The fixed-particle-number problem

Let us begin with the three-dimensional Coulomb case. Represent the positions of particles by a symmetric probability measure on , and suppose that its one-body measure has a density of total mass . Its expected pair interaction is

The Hartree, or direct, energy associated with the same one-body density is

The Lieb–Oxford inequality concerns the indirect Coulomb energy, defined with the sign convention

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 , set

Given a reference probability measure on , define the normalized, diagonal-excluded Coulomb modulated energy by

If , a direct expansion gives the identity

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 and every bounded probability density ,

whereas, when and , the Lieb–Oxford inequality gives

The two estimates have the same microscopic scale. The first holds configuration by configuration but uses the worst-case density norm ; the second is an averaged inequality tied to the one-body density of , 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 . 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 , improving the published 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 in every dimension and throughout the locally integrable range .² For a density of mass , let

be the smallest possible pair interaction among all -particle probability measures with one-body density . Define the signed gain relative to the Hartree term by

Thus, is the largest reduction of the pair interaction below the Hartree value among plans with one-body density ; equivalently, is the smallest indirect Riesz energy at that density. The sharp fixed-particle-number constant is

Here, an optimizer is a density attaining this supremum. This should be distinguished from an optimal plan, which is an -particle probability measure attaining the minimum that defines . 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

They therefore form a natural finite-particle hierarchy inside the universal sharp-constant problem. In the three-dimensional Coulomb case, is the constant denoted by in the original paper of Lieb and Oxford.

What the paper proves

The main result is the following.

For every , every , and every finite , the supremum defining is attained. Moreover,

Combining existence with the recent compact-support theorem of Di Marino and Lelotte shows that every fixed- 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 , so 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 , 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- 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

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 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 -particle plans. After recentering around one density profile, only some of the 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 -particle sector and an -particle sector. In this sense, the limiting object obtained after recentering is grand-canonical even though every member of the original sequence has exactly 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:

The nonstrict monotonicity then gives for every , which supplies the strict lower-particle inequality needed at the next step. At there are no lower-particle constants, so the first condition is vacuous. The three ingredients therefore form a closed 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 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

are qualitative. The proof of 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 . 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 is determined by precisely such local correlations. Testing that conjecture through fixed- 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

[CCH17] V. Calvez, J. A. Carrillo, and F. Hoffmann, “Equilibria of homogeneous functionals in the fair-competition regime,” Nonlinear Analysis 159 (2017), 85–128. doi:10.1016/j.na.2017.03.008; arXiv:1610.00939.

[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.

[Lie79] E. H. Lieb, “A lower bound for Coulomb energies,” Physics Letters A 70 (1979), no. 5–6, 444–446. doi:10.1016/0375-9601(79)90358-X.

[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.

[Ser26] S. Serfaty, Lectures on Coulomb and Riesz Gases, Colloquium Publications 70, American Mathematical Society, Providence, RI, 2026. doi:10.1090/coll/070; arXiv:2407.21194.

[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.

1 The problem and a first uniform theorem

For and a fixed inverse temperature , consider the canonical Gibbs ensemble on the full labeled configuration space . Here, “canonical” means that the particle number is fixed. We write

for the planar logarithmic Coulomb kernel, and define

Equivalently,

Since as , the interaction is repulsive and the density vanishes at collisions.

The factor in (1.3) places the system at the diffusive temperature scale. The reversible overdamped Langevin dynamics is formally

and the noise amplitude remains of order one as . This differs from the usual random-matrix scaling (see, e.g., [Ser24]), where the Gibbs exponent is of order and the effective noise vanishes with the particle number.

For a probability measure , set

Our convention for the logarithmic Sobolev constant is

The domain in (1.6) requires some care because the density vanishes at collisions. Let

and, initially for , set

We denote the closed form domain by

The second equality follows from the zero-capacity argument in Section 2. Thus, is the full closed weighted Sobolev domain, with no boundary condition imposed at collisions.

The question is whether, for each fixed , one can find

We impose no symmetry or other structural assumption on the observable .

