Discrete-to-continuum limits for random atomistic lattices: quantum (Thomas–Fermi-type) elastic energies and survival of lattice symmetry

A 2006 note by Blanc, Le Bris and Lions proved that the energy of a random lattice of atoms interacting through pair potentials converges, under an Ansatz forcing the atoms to follow a macroscopic deformation $u$, to a deterministic hyperelastic energy $\int_D W(\nabla u)\,dx$, and listed open extensions together with remarks on the limit energy. Two of the extensions are now substantially resolved: the variational (Γ-convergence) removal of the Ansatz for classical pair and volumetric interactions, by Alicandro, Cicalese and Gloria (2011) and successors, and the increment-stationarity technicalities, by Gloria (2014). Multi-body interactions are understood on periodic lattices, and thermodynamic limits (energy per unit volume, without deformation) of quantum Coulomb models with random nuclei are known. What remains open is the announced but never published macroscopic limit for microscopic energies of quantum origin — Thomas–Fermi-type electronic-structure models on stationary ergodic stochastic lattices — in both the exact-deformation sense of the source and the Γ-convergence sense, together with the structural question raised in the same note: which lattice material symmetries (the $GL_d(\mathbb{Z})$ invariance underlying the Cauchy–Born rule) survive stochastic perturbation of the lattice. The note also observes that the random reference configuration is generically not an energy minimum, so that the macroscopic energy is not stress-free at the identity; the questions of natural states, stability and linearization this suggests go beyond the note and beyond the existing deterministic atomistic theory.

Background

A central problem of the mathematical theory of elasticity is the derivation of macroscopic (continuum) energy laws of solids from microscopic (atomistic) models. In the simplest classical setting a crystal is a periodic lattice such as $\mathbb{Z}^d$ whose atoms interact through a pair potential $W_0$, a function of the bond vector between two atoms, and the discrete energy of a configuration is the sum of $W_0$ over interacting pairs. The expected continuum limit is a hyperelastic energy $\int_D W(\nabla u(x))\,dx$, where $D$ is the reference domain, $u$ the macroscopic deformation and $W$ the stored-energy density. The Cauchy–Born rule is the classical recipe for computing $W$ in the periodic case: it postulates that under a constant macroscopic deformation gradient $F$ every atom moves to its affinely deformed position, so that $W(F)$ is the energy per unit cell of the lattice uniformly stretched by $F$. Friesecke and Theil exhibited a two-dimensional mass-spring lattice in which this rule fails: the uniform deformation need not minimize the microscopic energy even when the continuum density would be minimized by it.

For crystals with defects or disorder the reference positions themselves are random. A stochastic lattice is a random, locally finite set of points in $\mathbb{R}^d$; it is stationary if its statistics are invariant under all translations, and ergodic if, in addition, spatial averages along one realization coincide almost surely with ensemble averages, so that the ergodic theorem applies. The source note (Blanc, Le Bris, Lions, 2006) proved that for a stationary ergodic stochastic lattice with suitable moment bounds, and for pair potentials that are Lipschitz and integrable at infinity, the energy per particle of the lattice scaled by a small factor $\varepsilon$ and deformed by a macroscopic field $u\in W^{1,p}(D,\mathbb{R}^d)$, $p>d$, satisfying the one-sided quantitative injectivity bound $|u(x)-u(y)|\ge\alpha|x-y|$ for some $\alpha>0$, converges almost surely, as $\varepsilon\to0$, to a deterministic hyperelastic energy whose density is an explicit expectation over lattice pairs. The proof, however, imposes the exact-deformation Ansatz: each atom is forced to sit at the deformed point $u(\varepsilon x_i)$ instead of being free to relax. The authors call this hypothesis questionable and identify Γ-convergence — the standard variational convergence of sequences of functionals, which guarantees that (almost-)minimizers of the discrete energies converge to minimizers of the limit functional — as the correct framework they were unable to implement; the one-dimensional variational precursor for statistically homogeneous (that is, stationary) systems is due to Iosifescu, Licht and Michaille.

Section 5 of the note collects natural extensions of the work together with remarks on the limit energy. First, remove the Ansatz and develop a genuine Γ-convergence theory for stochastic lattices. Second, extend the analysis from pair potentials to many-body interactions and to energies of quantum origin, in the spirit of the Thomas–Fermi–von Weizsäcker (TFW) model — the simplest reduced electronic-structure model, in which electrons are described only by a density $\rho$ carrying a kinetic energy of the form $\int|\nabla\sqrt{\rho}|^2+\int\rho^{5/3}$ and interacting electrostatically (through Coulomb forces) with the nuclei under a charge-neutrality constraint; for such models the authors' companion paper established the thermodynamic limit, meaning the existence of the ground-state energy per unit volume as the system grows, but not the macroscopic limit under deformation. Third, decide which symmetry properties of the limit density survive: on a deterministic periodic lattice the density is invariant under the group $GL_d(\mathbb{Z})$ of integer matrices with determinant $\pm1$ that maps the lattice to itself, a fact underlying the Cauchy–Born rule, whereas the density produced by the stochastic formula loses this invariance except in very special cases. Fourth, the note observes that the random reference configuration is not an energy minimum, so the macroscopic energy is not stress-free at the identity deformation; it draws no further program from this remark, and questions of stability and of the linearization of the energy around such reference states go beyond it and beyond the existing deterministic atomistic theory. Fifth, treat increment-stationary lattices, a weaker statistical symmetry in which only the increments between points, not the point set itself, are required to be stationary.

