Fixed-Particle-Number Optimizers for the Lieb–Oxford Inequality
I have posted my paper Fixed-Particle-Number Optimizers for the Lieb–Oxford Inequality on arXiv. The paper studies a basic existence question behind the Lieb–Oxford inequality: if the number of particles is fixed, is the best constant in the inequality actually achieved by some density? I prove that the answer is yes for every finite particle number. The same argument shows that the sharp fixed-particle-number constants increase strictly with the number of particles.
The main difficulty in proving attainment at fixed particle number is loss of compactness. A maximizing sequence can separate into pieces that move arbitrarily far apart, so that, after recentering, some particles remain visible while others escape to infinity. The proof keeps track of the resulting particle-number sectors and uses two strict comparison arguments to show that no such loss can occur at the optimal value. Those comparisons also close an induction in the particle number.
The fixed-particle-number problem
Let us begin with the three-dimensional Coulomb case. Represent the positions of \(N\) particles by a symmetric probability measure \(P\) on \((\mathbb R^3)^N\), and suppose that its one-body measure has a density \(\rho_P\) of total mass \(N\). Its expected pair interaction is
\[ \int_{(\mathbb R^3)^N}\sum_{1\leq i<j\leq N}\frac{1}{|x_i-x_j|}\,\mathrm{d}P. \]
The Hartree, or direct, energy associated with the same one-body density is
\[ \frac12 D(\rho_P) :=\frac12\iint_{\mathbb R^3\times\mathbb R^3} \frac{\rho_P(x)\rho_P(y)}{|x-y|}\,\mathrm{d}x\,\mathrm{d}y. \]
The Lieb–Oxford inequality concerns the indirect Coulomb energy, defined with the sign convention
\[ \begin{aligned} \mathcal E_{\mathrm{ind}}[P] &:=\int_{(\mathbb R^3)^N}\sum_{1\leq i<j\leq N} \frac{1}{|x_i-x_j|}\,\mathrm{d}P -\frac12D(\rho_P)\\ &\geq -c_{\mathrm{LO}}(1,3) \int_{\mathbb R^3}\rho_P(x)^{4/3}\,\mathrm{d}x. \end{aligned} \]
Thus, the indirect energy is the actual pair interaction minus the Hartree energy. It is the part of the interaction that is not captured by the direct term and is closely connected with the exchange-correlation energy in density-functional theory (see, e.g., [PS22] and [LLS23, Sections 2.4 and 5]). Lieb first proved a universal lower bound of this form, and Lieb and Oxford subsequently improved the constant [Lie79, LO81].
There is also a direct connection with the modulated energies that appear in the mean-field theory of Coulomb and Riesz systems. For a pairwise distinct point configuration \(X=(x_1,\ldots,x_N)\in(\mathbb R^3)^N\), set
\[ \mu_X:=\frac1N\sum_{i=1}^N\delta_{x_i}. \]
Given a reference probability measure \(\mu\) on \(\mathbb R^3\), define the normalized, diagonal-excluded Coulomb modulated energy by
\[ \mathsf{F}_N(X,\mu) :=\frac12\iint_{(\mathbb R^3)^2\setminus\Delta} \frac{\mathrm{d}(\mu_X-\mu)(x)\,\mathrm{d}(\mu_X-\mu)(y)}{|x-y|}, \qquad \Delta:=\{(x,x):x\in\mathbb R^3\}. \]
If \(X\sim P\), a direct expansion gives the identity
\[ \mathcal E_{\mathrm{ind}}[P] =N^2\,\mathbb E_{X\sim P} \left[\mathsf{F}_N\left(X,\frac{\rho_P}{N}\right)\right]. \]
This is the same normalized modulated energy that recurs in much of my work on mean-field limits for Coulomb and Riesz systems; see [Ser26, Chapter 4] for a comprehensive introduction and [Ros25] for a survey of its role in sharp commutator estimates and mean-field convergence.
