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Absence of non-compactly supported minimisers for the Lieb-Oxford bound

This paper proves that any minimizers of the Lieb-Oxford bound for a fixed finite number of particles must be compactly supported, thereby extending a previous result established by Lieb and Oxford for the one-particle case.

Original authors: Simone Di Marino, Rodrigue Lelotte

Published 2026-07-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Simone Di Marino, Rodrigue Lelotte

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, invisible cloud of charged particles floating in space. These particles don't just sit there; they push and pull on each other with a force that gets weaker the farther apart they are, kind of like how a magnet's pull fades as you move it away. Physicists have a famous rule, called the Lieb–Oxford bound, that acts like a safety net. It says, "No matter how you arrange these particles, the total energy of their messy interactions can never drop below a certain floor."

For a long time, scientists wondered about the "perfect" arrangement of these particles—the specific shape of the cloud that would hit that energy floor exactly. In the simplest case, with just one particle, we already knew the answer: the cloud has to be a neat, finite blob. It can't stretch out forever into the infinite void. But what about when you have many particles, say $N$ of them?

Some chemists had a hunch that even for a crowd of particles, the perfect cloud would still be a neat, finite blob. Others weren't so sure. Maybe the best shape was a fuzzy cloud that stretched out forever, getting thinner and thinner but never quite disappearing?

In this paper, Simone Di Marino and Rodrigue Lelotte put that fuzzy-cloud idea to the test. They didn't just guess; they built a mathematical proof to see if a "never-ending" cloud could ever be the perfect solution.

The Big Discovery
Their main finding is a definitive "no." They proved that if a perfect arrangement (a "minimiser") exists for any fixed number of particles $N \ge 1$ , that arrangement must be compactly supported. In plain English: the cloud of particles has to have a hard edge. It must stop existing at some point. It cannot stretch out infinitely.

What They Ruled Out
The paper explicitly argues against the idea that the optimal shape could be a density that spreads out forever. They show that if you try to imagine a solution where the particles are spread over an infinite area (an "unbounded support"), the math breaks down. The energy calculations simply don't add up for an infinite cloud. The authors demonstrate that such a shape would lead to a contradiction, meaning it's impossible for the "perfect" cloud to be infinite.

How Sure Are They?
The authors are not just suggesting this might be true; they have proved it. They didn't run a computer simulation or measure a real experiment; they used rigorous mathematical logic to show that the infinite case is impossible. However, there is one tiny "if" in their statement: they prove that if a perfect shape exists, it must be finite. They note that whether a perfect shape actually exists for every number of particles is still an open question in the math world, but they have settled the question of what that shape would look like if it were found.

The Magic of the Proof
To crack this code, the authors used a clever tool from a field called "optimal transport." Think of this like a logistics problem: imagine you have to move a pile of sand (the particles) from one place to another with the least amount of effort. The "cost" of moving them depends on how far they have to go.

The authors looked at the "price tag" (mathematicians call this a "potential") attached to every point in space for this moving process. They discovered that if the sand cloud stretched out forever, the price tag at the very edge of the universe would behave in a weird, impossible way. It would have to be both high and low at the same time, which is a logical contradiction.

By showing that an infinite cloud forces the math to scream "impossible," they proved that the only way the math can stay calm and consistent is if the cloud has a definite boundary. The particles must huddle together in a finite space, leaving the rest of the universe empty.

So, the next time you imagine a perfect cloud of particles, picture it as a cozy, finite island in the ocean of space, not a mist that fades into infinity. That's the only way the universe's energy rules allow it to be perfect.

Technical Summary: Absence of Non-Compactly Supported Minimisers for the Lieb–Oxford Bound

Problem Statement
The paper addresses the mathematical properties of minimisers for the Lieb–Oxford (LO) bound, a fundamental lower bound for the indirect part of the Coulomb or Riesz energy in systems of charged particles. The LO inequality asserts the existence of a universal constant $c_{LO}(s, d)$ such that for any probability measure $P$ on $(\mathbb{R}^d)^N$ with one-body density $\rho_P$ , the difference between the many-body interaction energy and the Hartree energy is bounded below by $-c_{LO} \int \rho_P^{1+s/d}$ .