To identify where particle-number dependence can enter, we first separate the center of mass. In the present quadratic setting, this separation is exact. Set

Then,

Thus, factors into a two-dimensional Gaussian proportional to and a Gibbs measure on the relative hyperplane . The Gaussian factor has Poincaré cost and logarithmic Sobolev rate ; all of the nontrivial work lies in the relative coordinates.

For each fixed , a qualitative logarithmic Sobolev inequality follows from the existing Lyapunov theory, as we explain in Section 2. The difficulty is quantitative: the fixed-dimensional argument gives no reason for its constant to remain positive while the ambient dimension and the number of collision strata grow with .

Set

The subscript records the integrated Bochner estimate from which this particular number arises.

The main result of this post is the following uniform logarithmic Sobolev inequality at small inverse temperature.

A later refinement of the same perturbative strategy extends Theorem 1.1 to every . We will explain that improvement in Section 6. The first proof remains worth presenting because its numerical threshold is determined by a single inequality needed to close the estimate, while the rest of the argument already has the form used later.

The strategy of the proof is classical: one proves a uniform Poincaré inequality and, independently, a defective logarithmic Sobolev inequality, and then removes the defect by Rothaus’s centering argument [Rot85]. Uniformity comes from the two input estimates, not from the tightening step itself.

2 Relation with existing work

The relevant literature is best understood in terms of proof mechanisms rather than as a linear progression of increasingly weak hypotheses. The classical dimension-free theory begins with the Gaussian logarithmic Sobolev inequality, tensorization, hypercontractivity, the Bakry–Émery curvature criterion, and bounded perturbation; see Gross [Gro75], Holley–Stroock [HS87], and Bakry–Gentil–Ledoux [BGL14]. For broader treatments of functional inequalities and concentration, see also [ABC+00, Led01]. Weakly coupled and multiscale Gibbs systems are instead treated by assembling one-site or block inequalities with a quantitative interaction estimate, as in the Dobrushin–Zegarlinski theory, the Otto–Reznikoff criterion, and two-scale decompositions [GZ03, OR07, GOVW09].

For regular mean-field particle systems, recent proofs use flat convexity, conditional coercivity, reverse-heat-flow transport, uniform spectral gaps, or a free-energy decomposition; representative works include [GLWZ22, Wan24, CNZ24, Mon24, BBD25]. These theories typically require global control of the interaction field, an bound for its Hessian, or a worst-case influence matrix. The logarithmic kernel presents two distinct obstructions: its gradient and Hessian diverge at collisions, while at every nonzero separation its Hessian has one positive and one negative eigenvalue. Thus, the interaction is neither convex nor concave even off the collision set.

Singular one-dimensional systems admit a different mechanism. Chafaï–Lehec [CL20] and my work with Serfaty [RS25] exploit ordering, convexity on a Weyl chamber, and one-dimensional transport or rigidity. In the plane, particles can pass around one another without crossing the collision set, so there is no analogous global chamber decomposition.

The closest dynamical antecedents are the planar Coulomb works of Bolley–Chafaï–Fontbona [BCF18] and Lu–Mattingly [LM20]. The former study the overdamped dynamics and prove fixed- functional and ergodic estimates. The latter write down the overdamped system, identify the potential energy as its natural Lyapunov function, and develop the collision-sensitive force estimates in the kinetic setting. These results make the following qualitative statement essentially part of the existing Lyapunov folklore, even though the exact formulation does not seem to have been written down explicitly in the literature.

We make no novelty claim for Proposition 2.1. Bolley–Chafaï–Fontbona explicitly point to the local-to-global Lyapunov route and the collision-domain difficulty [BCF18, Section 1.4.5]; the abstract mechanism is developed in [CG14, CGWW09, CG17]. Here is a sketch of the proof.

Finiteness and capacity at collisions. Here, capacity means the variational -capacity associated with the weighted Dirichlet form and its form norm; see [FOT11, Section 2.1]. Fix , set , and choose a smooth function equal to one on and zero on . For , the cutoff equals one near the collision divisor and satisfies . Writing , the Gibbs density contributes the factor , while polar coordinates contribute the Jacobian . The remaining variables have an -independent integrable majorant, owing to the Gaussian confinement. Therefore, Likewise, Thus, every collision divisor has zero weighted capacity. Since the full collision set is a finite union of these divisors, it also has zero capacity. Equivalently, cutoffs vanishing near the full collision set converge to one in the form norm, which proves the second equality in (1.9).