The identity also clarifies the comparison with the standard pointwise lower bound for modulated energy. For every such configuration \(X\) and every bounded probability density \(\mu\),
\[ \mathsf{F}_N(X,\mu) \geq-C\|\mu\|_{L^\infty}^{1/3}N^{-2/3}, \]
whereas, when \(X\sim P\) and \(\mu=\rho_P/N\), the Lieb–Oxford inequality gives
\[ \mathbb E_{X\sim P}[\mathsf{F}_N(X,\mu)] \geq-c_{\mathrm{LO}}(1,3)N^{-2/3} \int_{\mathbb R^3}\mu(x)^{4/3}\,\mathrm{d}x. \]
The two estimates have the same microscopic \(N^{-2/3}\) scale. The first holds configuration by configuration but uses the worst-case density norm \(\|\mu\|_{L^\infty}\); the second is an averaged inequality tied to the one-body density of \(P\), but it uses the finer local-density integral. For Coulomb and super-Coulomb Riesz interactions, the pointwise lower bound and its optimal order are discussed in [Ser26, Sections 4.2 and 12.4]. The same optimal-order lower bound in the sub-Coulomb regime was proved by Hess-Childs, Serfaty, and me [HCRS25, Remark 2.13].1
The constant is universal in the sense that it is independent of \(N\). Its exact sharp value remains unknown despite substantial work on successively improving upper and lower bounds; see, for example, [LLS22, LLS23] and the references therein. In separate computer-assisted work, I certified the slightly sharper bound \(c_{\mathrm{LO}}(1,3)\leq1.575524158829\), improving the published \(1.58\) bound of Lewin, Lieb, and Seiringer. The certificate report is publicly available here. Determining the sharp constant is important both for the variational problem itself and for quantifying a fundamental constraint on density-functional approximations.
The same problem makes sense for the Riesz interaction \(|x-y|^{-\mathsf{s}}\) in every dimension \(\mathsf{d}\geq1\) and throughout the locally integrable range \(0<\mathsf{s}<\mathsf{d}\).2 For a density \(\rho\geq0\) of mass \(N\), let
\[ \mathcal W_N(\rho) :=\min_{\substack{P\ \mathrm{symmetric}\\ \rho_P=\rho}} \int_{(\mathbb R^{\mathsf{d}})^N} \sum_{1\leq i<j\leq N}\frac{1}{|x_i-x_j|^{\mathsf{s}}}\,\mathrm{d}P \]
be the smallest possible pair interaction among all \(N\)-particle probability measures with one-body density \(\rho\). Define the signed gain relative to the Hartree term by
\[ \mathcal G_N(\rho):=\frac12D(\rho)-\mathcal W_N(\rho), \qquad D(\rho):=\iint_{\mathbb R^{\mathsf{d}}\times\mathbb R^{\mathsf{d}}} \frac{\rho(x)\rho(y)}{|x-y|^{\mathsf{s}}}\,\mathrm{d}x\,\mathrm{d}y. \] Thus, \(\mathcal G_N(\rho)\) is the largest reduction of the pair interaction below the Hartree value among plans with one-body density \(\rho\); equivalently, \(-\mathcal G_N(\rho)\) is the smallest indirect Riesz energy at that density. The sharp fixed-particle-number constant is
\[ \Lambda_N(\mathsf{s},\mathsf{d}) :=\sup_{\substack{\rho\geq0,\ \int_{\mathbb R^{\mathsf{d}}}\rho(x)\,\mathrm{d}x=N\\ \rho\in L^1(\mathbb R^{\mathsf{d}})\cap L^{1+\frac{\mathsf{s}}{\mathsf{d}}}(\mathbb R^{\mathsf{d}})}} \frac{\mathcal G_N(\rho)} {\displaystyle\int_{\mathbb R^{\mathsf{d}}}\rho(x)^{1+\mathsf{s}/\mathsf{d}}\,\mathrm{d}x}. \]
Here, an optimizer is a density \(\rho\) attaining this supremum. This should be distinguished from an optimal plan, which is an \(N\)-particle probability measure attaining the minimum that defines \(\mathcal W_N(\rho)\). The paper proves existence of optimizing densities; optimal plans at a prescribed admissible density already follow from the direct method.