While the constant $c_{LO}(s, d)$ is universal (independent of the particle number $N$ ), the authors investigate the specific case where $N$ is fixed and finite. They consider the optimization problem of finding the smallest constant $c_{LO}(s, d, N)$ for which the inequality holds for all $N$ -particle distributions. The central question is the structural nature of the density $\rho$ that minimizes this functional. Specifically, the paper seeks to determine whether such minimisers must be compactly supported. This property was previously established only for the single-particle case ( $N=1$ ) by Lieb and Oxford, while heuristic arguments in the chemistry literature suggested it might hold for general $N$ .

Methodology
The proof relies on a variational analysis of the functional $F(\eta) = C(\eta) - \frac{1}{2}D(\eta) + c_{LO} \int \eta^{1+s/d}$ , where $C(\eta)$ represents the optimal transport cost (multimarginal optimal transport with Riesz cost) and $D(\eta)$ is the Hartree energy.

Optimal Transport Duality: The authors utilize the Kantorovich duality for the multimarginal optimal transport problem. They work with a "strong" formulation of the duality (denoted KD'), where the Kantorovich potential (KP) $\phi$ is an element of a Lebesgue space rather than a strictly continuous function defined everywhere. They establish the existence of maximisers for this dual problem and characterize them via variational inequalities.
Variational Characterization: By treating the LO minimisation as a convex optimization problem, the authors derive a necessary condition for optimality. They show that the function $\phi_\rho = U_\rho - \beta_s c_{LO} \rho^{s/d}$ (where $U_\rho$ is the Riesz potential of $\rho$ and $\beta_s = 1+s/d$ ) acts as a Kantorovich potential for the dual problem.
Regularity Bootstrapping: A critical technical step involves proving the regularity of the density $\rho$ . Using the inner estimate derived from the optimality condition and the Hardy-Littlewood-Sobolev (HLS) inequality, the authors demonstrate that if $\rho$ is a minimiser, it must possess higher local integrability. Through an iterative argument, they prove that $\rho$ is continuous and vanishes on the boundary of its support.
Asymptotic Analysis: The authors leverage recent results regarding the asymptotic behavior of Kantorovich potentials. Specifically, they utilize the fact that for a potential associated with a density of unbounded support, the potential behaves asymptotically as $N-1/|x|^s + C_\phi$ at infinity.

Key Contributions and Results
The primary result of the paper is Theorem 1, which states:

Let $0 < s < d$ and let $\rho \in L^1 \cap L^{1+s/d}(\mathbb{R}^d, \mathbb{R}^+)$ with $\int \rho = N \ge 1$ be a minimiser for the Lieb-Oxford bound for a fixed number of particles $N$ . Then, $\rho$ must be compactly supported.

The proof proceeds by contradiction:

The authors assume the support of $\rho$ is unbounded.
They establish that the constant $c$ appearing in the optimality conditions equals the limit of the Kantorovich potential at infinity ( $C_\phi$ ).
They analyze the behavior of the Riesz potential $U_\rho$ and the potential $\phi$ at infinity.
By comparing the asymptotic lower bound of the Riesz potential (which scales as $N/|x|^s$ ) with the asymptotic behavior of the Kantorovich potential (which scales as $(N-1)/|x|^s$ ), they derive a contradiction. The inequality required for optimality cannot be satisfied if the support extends to infinity.

Additionally, the paper provides Theorem 2, which formalizes the existence and regularity properties of Kantorovich potentials for the strong duality formulation (KD'), showing they can be chosen to be $c$ -conjugated, Lipschitz-continuous, and uniformly bounded.

Significance and Claims
The paper claims to extend a known result from the one-particle case to any finite number of particles $N \ge 1$ . The authors explicitly state that they do not prove the existence of such minimisers, noting that this remains an open question. Instead, their contribution is a structural characterization: if a minimiser exists for a fixed $N$ , it must be compactly supported.

This result rules out the possibility of non-compactly supported minimisers, thereby narrowing the search space for the optimal density in the Lieb-Oxford problem. The authors emphasize that their proof relies on the asymptotic behavior of Kantorovich potentials, a property recently established in the literature, and that the result holds for general Riesz interactions ( $0 < s < d$ ) in any dimension $d$ . The work provides a rigorous mathematical foundation for assumptions often made in the physics and chemistry literature regarding the localization of optimal densities in density functional theory.

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