Lyapunov coercivity. Let

On , if with , then

Because is harmonic away from the origin,

The singular Coulomb forces remain collectively large as a bounded configuration approaches the collision set, despite possible cancellations among individual pair forces. Together with quadratic confinement at infinity, this makes the energy gradient coercive outside a compact subset of the collision-free configuration space. The simpler identity

shows the tail mechanism directly: for ,

This radial Lyapunov function does not control collisions by itself, which is why the singular force estimate is still needed.

From local to global. The region where the Lyapunov drift is not coercive is compactly contained in . On a connected neighborhood of that compact set, the density is smooth and bounded above and below, so ordinary local Poincaré and Sobolev inequalities apply. The local-to-global argument yields a defective logarithmic Sobolev inequality and a fixed- Poincaré inequality; Rothaus tightening then gives the stated LSI.

We emphasize that every constant in the preceding argument may depend on the ambient dimension , the number of collision divisors, the compact set on which local coercivity is invoked, and the Lyapunov threshold. Thus, the uniform LSI we will show should not be viewed as an optimization of Proposition 2.1. Rather, it requires a new approach in which the particle-number dependence cancels explicitly.

3 Conditional weighted inequalities and labeled fluctuations

The logarithmic interaction has an important structural property: after conditioning on all but one particle, or all but one pair, the remaining density is a Gaussian multiplied by positive powers of distances to finitely many points or affine subspaces. The number of factors grows with , but, since is held fixed as , their exponents add to a quantity of order one. Up to this point, only the logarithmic form of the interaction has been used and not the fact that we are in a planar setting. The latter enters later through the circle estimate, complex analysis, and the two-dimensional weighted theory discussed in Section 6.

Condition first on . The one-site conditional law is

whose distance factors have total exponent

Here and below, the total exponent means the sum of the powers appearing in a product of distance factors. In the sequel, we will refer to the -regularization of a factor , meaning its replacement by , with .

4 The uniform Poincaré inequality

The variance decomposition in (3.18) now dictates the proof. We first control the permutation-invariant component by combining the integrated Bochner identity with a conditional weighted Wirtinger inequality. We then control the label-dependent component using the spectral-gap inequality for the random-transposition walk together with the two-particle conditional Poincaré inequality.

5 A defective logarithmic Sobolev inequality

The Poincaré argument exploits an orthogonal decomposition into permutation-invariant and label-dependent components. That decomposition does not by itself yield the entropy estimate needed here, so we work directly with an arbitrary labeled density. After recentering the canonical ensemble at its thermal equilibrium, the one-site logarithmic Sobolev inequality can be summed using a conditional entropy estimate, while a repulsive partition bound prevents the resulting zero-order defect from growing with .

The final passage from this defective inequality and the Poincaré estimate of Section 4 to a genuine LSI is the classical Rothaus argument. This strategy appears in many later Lyapunov, multiscale, metastable, and mean-field developments; see, for example, [CGWW09, BGL14, MS14, Wan24].

6 Beyond the Bochner threshold

The entropy argument already works for every , so the first obstruction lies entirely in the Poincaré estimate. It turns out that the Bochner calculation for permutation-invariant observables can be replaced by a weighted estimate, while the full two-particle conditional LSI can be replaced by the relative-coordinate Poincaré inequality of Proposition 3.3. These two changes extend the perturbative theorem to the whole range .

7 Consequences, neighboring models, and the problem behind Part II

The most distinctive consequence of Theorem 1.1 is the space on which the inequality holds. It acts on every labeled observable, not only on functions of the empirical measure. For example, exchangeability gives , and the Poincaré consequence of the theorem yields

For nonconstant , the tagged observable is not determined by the empirical measure. Consequently, an inequality restricted to test functions of the empirical measure does not apply to this observable directly. Dynamically, the same distinction means that entropy decay applies to finite-entropy initial laws that are not exchangeable, and therefore controls perturbations carrying information in the particle labels.