The fixed-particle constants recover the universal sharp constant through
\[ c_{\mathrm{LO}}(\mathsf{s},\mathsf{d})=\sup_{N\geq1}\Lambda_N(\mathsf{s},\mathsf{d}). \]
They therefore form a natural finite-particle hierarchy inside the universal sharp-constant problem. In the three-dimensional Coulomb case, \(\Lambda_N(1,3)\) is the constant denoted by \(C_N\) in the original paper of Lieb and Oxford.
What the paper proves
The main result is the following.
For every \(\mathsf{d}\geq1\), every \(0<\mathsf{s}<\mathsf{d}\), and every finite \(N\geq1\), the supremum defining \(\Lambda_N(\mathsf{s},\mathsf{d})\) is attained. Moreover, \[ 0<\Lambda_1(\mathsf{s},\mathsf{d})<\Lambda_2(\mathsf{s},\mathsf{d})<\Lambda_3(\mathsf{s},\mathsf{d})<\cdots. \]
Combining existence with the recent compact-support theorem of Di Marino and Lelotte shows that every fixed-\(N\) optimizer is compactly supported [DML26]. Since the universal constant is the supremum of a strictly increasing sequence, it is not attained at any finite particle number.
The one-particle case has a special structure. There is no pair interaction when \(N=1\), so \(\mathcal W_1=0\) and the problem reduces to a scale-invariant quotient between the Riesz-Hartree energy and a local density power. In the three-dimensional Coulomb case, Lieb and Oxford proved that the supremum is attained by a symmetric decreasing, compactly supported density. Calvez, Carrillo, and Hoffmann later proved attainment throughout the full Riesz range and obtained a radially non-increasing, compactly supported, bounded optimizer [CCH17].
For \(N\geq2\), however, the interaction is a genuine multimarginal optimal transport problem. Previous work provided numerical bounds and reformulations of the fixed-particle constants and explored their density-functional applications [SVGG16], but, to my knowledge, attainment of the exact fixed-\(N\) variational problem remained open. Very recently, Di Marino and Lelotte proved that any fixed-particle optimizer, if it exists, must be compactly supported, but their theorem did not establish existence [DML26]. Their paper directly inspired the present work.
There was also a basic monotonicity result already in Lieb and Oxford’s work. By adding an independent particle whose position has a smooth density dilated to an increasingly large spatial scale (a diffuse particle at infinity), one obtains
\[ \Lambda_N(\mathsf{s},\mathsf{d})\leq\Lambda_{N+1}(\mathsf{s},\mathsf{d}). \]
Their argument was written for the Coulomb interaction, but it uses only homogeneity and extends to the full Riesz range. What was missing was strictness and, with it, a mechanism for excluding loss of compactness at every finite particle number.
From particles at infinity to a closed induction
A first attempt at existence would start with a maximizing sequence of densities and try to extract a convergent subsequence. The quotient defining \(\Lambda_N\) is invariant under translations and dilations, so one must first fix those symmetries. Even after doing so, a maximizing sequence can split into two spatially separated nontrivial pieces (the dichotomy alternative in Lions’s concentration–compactness principle [Lio84]). Repeating the extraction can produce several profiles whose mutual distances diverge.
There is an additional issue at the level of the \(N\)-particle plans. After recentering around one density profile, only some of the \(N\) particle coordinates may remain in bounded sets. The remaining coordinates escape to infinity. The limiting object can therefore assign positive probability to several particle-number sectors—for example, to a \(k\)-particle sector and an \(\ell\)-particle sector. In this sense, the limiting object obtained after recentering is grand-canonical even though every member of the original sequence has exactly \(N\) particles. This point of view is closely related to the grand-canonical optimal transport formalism developed by Di Marino, Lewin, and Nenna [DMLN25].
