Thermodynamic limits for microscopic stochastic lattices under minimal hypotheses: point nuclei, weak geometric bounds, and increment-stationarity

The 2007 paper of Blanc, Le Bris and Lions defined, via thermodynamic limits, a consistent infinite-volume energy for 'microscopic stochastic lattices' - stationary ergodic random point sets and stationary random nuclear charge measures - for classical pair-interaction systems and for the quantum Thomas-Fermi (TF) and Thomas-Fermi-von Weizsäcker (TFW) electronic models. The construction required restrictive hypotheses: absolute stationarity of the geometry, $L^{5/3}$ (TF) or $L^{6/5}$ (TFW) integrability of the smeared nuclear measure, pointwise Delaunay-type local upper and lower mass bounds for uniqueness and density convergence, point nuclei only on randomly perturbed lattices, and sufficient integrability conditions tying the pair potential to the stochastic geometry. The authors listed the natural open generalizations, and later work settled part of the program: the announced macroscopic (discrete-to-continuum, stored-energy) limits were carried out by the authors themselves and by Alicandro-Cicalese-Gloria; existence of the thermodynamic limit for many-body quantum Coulomb systems with random point nuclei was proved by Blanc and Lewin; the stationary reduced Hartree-Fock theory for disordered crystals was developed by Cancès, Lahbabi and Lewin for smeared charges; and Gloria determined when increment-stationary point sets are genuinely more general than stationary ones. The surviving open core is the minimal-hypothesis thermodynamic-limit theory, a set of independent open questions from the source program: (A) identification of the TF/TFW (and mean-field) limit, with Coulomb renormalization, uniqueness of the stationary ground-state density, and density convergence, for stationary nuclear measures with Dirac (point-nucleus) components beyond perturbed lattices; (B) the same conclusions under geometric hypotheses weaker than pointwise Delaunay-type local mass bounds (e.g. averaged bounds); (C) a rigorous treatment of the quantum models under increment-stationarity rather than stationarity; and (D) weaker (minimal) sufficient conditions on the interaction potential relative to the random geometry, and broader classes of stochastic lattices for which a deterministic thermodynamic limit exists.

Background

Crystalline matter is modeled mathematically as an infinite set of nuclei (classical particles carrying positive charge) together with an electronic structure, and its macroscopic energetics is extracted through a thermodynamic limit: the ground-state energy of the system confined to a bounded region $D$ is divided by the volume $|D|$, and one studies the limit as $D$ exhausts space. For nuclei clamped on a periodic lattice this program is classical, for pair-interaction ('classical') energies as well as for mean-field quantum models of electronic structure: the limit exists and is identified with a variational problem on the periodicity cell. Two such mean-field models are central here. In Thomas-Fermi (TF) theory the interacting quantum electrons are replaced by a non-negative charge density $\rho$ whose kinetic energy is a local functional $C^{TF}\int\rho^{5/3}$, interacting with the nuclear charge and with itself through the Coulomb potential $1/|x|$. The Thomas-Fermi-von Weizsäcker (TFW) model adds a gradient ('von Weizsäcker') term $C^{W}\int|\nabla\sqrt{\rho}|^{2}$, which restores some quantum-mechanical oscillatory structure and makes ground states unique. The reduced Hartree-Fock (rHF) model is a mean-field model in which electrons are described by a one-particle density matrix with kinetic energy given by the trace of $-\Delta$ times that matrix, without an exchange term; it is closer to Kohn-Sham density functional theory used in materials computation.

Real solids - alloys, doped semiconductors, thermally disordered crystals - are not periodic. Blanc, Le Bris and Lions (2007) extended the thermodynamic-limit construction to random geometries. Their framework is probabilistic: a stochastic lattice is a random, locally finite set of points in $\mathbb{R}^{d}$ whose law is compatible with an ergodic measure-preserving action $(\tau_k)_{k\in\mathbb{Z}^{d}}$ of the lattice on an underlying probability space (ergodic meaning that every event invariant under the action has probability 0 or 1); stationarity means the point set is shifted, in law, by integer translations, $\ell(\tau_k\omega)=\ell(\omega)+k$ up to sign conventions. A typical special case is a randomly perturbed lattice $x_k=k+X_k$ with stationary, centered displacements $(X_k)$; the nuclear charge is a stationary random measure $m(x,\omega)=\mathcal{M}(\tau_x\omega)$. 'Delaunay-type' (or admissibility) hypotheses are uniform geometric bounds on the point set - a hard-core radius (no two points closer than $r$) and a covering radius (every point of space within $R$ of the set, so no large holes) - which, via the Delaunay tessellation (the triangulation joining points whose circumscribing cells contain no other points), control the local mass of the configuration.