The usual consequences now remain uniform as the dimension grows. The physical generator is

and the LSI gives

Thus, the entropic relaxation time does not diverge solely because increases. This does not imply an -independent total-variation mixing time from arbitrary initial data, since the initial entropy may grow like or be infinite.

Similarly, for a Lipschitz function and

one has

Herbst’s argument therefore gives the natural order- concentration exponent

A fixed- inequality with would not retain either (7.3) or (7.6) at these scales.

Uniform finite-particle LSIs are also the natural microscopic coercivity estimate in the modulated logarithmic Sobolev approach to generation of chaos [RS25]. The equilibrium theorem proved here is not by itself a complete dynamical propagation-of-chaos result: one must still compare the particle law with the evolving mean-field reference and control the corresponding modulated entropy production. It does, however, supply the full labeled equilibrium endpoint rather than only a symmetric or empirical-measure inequality.

Energy kernels as transport energies

Fix a target probability distribution on . For , we consider the kernel

When the relevant moments are finite, the resulting squared Maximum Mean Discrepancy (MMD) is

For , this is the energy distance used in statistics [1, 2]. More generally, it is, up to a constant, the squared homogeneous Sobolev norm . We fix and evolve by the Wasserstein gradient flow of this discrepancy. Formally, is transported by the nonlocal velocity field generated by .

Energy-kernel MMDs combine sensitivity to large-scale separation with dimension-free empirical approximation: the MMD between a distribution and the empirical measure of independent samples is typically of order , independently of the ambient dimension. This also connects the problem to deterministic quantization. Fornasier and Hütter established qualitative consistency of equal-weight power-kernel quantizers, while Colasanto, Focardi, Fornasier, and Mattesini recently obtained sharp empirical-quantization rates for power kernels [3, 4]. In joint work with Hess-Childs and Serfaty, we have also obtained sharp empirical-quantization rates for a broader class of homogeneous and screened Riesz MMDs and for their singular diagonal-excluded counterparts, known as modulated energies [5].

From continuum flow to particle dynamics

The nonsmoothness that makes these kernels useful also puts the dynamics outside standard theory. The interaction force is not Lipschitz at the particle diagonal, and in dimensions the energy is not displacement semiconvex in Wasserstein space. The continuum equation and the particle system therefore have to be constructed separately.

In the range , we prove global existence and uniqueness for probability densities in subcritical spaces, with finite moments; we also include the endpoint , whose special structure makes it explicitly solvable. For the diagonal-free particle system, we prove global noncollision and convergence, at fixed , to the collision-free critical set. We construct explicit collision-free saddle equilibria, implying the particles do not in general evolve to an optimal empirical quantizer of the target.

We then connect the two descriptions through a modulated-energy estimate, which is a type of weak-strong stability estimate widely used in the equilibrium and nonequilibrium analysis of Coulomb/Riesz gases (see, e.g., [6]). It gives a quantitative mean-field limit on every fixed time interval, and for well-prepared random initial data it propagates the dimension-free MMD scale from the initial empirical approximation to the evolving particle system.

The long-time picture is subtler. In most of the parameter range, an absolutely continuous Lagrangian critical point must equal the target. The exceptions at the natural moment level are confined to in dimensions one and three: in dimension three we recover rigidity under several additional hypotheses, while in dimension one, outside our present continuum well-posedness range, we construct compactly supported non-minimizing critical points. Energy dissipation gives convergence toward the critical set without additional long-time bounds for , and with uniform moment bounds and subcritical bounds for . In the rigid regimes, those bounds also give convergence to the target.

There is overlap here with recent work of Fornasier, Huang, and Sun on a broader attraction–repulsion equation [7]. When their attractive and repulsive exponents agree, their model contains, as a special case, the energy-kernel MMD flow for ; their main emphasis is the unequal-exponent regime, and their results provide a complementary Lagrangian well-posedness and stationary-state theory.