The proof is organized around the following schematic:
\[ \begin{array}{c} \boxed{\displaystyle \Lambda_N>\max_{1\leq k<N}\Lambda_k} \\[6pt] {\scriptstyle \Downarrow\quad \text{via concentration--compactness and strict completion}} \\[6pt] \boxed{\displaystyle \Lambda_N\ \text{is attained}} \\[6pt] {\scriptstyle \Downarrow\quad \text{via compact support and strict one-particle extension}} \\[6pt] \boxed{\displaystyle \Lambda_{N+1}>\Lambda_N}. \end{array} \]
The nonstrict monotonicity then gives \(\Lambda_k\leq\Lambda_N<\Lambda_{N+1}\) for every \(k\leq N\), which supplies the strict lower-particle inequality needed at the next step. At \(N=1\) there are no lower-particle constants, so the first condition is vacuous. The three ingredients therefore form a closed induction.
Completing a fluctuating profile
Decompose the limiting profile into its \(k\)-particle sectors and let \(K\in\{0,\ldots,N\}\) be the associated particle-number random variable, so that \(\mathbb P(K=k)\) is the total probability mass of the \(k\)-particle sector. The associated sector-weighted one-body density therefore has total mass \(\mathbb E[K]\). Suppose that \(\mathbb E[K]<N\). If \(K=k<N\) almost surely, then the strict inequality \(\Lambda_k<\Lambda_N\) rules out the profile at the optimal value. The more delicate case is when \(K\) is not constant and hence fluctuates among several sectors.
For such a truncated grand-canonical profile \(\Gamma\), write
\[ Q(\Gamma):= \frac{\frac12D(\rho_\Gamma)-\mathcal W(\Gamma)} {\displaystyle\int_{\mathbb R^{\mathsf d}} \rho_\Gamma(x)^{1+\mathsf s/\mathsf d}\,\mathrm{d}x}, \]
where \(\mathcal W(\Gamma)\) is the sector-averaged pair interaction. This is the grand-canonical Lieb–Oxford quotient evaluated at a profile \(\Gamma\) arising from the profile decomposition of a fixed-\(N\) maximizing sequence.
To compare such a profile with the fixed-\(N\) problem, every \(k\)-particle sector is completed to exactly \(N\) particles and then symmetrized. An exact identity shows that the fluctuation of the particle number produces a signed interaction defect whose total mass is
\[ \operatorname{Var}(K)>0. \]
The missing particles are added on two spatial scales. A fraction \(\varepsilon\) of the added density is placed on a large remote annulus of radius \(R\). At that distance, its leading Riesz interaction with the signed defect depends only on the defect’s total mass \(\operatorname{Var}(K)\) and is therefore positive, giving a contribution of order \(\varepsilon R^{-\mathsf s}\). The remaining fraction is spread on a much larger annulus so that its Riesz interaction and local-density costs are negligible. With the first annulus fixed, the gain is of order \(\varepsilon\), whereas the relevant local-power cost is of order \(\varepsilon^{1+\mathsf{s}/\mathsf{d}}=o(\varepsilon)\).
Writing \(\widetilde\rho_\varepsilon\) for the one-body density of the completed \(N\)-particle state, the strict comparison is
\[ Q(\Gamma) <\frac{\mathcal G_N(\widetilde\rho_\varepsilon)} {\displaystyle\int_{\mathbb R^{\mathsf d}} \widetilde\rho_\varepsilon(x)^{1+\mathsf s/\mathsf d}\,\mathrm{d}x} \leq\Lambda_N. \]
Thus, \(Q(\Gamma)<\Lambda_N\), so a profile with fluctuating particle number cannot account for the full limiting value of a fixed-\(N\) maximizing sequence.
This strict completion, combined with the profile decomposition, leaves only one possibility for a maximizing sequence: after suitable translations and dilations, one profile retains all \(N\) particles. That profile gives an optimizing density.
Adding one more particle strictly
Existence at particle number \(N\) is only half of the induction. To continue, one must improve the nonstrict inequality \(\Lambda_N\leq\Lambda_{N+1}\) to a strict one.
