbb R^3$ for which an optimal joint distribution exists, but no pair of density-preserving Borel maps is optimal. Nevertheless, maps can approach the optimal cost arbitrarily closely. The failure is attainment, not a positive gap between the best possible cost values.

2. Background and essential definitions

Marginals describe individual particles; couplings describe correlation

Let $\mu=\rho(x)\,dx$ be a probability measure. A coupling $\pi$ is a joint probability distribution for $(x_1,x_2,x_3)\in(\mathbb R^3)^3$ such that each coordinate separately has distribution $\mu$.

This leaves enormous freedom. Independent particles have coupling $\mu\otimes\mu\otimes\mu$, while correlated particles can avoid configurations that put them close together.

The Coulomb cost is

$$c(x_1,x_2,x_3)=\frac1{|x_1-x_2|} +\frac1{|x_1-x_3|} +\frac1{|x_2-x_3|}.$$

It is infinite when two positions coincide. The Kantorovich problem allows every coupling with the prescribed marginals:

$$E_*(\mu)=\inf_{\pi\in\Pi(\mu,\mu,\mu)}\int c\,d\pi.$$

The symbol $\Pi(\mu,\mu,\mu)$ denotes this set of admissible joint distributions. The objective minimizes expected repulsion, not the distance traveled from an initial configuration.

Deterministic maps are a restricted class

A Borel map is measurable with respect to the sets generated by open sets; no continuity or differentiability is required. Its pushforward satisfies

$$T_\#\mu(A)=\mu(T^{-1}(A)).$$

The condition $T_\#\mu=\mu$ means that $T$ preserves the probability distribution.

A Monge coupling is

$$\pi=(\mathrm{Id},T_2,T_3)_\#\mu, \qquad (T_2)_\#\mu=(T_3)_\#\mu=\mu.$$

Conditioned on the first coordinate $x$, there is no remaining randomness: the other two coordinates are fixed. A general coupling can instead divide the conditional mass among several possible pairs.

The theorem concerns every preserving Borel pair. It does not merely reject smooth maps, a particular radial recipe, or cyclic maps satisfying an additional relation such as $T_3=T_2^2$.

Infimum and minimum need not coincide

An infimum is the greatest lower bound of attainable costs. A minimum is an attainable cost equal to that bound. For example, positive numbers have infimum zero, but no smallest positive number.

Here the set of deterministic plans is rich enough to approximate the best cost, yet can fail to contain an optimizer. Limits of deterministic plans may acquire conditional randomness through increasingly fine rearrangements.

3. Development history and previous partial results

Coulomb multi-marginal transport arises in the strong-interaction limit of density functional theory. Seidl, Gori-Giorgi and Savin developed the co-motion formulation for strictly correlated electrons. The prescribed one-particle density is retained while correlation minimizes repulsion.

For two particles, Cotar, Friesecke and Klüppelberg established existence and uniqueness of optimal maps for absolutely continuous marginals. This is an important positive precedent, but adding a third particle changes the geometry of the correlation problem.

For any number of particles on the line, Colombo, De Pascale and Di Marino constructed optimal cyclic maps for identical nonatomic marginals with finite transport value. The higher-dimensional three-particle question is therefore different from the one-dimensional theory.

Colombo and Di Marino proved equality between the Coulomb Kantorovich minimum and the infimum over measure-preserving cyclic maps for identical nonatomic marginals. That theorem supplies approximating maps; equality of values does not assert existence of a minimizing map.

Several earlier negative results concerned narrower assertions. Pass constructed non-Monge Coulomb optimizers for radial data, but the existence of one non-Monge optimizer does not rule out another optimizer of Monge form. Colombo and Stra disproved specific radial quantile-cyclic prescriptions. Bindini, De Pascale and Kausamo constructed smooth densities for a reduced radial three-particle cost for which four prescribed maps fail. That reduced cost minimizes over angular variables and is not the Coulomb cost on the line.

Gerolin, Kausamo and Rajala established nonexistence of optimal maps for a different interaction, the repulsive harmonic cost. Their shared-component mass-splitting mechanism is a methodological predecessor of the present construction.

The general higher-dimensional map-attainment question is discussed in the Friesecke–Gerolin–Gori-Giorgi account, and the three-particle Coulomb case remained explicitly posed in Friesecke’s January 2026 preprint. Family 373 claims to exclude every deterministic preserving pair for one smooth identical marginal, rather than only selected candidate maps.

4. What the new manuscript claims

The manuscript constructs a nonnegative density $\rho$ with

$$\int_{\mathbb R^3}\rho\,dx=1, \qquad \rho\in C_c^\infty(\mathbb R^3), \qquad \sqrt\rho\in C_c^\infty(\mathbb R^3).$$

The notation $C_c^\infty$ means smooth with compact support. Smoothness of the square root is an extra condition; it is not automatic for every smooth nonnegative function near its zeros.

For $\mu=\rho\,dx$, the Kantorovich value is finite and attained. Yet every preserving Borel pair satisfies

$$\int c(x,T_2(x),T_3(x))\,d\mu(x)>E_*(\mu).$$

The left side is allowed to be infinite. The theorem therefore rules out every Monge optimizer, regardless of regularity or cyclic relations.

At the same time,

$$\inf_{(T_2)_\#\mu=(T_3)_\#\mu=\mu} \int c(x,T_2(x),T_3(x))\,d\mu(x)=E_*(\mu).$$

The source directly constructs finite-cost maps approaching the optimum for its example.

Symmetrizing an admissible Monge plan by averaging over particle permutations does not restore optimality. Permutation averaging preserves both the marginals and the symmetric cost, so it preserves the original strict excess cost as well.