Transport first, relaxation second

One of the main messages of the paper is that convergence has two distinct scales. If a source is initially placed a distance from a compactly supported target, then the far-field velocity is of order . The source therefore needs a time of order merely to reach the target region. On the whole space, this rules out both a global Polyak–Łojasiewicz inequality and any multiplicative MMD decay rate that is uniform over all initial data. It does not rule out rapid relaxation after a waiting time determined by the initial geometry.

The distinction becomes completely explicit for the one-dimensional energy distance. The ordered particle equations decouple through the target cumulative distribution function. A particle outside the target support first moves toward it at constant velocity. Once it enters, it converges exponentially to its equilibrium quantile provided the target density is bounded below on its support. Thus a finite entrance time separates a transport phase from a relaxation phase. The lower bound on the target density is essential. For example, if a target density vanishes quadratically at its midpoint, then a displaced middle particle relaxes only like , and its excess energy decays like . The continuum flow displays the same mechanism quantile by quantile, although for a source with unbounded support there need not be a finite time by which every quantile has entered the target region.

Earlier work of Di Francesco, Fornasier, Hütter, and Matthes analyzed the same balanced one-dimensional power flow through pseudo-inverse distribution functions and proved well-posedness and long-time convergence in several regimes [8]. The distance-kernel flow was also recently studied through quantiles by Duong, Stein, Beinert, Hertrich, and Steidl [9]. Our example isolates the transport and post-entry relaxation scales and shows explicitly how degeneracy of the target changes the second scale. We expect this two-timescale mechanism to be much more general than the example in which we can presently prove it.

What remains open

Three questions come to mind immediately.

First, can the moment bounds and subcritical bounds needed to identify long-time limits be propagated uniformly in time? In the rigid regimes, a positive answer would turn our conditional convergence theory into unconditional convergence to the target.

Second, does rigidity at the natural moment level also hold in the remaining regime , , or can there be an absolutely continuous non-minimizing critical point there?

Third, can the two-timescale picture be established for the full family of energy-kernel flows in arbitrary dimension? This would require a theory that separates the datum-dependent time needed to transport mass into the relevant target region from the subsequent local relaxation and that explains how the latter depends on target positivity and degeneracy. On the torus, Chizat, Colombo, Colombo, and Fernández-Real prove polynomial convergence near sufficiently regular positive targets in the corresponding non-Coulomb Sobolev regime [10]. Their local-in-data theory complements our global obstruction: the failure of a global PL inequality does not preclude local decay, but it explains why such a local argument does not yield a global exponential theory. Understanding the analogous local theory on the whole space and its interaction with particle approximation and the long-time mean-field limit is part of the same problem.

The paper is available on arXiv; the PDF can be downloaded here.

I thank Massimo Fornasier for bringing his related work on this topic to our attention.

References

1. G. J. Székely and M. L. Rizzo, “A new test for multivariate normality,” Journal of Multivariate Analysis 93 (2005), no. 1, 58–80. doi:10.1016/j.jmva.2003.12.002.

2. G. J. Székely and M. L. Rizzo, “Energy statistics: A class of statistics based on distances,” Journal of Statistical Planning and Inference 143 (2013), no. 8, 1249–1272. doi:10.1016/j.jspi.2013.03.018.

3. M. Fornasier and J.-C. Hütter, “Consistency of probability measure quantization by means of power repulsion–attraction potentials,” Journal of Fourier Analysis and Applications 22 (2016), no. 3, 694–749. doi:10.1007/s00041-015-9432-z.

4. F. Colasanto, M. Focardi, M. Fornasier, and F. Mattesini, “Sharp rates of MMD empirical estimation with power kernels,” arXiv:2605.18497 (2026). arXiv.

5. E. Hess-Childs, M. Rosenzweig, and S. Serfaty, “Optimal quantization for Riesz MMDs,” manuscript in preparation (2026).

6. S. Serfaty, Lectures on Coulomb and Riesz Gases, Colloquium Publications 70, American Mathematical Society, Providence, RI, 2026. doi:10.1090/coll/070.