Let \(\rho\) be an optimizing density at particle number \(N\), and let \(P\) be an associated optimal plan. The theorem of Di Marino and Lelotte [DML26] implies that \(\rho\) is compactly supported, and hence \(P\) is supported on configurations in a fixed compact set. Adding an independent particle whose density is spread over an increasingly large spatial scale gives only the nonstrict comparison.
For strictness, a small portion of the added particle’s density is placed in a fixed region outside the support of \(\rho\), and its position is coupled non-independently with the original \(N\)-particle configuration. If \(X=(x_1,\ldots,x_N)\) is a configuration sampled from \(P\) and \(y\) is the added particle, their cross-interaction is
\[ W(X,y):=\sum_{i=1}^N\frac1{|x_i-y|^{\mathsf s}}. \]
If the product coupling were optimal, its support would satisfy the two-cycle case of cyclical monotonicity [GM96, Theorem 2.3]:
\[ W(X,y)+W(X',y')\leq W(X,y')+W(X',y). \]
Because a product support contains both pairings, the reverse inequality also holds, forcing equality. It follows that \(W(X,\cdot)-W(X',\cdot)\) is constant on the exterior region. Exterior uniqueness for Riesz potentials then forces \(X\) and \(X'\) to have the same empirical counting measure, which is incompatible with the fact that their average counting measure is the absolutely continuous density \(\rho\). A non-product perturbation therefore lowers the interaction between the original configuration and the added particle while preserving both marginals.
The rest of the added density is again sent to a much larger scale. The strict interaction improvement is first order in the amount of locally added mass, while the new local-density contribution is of higher order. This yields
\[ \Lambda_{N+1}(\mathsf{s},\mathsf{d})>\Lambda_N(\mathsf{s},\mathsf{d}), \]
which closes the induction.
What remains open
The theorem proves that optimizing densities exist, but it does not characterize them. Even after quotienting out the natural translation, rotation, and dilation symmetries, uniqueness is open. The proof gives no symmetry statement, no description of the geometry of the support, and no regularity beyond membership in the natural \(L^1\cap L^{1+\frac{\mathsf{s}}{\mathsf{d}}}\) class together with compact support. It also does not characterize the associated multimarginal optimal plans, whose structure and uniqueness form a separate problem. An Euler–Lagrange or dual characterization of the optimizing densities may provide a route to these questions.
The strict gaps
\[ \Lambda_{N+1}(\mathsf{s},\mathsf{d})-\Lambda_N(\mathsf{s},\mathsf{d})>0 \]
are qualitative. The proof of \(\Lambda_{N+1}>\Lambda_N\) starts from a particular optimizing density and an associated optimal plan, and the resulting reduction in the cross-interaction is not controlled uniformly over those choices or uniformly in \(N\). Quantitative lower bounds for the successive gaps would require estimates on a suitably normalized family of optimizers after fixing translation and dilation. Such estimates could also illuminate how quickly the fixed-particle constants approach the universal constant.
Finally, after centering and scaling, ordinary weak subsequential limits of the normalized empirical measures are not difficult to obtain. The substantive questions are whether a natural normalization and selection of optimizers produce a canonical macroscopic limit, and whether microscopic blow-ups of the associated optimal plans converge to a stationary infinite-volume point process. Weak empirical-measure convergence alone forgets the pair correlations responsible for the indirect energy. In the three-dimensional Coulomb case, the widely discussed conjectural uniform-electron-gas value for \(c_{\mathrm{LO}}(1,3)\) is determined by precisely such local correlations. Testing that conjecture through fixed-\(N\) optimizers would therefore require a microscopic blow-up to unit local density and convergence of the associated indirect energy, not merely a macroscopic weak limit [LLS22].
The paper is available on arXiv; the PDF can be downloaded here.
References
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[DML26] S. Di Marino and R. Lelotte, “Absence of non-compactly supported minimisers for the Lieb–Oxford bound,” arXiv:2607.11440 (2026). arXiv:2607.11440.
