Math Conjectures

Three-dimensional Coulomb crystallization

OPENMajorConjectureProposed c. 1934 · Canonical special case

Canonical statement

Let ΛL=(R/LZ)3\Lambda_L=(\mathbb R/L\mathbb Z)^3, let N=L3∈NN=L^3\in\mathbb N, and let GLG_L be the zero-mean periodic Coulomb Green function satisfying −ΔGL=4π(δ0−L−3)-\Delta G_L=4\pi(\delta_0-L^{-3}). For pairwise distinct x1,…,xN∈ΛLx_1,\ldots,x_N\in\Lambda_L, define the neutral-jellium energy, with the point self-energies omitted, by
EL(x1,…,xN)=∑1≤i<j≤NGL(xi−xj)−∑i=1N∫ΛLGL(xi−y) dy+12∬ΛL2GL(x−y) dx dy. E_L(x_1,\ldots,x_N)=\sum_{1\le i<j\le N}G_L(x_i-x_j)-\sum_{i=1}^N\int_{\Lambda_L}G_L(x_i-y)\,dy+\frac12\iint_{\Lambda_L^2}G_L(x-y)\,dx\,dy.
Then
lim⁡L→∞L3∈NL−3inf⁡x1,…,xL3∈ΛLxi≠xj (i≠j)EL(x1,…,xL3)=eBCC, \lim_{\substack{L\to\infty\\L^3\in\mathbb N}}L^{-3}\inf_{\substack{x_1,\ldots,x_{L^3}\in\Lambda_L\\x_i\ne x_j\ (i\ne j)}}E_L(x_1,\ldots,x_{L^3}) =e_{\mathrm{BCC}},
where eBCCe_{\mathrm{BCC}} is the thermodynamic energy per unit volume of the unit-density body-centered-cubic lattice under the same normalization.
View source LaTeX
Let \(\Lambda_L=(\mathbb R/L\mathbb Z)^3\), let \(N=L^3\in\mathbb N\), and let \(G_L\) be the zero-mean periodic Coulomb Green function satisfying \(-\Delta G_L=4\pi(\delta_0-L^{-3})\). For pairwise distinct \(x_1,\ldots,x_N\in\Lambda_L\), define the neutral-jellium energy, with the point self-energies omitted, by \[ E_L(x_1,\ldots,x_N)=\sum_{1\le i<j\le N}G_L(x_i-x_j)-\sum_{i=1}^N\int_{\Lambda_L}G_L(x_i-y)\,dy+\frac12\iint_{\Lambda_L^2}G_L(x-y)\,dx\,dy. \] Then \[ \lim_{\substack{L\to\infty\\L^3\in\mathbb N}}L^{-3}\inf_{\substack{x_1,\ldots,x_{L^3}\in\Lambda_L\\x_i\ne x_j\ (i\ne j)}}E_L(x_1,\ldots,x_{L^3}) =e_{\mathrm{BCC}}, \] where \(e_{\mathrm{BCC}}\) is the thermodynamic energy per unit volume of the unit-density body-centered-cubic lattice under the same normalization.
MATH-PHYS-005:[Wigner1934InteractionElectrons][BlancLewin2015CrystallizationReview][CotarPetrache2019JelliumUEG][PetracheSerfaty2020CoulombCrystallization]
Mathematical physicsProbability theoryDiscrete geometry

Notes

This is the three-dimensional Coulomb, or jellium, crystallization conjecture: the mathematical form of the prediction that electrons in a uniform neutralizing background arrange themselves into a crystal. For N=L3N=L^3 point charges on the torus (R/LZ)3(\mathbb R/L\mathbb Z)^3 interacting through the periodic Coulomb potential against a uniform background, the conjecture states that the minimal energy per unit volume converges, as L→∞L\to\infty, to the corresponding energy of unit-density body-centered-cubic configurations. The date is approximate: the BCC prediction developed from Wigner's 1934 study of the interaction of electrons in metals [Wigner1934InteractionElectrons].

Rigorous crystallization results are scarce: they exist for selected one-dimensional interactions and for exceptional dimensions and potentials, but not for the three-dimensional Coulomb gas; see [BlancLewin2015CrystallizationReview] for a broad review. The jellium energy is known to agree with that of the uniform electron gas at next order [CotarPetrache2019JelliumUEG], and Petrache and Serfaty showed that crystallization for Coulomb and Riesz interactions would follow from the Cohn–Kumar conjecture on the universal optimality of lattices [PetracheSerfaty2020CoulombCrystallization].

Unconditionally, even the equality of the ground-state energy density with the BCC Madelung value is unknown — and convergence of the minimizers themselves is a stronger statement still — so the conjecture remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.