7. M. Fornasier, H. Huang, and L. Sun, “The nonlocal attraction-repulsion transport equation with power kernels,” arXiv:2607.04424 (2026). arXiv.

8. M. Di Francesco, M. Fornasier, J.-C. Hütter, and D. Matthes, “Asymptotic behavior of gradient flows driven by nonlocal power repulsion and attraction potentials in one dimension,” SIAM Journal on Mathematical Analysis 46 (2014), no. 6, 3814–3837. doi:10.1137/140951497.

9. R. Duong, V. Stein, R. Beinert, J. Hertrich, and G. Steidl, “Wasserstein gradient flows of MMD functionals with distance kernel and Cauchy problems on quantile functions,” ESAIM: Control, Optimisation and Calculus of Variations 32 (2026), Paper No. 10. doi:10.1051/cocv/2025097.

10. L. Chizat, M. Colombo, R. Colombo, and X. Fernández-Real, “Quantitative convergence of Wasserstein gradient flows of kernel mean discrepancies,” arXiv:2603.01977 (2026). arXiv.

Coulomb discrepancy as a transport energy

The Maximum Mean Discrepancy (MMD) uses a kernel to measure the difference between two probability distributions. In our setting, the kernel is the Coulomb potential , and, whenever this energy is finite, the resulting squared discrepancy is the negative-order Sobolev energy

We fix a target distribution and evolve a source distribution by the Wasserstein gradient flow of this functional. The resulting equation is a nonlocal transport equation in which moves under the Coulomb field generated by the discrepancy . It can be viewed as a distribution-matching dynamics, but also as the mean-field evolution of particles interacting through Coulomb forces in a fixed background of opposite charge.

The Coulomb structure makes the problem unusually explicit: the discrepancy potential solves a Poisson equation, so the energy is directly related to the field it generates. At the same time, this does not make convergence automatic. The source density acts as the mobility in the energy dissipation, so vacuum regions can weaken the usual coercivity argument. On the whole space, mass can also begin arbitrarily far from the target. Much of the paper concerns how these two effects shape the long-time behavior.

What the first version established

We first develop a Cauchy theory that allows rough initial data. When the target has a bounded density, we construct global weak solutions starting from an arbitrary Borel probability measure. In particular, the source may initially contain atoms. Coulomb repulsion produces an instantaneous regularization, which we call ultracontractivity by analogy with the -to- smoothing of classical diffusion semigroups [1, 2]. At every positive time, the solution has a bounded density, with an estimate independent of the initial data. Uniqueness is recovered when the initial density is bounded. This smoothing effect does not extend to higher regularity: even for smooth initial data and targets, the Hölder seminorm of the solution can grow exponentially in time.

On the flat torus, we prove exponential decay of the squared MMD toward any bounded, uniformly positive target without assuming that the initial source is bounded away from zero. The point is that the standard Polyak–Łojasiewicz (PL) argument sees the vacuum in the evolving density. We replace it with a defective PL inequality whose defect explicitly accounts for this vacuum and can still be controlled along the flow.

Independent and overlapping results were obtained by Lénaïc Chizat, Maria Colombo, Roberto Colombo, and Xavier Fernández-Real [3] in a broader study of Sobolev and Riesz kernel discrepancies on the torus. At the Coulomb endpoint, they prove global weak-* convergence to an arbitrary bounded target from bounded initial densities and exponential energy decay for uniformly positive targets even when the initial density has vacuum regions. Their exponential-convergence argument fills those regions through a maximum principle and then applies the usual PL inequality; our argument instead retains the vacuum as an explicit defect, and our construction can start from an arbitrary Borel probability measure before entering the bounded-density class at positive times.

On Euclidean space, the answer depends strongly on geometry. For radial data, if the target has connected support, is bounded above and below on that support, and contains the support of the source, we obtain a PL inequality and exponential convergence. On the unrestricted whole-space class, there is a simple but decisive obstruction: a localized source placed a distance from the target retains a fixed fraction of its initial squared MMD for times of order . Thus neither a global PL inequality nor a multiplicative convergence rate uniform over all initial data can hold in that setting.