[DMLN25] S. Di Marino, M. Lewin, and L. Nenna, “Grand-canonical optimal transport,” Archive for Rational Mechanics and Analysis 249 (2025), Paper No. 12. doi:10.1007/s00205-024-02080-x; arXiv:2201.06859.
[GM96] W. Gangbo and R. J. McCann, “The geometry of optimal transportation,” Acta Mathematica 177 (1996), no. 2, 113–161. doi:10.1007/BF02392620.
[HCRS25] E. Hess-Childs, M. Rosenzweig, and S. Serfaty, “A sharp commutator estimate for all Riesz modulated energies,” arXiv:2511.13461 (2025). arXiv:2511.13461.
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[Lio84] P.-L. Lions, “The concentration–compactness principle in the calculus of variations. The locally compact case. Part I,” Annales de l’Institut Henri Poincaré. C, Analyse non linéaire 1 (1984), no. 2, 109–145. doi:10.1016/S0294-1449(16)30428-0.
[LLS22] M. Lewin, E. H. Lieb, and R. Seiringer, “Improved Lieb–Oxford bound on the indirect and exchange energies,” Letters in Mathematical Physics 112 (2022), Paper No. 92. doi:10.1007/s11005-022-01584-5; arXiv:2203.12473.
[LLS23] M. Lewin, E. H. Lieb, and R. Seiringer, “Universal functionals in density functional theory,” in Density Functional Theory: Modeling, Mathematical Analysis, Computational Methods, and Applications, Springer, 2023, 115–182. doi:10.1007/978-3-031-22340-2_3; arXiv:1912.10424.
[LO81] E. H. Lieb and S. Oxford, “Improved lower bound on the indirect Coulomb energy,” International Journal of Quantum Chemistry 19 (1981), no. 3, 427–439. doi:10.1002/qua.560190306.
[PS22] J. P. Perdew and J. Sun, “The Lieb–Oxford lower bounds on the Coulomb energy, their importance to electron density functional theory, and a conjectured tight bound on exchange,” in The Physics and Mathematics of Elliott Lieb, European Mathematical Society Press, 2022, 165–178. doi:10.4171/90-2/36; arXiv:2206.09974.
[Ros25] M. Rosenzweig, “Commutators, mean-field, and supercritical mean-field limits for Coulomb/Riesz gases,” Journées équations aux dérivées partielles (2025), Talk no. 7, 1–32. doi:10.5802/jedp.698; article page.
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[SVGG16] M. Seidl, S. Vuckovic, and P. Gori-Giorgi, “Challenging the Lieb–Oxford bound in a systematic way,” Molecular Physics 114 (2016), no. 7–8, 1076–1085. doi:10.1080/00268976.2015.1136440; arXiv:1508.01715.
Footnotes
Serfaty uses the unnormalized modulated energy \(N^2\mathsf{F}_N\). For general Riesz interactions, both [Ser26] and [HCRS25] use the kernel \(\mathsf{s}^{-1}|x|^{-\mathsf{s}}\) for \(\mathsf{s}>0\); this agrees with the three-dimensional Coulomb convention \(\mathsf{s}=1\) and otherwise differs from the convention of this post only by a constant factor.↩︎
The endpoint \(\mathsf{s}=0\) is different: the literal kernel \(|x-y|^0\) is constant, while the two-dimensional Coulomb kernel \(-\log|x-y|\) arises only after the renormalized limit \[ \frac{|x-y|^{-\mathsf{s}}-1}{\mathsf{s}} \longrightarrow-\log|x-y|. \] Moreover, the strict comparison used here relies on \(\varepsilon^{1+\mathsf{s}/\mathsf{d}}=o(\varepsilon)\), which fails at \(\mathsf{s}=0\).↩︎
Citation
@online{rosenzweig2026,
author = {Rosenzweig, Matthew},
title = {Fixed-Particle-Number {Optimizers} for the {Lieb–Oxford}
{Inequality}},
date = {2026-08-18},
url = {https://matthewrosenzweigwork-max.github.io/posts/fixed-particle-number-optimizers-lieb-oxford/},
langid = {en}
}