Within this framework the paper proved: almost-sure convergence of the truncated pair energies per particle to an explicit expectation, for radial pair potentials $W$ decaying like $|W(r)|\le C/(1+r)^{p}$ with $p>d$, under an integrability condition linking $W$ and the geometry (essentially summability over a unit cell of the pair sums, in $L^{1}$ of the probability space), and, for perturbed lattices, under local absolute-continuity and summability conditions on the laws of the increments $k+X_k-X_0$; almost-sure convergence and identification of the TF thermodynamic limit with a stationary variational problem for $\mathcal{M}\in L^{5/3}(\Omega)$; the same for TFW under $\mathcal{M}\in L^{6/5}(\Omega)$ plus an almost-sure pointwise local upper mass bound, with uniqueness of the stationary ground-state density and convergence of finite-volume densities on interior regions requiring in addition an almost-sure pointwise local lower mass bound (no large holes); and, for point nuclei on a randomly perturbed lattice, a renormalized TFW energy - the divergent Coulomb self-energy of the point charges being subtracted and screened with a fixed mollifier - with convergence of the finite-volume energies. Coulomb renormalization is the delicate point: nucleus-nucleus and electron-nucleus $1/|x|$ interactions are both singular and long-range, and without periodicity the screening must be controlled probabilistically rather than by cell symmetry.

The authors explicitly listed the generalizations they could not prove: broader classes of stochastic lattices and weaker stationarity notions - in particular the treatment of the quantum models under 'increment-stationarity', where the group action moves the point set by random translations, $\ell(\tau_k\omega)=\ell(\omega)+Y_k(\omega)$, so that the law of intersite distances (increments) rather than of absolute positions is translation-invariant (the 2007 paper had already proved the classical two-body case in that setting, as later recalled by Gloria); geometric hypotheses weaker than the Delaunay-type local mass bounds; nuclear charge measures that are only locally finite measures, including Dirac point-nucleus masses, with fully rigorous Coulomb renormalization; minimal hypotheses relating $W$ to the stochastic geometry; and the passage to macroscopic stored-energy densities $E(\nabla u)$. Later literature has settled the macroscopic-limit direction and parts of the point-nucleus and stationarity questions, so the open problem below is the surviving core: the same thermodynamic-limit construction under essentially weaker hypotheses.

Problem Statement

Establish the thermodynamic-limit theory for the energy of microscopic stochastic lattices under hypotheses essentially weaker than those of Blanc, Le Bris and Lions (2007), staying within their framework: stationary (or increment-stationary) ergodic random point sets and stationary random nuclear charge measures, classical pair-potential energies, and quantum TF/TFW-type (and mean-field) electronic models. Four independent open questions of the source program share one objective - a consistent, deterministic infinite-volume energy for wider classes of disordered geometries, quantum models included:

(A) Point nuclei in the general stationary framework. Prove that for stationary ergodic nuclear charge measures that are merely locally finite measures with Dirac (point-nucleus) components - beyond the smeared $L^{5/3}$ (TF) and $L^{6/5}$ (TFW) cases and beyond the randomly perturbed lattices already treated in the source paper - the TF and TFW energies per unit volume converge (almost surely or in expectation, as in the source framework) and are identified with the minimum of an explicit stationary variational problem, with fully rigorous control of the Coulomb renormalization (self-energy subtraction/screening), with uniqueness of the stationary ground-state density, and with convergence of the finite-volume densities on interior domains. Existence without identification is known for the many-body quantum model (Blanc and Lewin 2012, under hypotheses its authors call far from optimal); the identification, uniqueness, and density-convergence conclusions for TF/TFW-type models with point nuclei are open.

(B) Weaker geometric hypotheses. Replace the pointwise Delaunay-type local upper and lower mass bounds (hard-core and covering conditions; almost-sure pointwise bounds $\sup_x m(x+B_1,\omega)<\infty$ and $\inf_x m(x+B_A,\omega)>0$) used to obtain TFW uniqueness and density convergence, by weaker conditions - for example averaged local lower mass bounds or moment/integrability conditions - while retaining the same conclusions; and determine where uniqueness or density convergence genuinely fails.

(C) Increment-stationarity. Give a rigorous treatment of the quantum models under increment-stationarity, the weaker stationarity notion in which the group action translates the point set by random vectors, $\ell(\tau_k\omega)=\ell(\omega)+Y_k(\omega)$, so that the law of intersite distances rather than of absolute positions is translation-invariant: prove existence and identification of the thermodynamic limit in this setting. The geometric relation between increment-stationarity and stationarity has been clarified (Gloria 2014: non-rigid in dimensions 1-2, rigid in dimensions $d>2$ under covariance decay), and the extension is expected to go through at the level of two-body energies, but no rigorous quantum-model treatment under increment-stationarity exists.

(D) Minimal potential-geometry conditions and broader lattice classes. Determine weaker (minimal) sufficient conditions relating the interaction potential $W$ and the stochastic geometry - beyond the sufficient integrability conditions on pair sums over unit cells and on the absolute continuity of increment laws used in the source paper - under which the pair energies per unit volume converge almost surely; and identify broader classes of stochastic lattices (in particular, which increment-stationary or otherwise weakly stationary point sets) for which a deterministic thermodynamic limit exists.

Completion rule: each of (A)-(D) is a self-contained open question; a rigorous proof of all the conclusions stated under any single one of them resolves the task, the remaining questions staying open, and establishing all four together amounts to the complete minimal-hypothesis thermodynamic-limit theory.