What is new in version 2

The first main addition is a genuinely global-in-the-source metric PL inequality on the torus. It applies to every source with finite Coulomb energy when the target is uniform or sufficiently close to uniform, in the sense that if and are the essential lower and upper bounds of the target density, then the coercivity constant is positive when . For the uniform target, the constant is . Beyond finite energy, no lower bound, smoothness, or proximity assumption is imposed on the source. Formulating the result using the descending Wasserstein metric slope is important here, since the Coulomb field need not have an intrinsic value on the support of a singular source. The inequality yields exponential convergence for the gradient flow after any positive time.

The proof begins with the uniform target, rewriting the dissipation as a Dirichlet term minus a cubic correction; a variational barrier argument controls this correction by evaluating the Euler–Lagrange equation at a minimum and using the Hessian trace inequality. Heat regularization, normalization, and a short-time flow argument then extend the estimate from smooth positive sources to the metric-slope statement for arbitrary finite-energy sources and to the near-uniform targets above.

The second addition concerns critical points and planar long-time behavior. We prove, in every dimension and on both the torus and Euclidean space, that a Lagrangian critical point must coincide with the target whenever the positive part of the discrepancy is absolutely continuous. In two dimensions, finite Coulomb energy also supplies uniform tightness through a logarithmic-capacity estimate. Combining this with dissipation, local compactness of the Coulomb fields, and the critical-point rigidity theorem gives qualitative convergence to the target for every solution constructed in the paper whose Coulomb energy is finite at some positive time. The convergence is narrow, weak-* in , and strong in for every .

The rigidity proof uses a fine locality theorem of Ambrosio, Ponce, and Rodiac [4, Theorem 1.1]: in dimensions , if the distributional Laplacian of a locally integrable function is a locally finite signed measure, then its absolutely continuous part vanishes almost everywhere on every vector level set of the gradient. Writing , criticality gives almost everywhere on , while the Poisson equation says that the absolutely continuous part of equals there. Applying their theorem to the level set forces ; in one dimension, the same conclusion follows from the analogous locality statement [5, Theorem 6.3]. Equality of the total masses then gives .

This last conclusion deliberately stops short of convergence in the Coulomb MMD itself, and it does not provide a rate. A vanishing amount of mass at an increasingly remote scale can disappear in the stated topologies while retaining nonzero Coulomb energy. Determining whether the dynamics can actually produce this mechanism remains open.

Looking ahead

Stay tuned as well for a forthcoming joint paper with Dejan Slepčev and Lihan Wang on Wasserstein gradient flows for MMDs generated by energy kernels, which we plan to post to arXiv this week. It studies analogous questions in a complementary kernel regime, with several overlapping results and some shared ideas and techniques. I will discuss that work in a separate blog post once it appears.

The Coulomb paper is available on arXiv, with the PDF available here.

References

1. E. B. Davies and B. Simon, “Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians,” Journal of Functional Analysis 59 (1984), no. 2, 335–395. doi:10.1016/0022-1236(84)90076-4.

2. E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, 1989. doi:10.1017/CBO9780511566158.

3. L. Chizat, M. Colombo, R. Colombo, and X. Fernández-Real, “Quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancies,” arXiv:2603.01977 (2026). arXiv.

4. L. Ambrosio, A. C. Ponce, and R. Rodiac, “Critical weak- differentiability of singular integrals,” Revista Matemática Iberoamericana 36 (2020), no. 7, 2033–2072. doi:10.4171/RMI/1190.

5. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised ed., Textbooks in Mathematics, CRC Press, Boca Raton, 2015.

What I will publish here

Posts may include:

• concise announcements of new research findings;
• useful observations that do not yet belong to a paper;
• explanations of the mechanism behind a theorem or proof;
• open questions and possible research directions; and
• updates connecting an earlier post to a completed preprint or publication.

The format is deliberately flexible. A short observation may need only a page, while a more substantial announcement can include definitions, numbered equations, theorems, proofs, references, and extensive .

An invitation

The blog is also a way to make research interests visible. If something here intersects with your work—or suggests a direction we might pursue together—please get in touch.