The construction extends to every dimension $d\ge2$ and every exponent $s>0$ for the inverse-power interaction

$$c_s(x_1,x_2,x_3)=\sum_{i<j}|x_i-x_j|^{-s}.$$

For each such pair $(d,s)$, the manuscript gives a smooth identical marginal with nonattainment by maps and equality of infima.

5. Why the advance matters

Importance: 5/5. If correct, the family resolves a central existence question for Coulomb multi-marginal transport and shows that smooth one-particle densities do not guarantee an exact deterministic co-motion description.

The practical distinction is subtle. Restricting to maps need not produce a wrong limiting energy, because the infima agree. But an exact optimizer may require mass splitting, so a method that expects a convergent pair of optimal maps can be asking for an object that does not exist.

With particle-density normalization $n=3\rho$, the example has total mass three and $\sqrt n\in H^1$. It therefore belongs to the usual density-regularity class discussed in the source’s density-functional setting. This does not assert realization as the ground-state density of a specified external potential.

6. The main idea

Arrange five small components

Take a small central ball $B$ around the origin and four disjoint outer regions near $e_1,-e_1,e_2,-e_2$, where $e_1,e_2$ are perpendicular unit vectors. Construct smooth diffeomorphisms $Y_k,W_k$, for $k=1,2$, from $B$ to the outer regions.

The two allowed families of optimal triples are

$$(x,Y_1(x),W_1(x)) \quad\text{and}\quad (x,Y_2(x),W_2(x)),\qquad x\in B,$$

together with all coordinate permutations. Each triple contains one central point and the two outer points belonging to the same branch.

Let $\nu$ be a smooth probability measure inside $B$. Define the common marginal by

$$\mu=\frac13\nu+ \frac16\sum_{k=1}^2\bigl((Y_k)_\#\nu+(W_k)_\#\nu\bigr).$$

The central component has mass $1/3$; each of the four outer components has mass $1/6$. Choose a central point with law $\nu$, select either branch with equal probability, then randomly permute the three positions. This gives a coupling with marginal $\mu$ in every coordinate.

Force every optimizer onto those families

The remaining geometric task is to show that the proposed coupling really is optimal and that every optimizer uses only the allowed families.

The source constructs a common supporting potential $\varphi$ on the five components such that

$$c(x_1,x_2,x_3)\ge \varphi(x_1)+\varphi(x_2)+\varphi(x_3),$$

with equality precisely on the prescribed triples. Integrating gives the same lower bound $3\int\varphi\,d\mu$ for every admissible coupling. The constructed coupling attains it.

The center triples $(0,e_k,-e_k)$ have equal cost and equal central gradient, making it possible for the two branches to share the same central potential. Strongly concave quadratic terms and the implicit function theorem produce the nearby smooth branches. Strict cost gaps exclude other triples of distinct component centers, while the Coulomb singularity excludes triples containing two points in one sufficiently small component.

Thus optimality forces a concrete geometric constraint, not just a favorable numerical comparison for one candidate plan.

A deterministic choice sends twice too much mass

Suppose a preserving map $T_2$ belonged to an optimal Monge pair. When its input lies in $B$, it must choose one of the four outer branch images. Fix one branch map $H$, and let $S\subset B$ be the measurable set where $T_2(x)=H(x)$.

By construction,

$$\mu(S)=\frac13\nu(S), \qquad \mu(H(S))=\frac16\nu(S).$$

But all of $S$ maps into $H(S)$. Measure preservation would require

$$\frac13\nu(S) \le\mu(T_2^{-1}(H(S))) =\mu(H(S)) =\frac16\nu(S).$$

Hence $\nu(S)=0$. The same reasoning applies to every branch, so no positive central mass can be assigned anywhere allowed by optimality.

Matching only the total mass of each outer component would miss this contradiction. The exact image-measure relation holds for every selection set $S$, and therefore excludes arbitrarily irregular Borel choices as well as continuous ones.

Fine rearrangements still approach the optimum

Partition the five components into increasingly small cells. Record how much mass the optimal coupling assigns to each product of three cells. Split each source cell into pieces of those masses, then map each piece to the required target-cell distributions.

These maps preserve the marginal exactly and reproduce the optimal coupling’s cell-level mass table. Their configurations need not lie exactly on the optimal branches, which is why the preceding obstruction does not apply to approximate optimality.

All active triples stay in three different components separated by a fixed positive distance. On this separated region the Coulomb cost varies continuously, with a bound proportional to cell diameter. Refining the cells makes the excess cost tend to zero.

The separation controls the singular cost on the approximating plans themselves. The argument does not assume that weak convergence alone allows passage through the Coulomb singularity.

7. Scope, limitations, and open questions

The result is existential: some smooth density has this behavior. It does not say that all smooth densities lack optimal maps. The construction uses several separated support components, so it does not resolve every version with connected support or strict positivity on a prescribed region.

There are three marginals, all identical. The theorem does not contradict two-particle map existence or the one-dimensional cyclic-map theory. The inverse-power extension still concerns three particles.

No positive gap separates deterministic and general optimal values. Near-optimal maps exist, but the source does not give a universal complexity bound for them or classify all optimal couplings.

The physical regularity condition on the density does not identify an external potential whose quantum ground state realizes it. Questions about such realizability, robust nonattainment under perturbations and favorable classes admitting maps remain distinct.

8. Sources and further reading

The primary papers linked above distinguish two-particle and one-dimensional existence, equality of infima, failures of prescribed radial maps and the repulsive harmonic predecessor. The Friesecke–Gerolin–Gori-Giorgi account explains the strong-interaction density-functional interpretation.

For optional background, optimal transport and pushforward measures explain the difference between maps and general couplings. The manuscript below supplies the smooth Coulomb nonattainment construction.

Source manuscripts

• M1. OpenAI, A counterexample to the Monge ansatz for the three-marginal Coulomb cost (September 25, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.