The original question's macroscopic-limit direction (the derivation of a stored-energy density $E(\nabla u)$) is not part of the target, because it has been settled for classical models by the companion works of the same authors and by Alicandro, Cicalese and Gloria.

References

1. Xavier Blanc, Claude Le Bris, Pierre-Louis Lions, 'The Energy of Some Microscopic Stochastic Lattices', Archive for Rational Mechanics and Analysis 184(2):303-339 (2007). DOI: 10.1007/s00205-006-0028-2, https://doi.org/10.1007/s00205-006-0028-2. (full text paywalled.)
2. Xavier Blanc, Claude Le Bris, Pierre-Louis Lions, 'Du discret au continu pour des modèles de réseaux aléatoires d'atomes', Comptes Rendus. Mathématique 342(8):627-633 (2006). DOI: 10.1016/j.crma.2005.12.033, https://doi.org/10.1016/j.crma.2005.12.033
3. Xavier Blanc, Claude Le Bris, Pierre-Louis Lions, 'Stochastic homogenization and random lattices', Journal de Mathématiques Pures et Appliquées 88(1):34-63 (2007). DOI: 10.1016/j.matpur.2007.04.006, https://doi.org/10.1016/j.matpur.2007.04.006 (HAL: hal-00140076)
4. Roberto Alicandro, Marco Cicalese, Antoine Gloria, 'Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity', Archive for Rational Mechanics and Analysis 200(3):881-943 (2011). DOI: 10.1007/s00205-010-0378-7, https://doi.org/10.1007/s00205-010-0378-7 (HAL: inria-00437765)
5. Xavier Blanc, Mathieu Lewin, 'Existence of the thermodynamic limit for disordered quantum Coulomb systems', Journal of Mathematical Physics 53(9):095209 (2012). DOI: 10.1063/1.4729052, arXiv:1201.4670, https://arxiv.org/abs/1201.4670
6. Éric Cancès, Salma Lahbabi, Mathieu Lewin, 'Mean-field models for disordered crystals', Journal de Mathématiques Pures et Appliquées 100(2):241-274 (2013). DOI: 10.1016/j.matpur.2012.12.003, arXiv:1203.0402, https://arxiv.org/abs/1203.0402
7. Antoine Gloria, 'When are increment-stationary random point sets stationary?', Electronic Communications in Probability 19, paper no. 30, 1-14 (2014). DOI: 10.1214/ECP.v19-3288, arXiv:1409.1156, https://arxiv.org/abs/1409.1156
8. A. I. Komech, 'On the crystal ground state in the Schrödinger-Poisson model with point ions', Mathematical Notes 99(5-6):886-894 (2016). DOI: 10.1134/S0001434616050278, arXiv:1310.3084, https://arxiv.org/abs/1310.3084 (arXiv preprint titled 'On crystal ground state in the Schrödinger-Poisson model')
9. Julian Fischer, Michael Kniely, 'Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model', Nonlinearity 33(11):5733-5772 (2020). DOI: 10.1088/1361-6544/ab9728, arXiv:1906.12245, https://arxiv.org/abs/1906.12245
10. David Gontier, Jianfeng Lu, Christoph Ortner, 'Thermodynamic Limits of Electronic Systems', chapter in É. Cancès and G. Friesecke (eds.), 'Density Functional Theory: Modeling, Mathematical Analysis, Computational Methods, and Applications', Springer, pp. 307-331 (2023; chapter published online 2022-12-19). DOI: 10.1007/978-3-031-22340-2_6, arXiv:2309.05118, https://arxiv.org/abs/2309.05118
11. Andrea Braides, Marco Caroccia, 'Asymptotic behavior of the Dirichlet energy on Poisson point clouds', Journal of Nonlinear Science 33(5), article 80 (2023). DOI: 10.1007/s00332-023-09937-7, arXiv:2203.16877, https://arxiv.org/abs/2203.16877
12. Alessandra Faggionato, 'Scaling limit of the directional conductivity of random resistor networks on point processes', Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 61(2) (2025). DOI: 10.1214/23-aihp1461, arXiv:2108.11258, https://arxiv.org/abs/2108.11258 (arXiv preprint titled '... on simple point processes')
13. Andrea Braides, Andrey Piatnitski, 'Homogenization of random convolution energies', Journal of the London Mathematical Society 104(1):295-319 (2021). DOI: 10.1112/jlms.12431, arXiv:1909.06832, https://arxiv.org/abs/1909.06832
14. Saad Benjelloun, Salma Lahbabi, Abdelqoddous Moussa, 'Homogenization of 2D materials in the Thomas-Fermi-von Weizsäcker theory' (2023). arXiv:2312.08067, https://arxiv.org/abs/2312.08067
15. Mathieu Lewin, 'Some open mathematical problems concerning charged quantum particles', Comptes Rendus. Physique 26(G1):369-380 (2025). DOI: 10.5802/crphys.249, https://doi.org/10.5802/crphys.249
16. Éric Cancès, David Gontier, Solal Perrin-Roussel, 'Kohn-Sham models for encapsulated two-dimensional materials' (2026). arXiv:2606.11785, https://arxiv.org/abs/2606.11785