Later work resolved large parts of this program for classical interactions. Alicandro, Cicalese and Gloria proved that on admissible stochastic lattices — random point sets satisfying a uniform covering condition (no large holes) and a uniform separation condition (no clusters) — pair-interaction and volumetric many-body energies Γ-converge, with freely relaxing discrete deformations, to a deterministic integral functional given by a cell-type formula; the same work establishes frame indifference (invariance $W(RF)=W(F)$ for rotations $R$), a stochastic-isotropy theorem (if the law of the point process and the interaction potentials are rotation-invariant, then the limit density is isotropic), non-interpenetrability of the limit deformations (injectivity of the limit deformations, so distinct material points are never mapped to the same point), and results on natural states, that is, stress-free shapes of the material. Successors treated random parking sets, spin systems on random lattices, nonconvex energies with degenerate growth, Poisson point clouds, and full statistical-mechanics (polymer-network) settings; many-body interactions on periodic multilattices — periodic lattices carrying several atoms per unit cell — are likewise well understood. On the quantum side, thermodynamic limits for disordered Coulomb systems and for reduced Hartree–Fock (mean-field) models — in which each electron feels only the average electrostatic field created by the others, the electronic state being a one-particle density matrix — with random nuclei are known, and special-quasirandom-structure approximations of random-lattice TFW energies — finite supercells standing in for the disordered crystal — have been rigorously justified; a 2025 homogenization theorem for two-dimensional TFW materials is, however, periodic in the nuclear density. The increment-stationarity question was substantially clarified by Gloria, who characterized when it implies stationarity, gave covariance-decay conditions ensuring this in dimensions $d>2$, and constructed one- and two-dimensional counterexamples.

No derivation of a deformation-dependent macroscopic energy for microscopic energies of quantum origin on stochastic lattices is known, in either the exact-deformation sense of the source note or the Γ-convergence sense; the continuation repeatedly announced by the authors in 2006–2007 was never published. The symmetry question has only partial analogues (stochastic isotropy; symmetry transmission in periodic linear elasticity), and the stability question has been studied only in deterministic atomistic settings. These gaps, at the level of generality of the source, form the open core stated below.

Problem Statement

Setting. Let $\ell(\omega)\subset\mathbb{R}^d$ be a stochastic lattice (a random, locally finite set of points) that is stationary and ergodic and satisfies quantitative admissibility conditions of covering type (no large holes) and separation type (no clusters), together with the moment bounds under which ergodic averages exist. Place nuclei at the points of the scaled lattice $\varepsilon\ell(\omega)$ and describe the electrons by a reduced quantum model of Thomas–Fermi–von Weizsäcker (TFW) type: a mean-field model in which nuclei and electrons interact only through Coulomb (electrostatic) forces under a charge-neutrality constraint, with a kinetic energy that is a simple functional of the electronic state — in TFW, $\int|\nabla\sqrt{\rho}|^2+\int\rho^{5/3}$ of the electronic density $\rho$ alone, the $\int\rho^{5/3}$ term being specific to TFW. For a macroscopic deformation $u$, let $E_\varepsilon(\omega)(u)$ denote the ground-state electronic energy, normalized per particle as in the source note (divided by the number $\#(\ell(\omega)\cap\varepsilon^{-1}D)$ of nuclei in the sample), of the configuration in which the nucleus sitting at $\varepsilon x_i$, for each point $x_i\in\ell(\omega)$, is displaced to $u(\varepsilon x_i)$; the pair-potential energies of the source note are the classical analogue.

Main objective. Establish the macroscopic limit of $E_\varepsilon$ for energies of quantum origin on stochastic lattices in both senses of the source note: (a) the exact-deformation form of the source note — for $u\in W^{1,p}(D,\mathbb{R}^d)$ with $p>d$ satisfying the source's one-sided quantitative injectivity bound $|u(x)-u(y)|\ge\alpha|x-y|$ for some $\alpha>0$ and all $x,y\in D$, an almost-sure convergence of $E_\varepsilon(\omega)(u)$ to $\int_D W(\nabla u(x))\,dx$ with a deterministic density $W$ given by an explicit ergodic-type formula; and (b) the variational form the authors regarded as the correct one — Γ-convergence of the energies, with atom positions free subject only to boundary data, on $L^p$ or $W^{1,p}$ to an integral functional whose density $W$ is quasiconvex in the sense of Morrey ($W(F)\le\frac{1}{|Q|}\int_Q W(F+\nabla\varphi(y))\,dy$ for every smooth $\varphi$ vanishing on the boundary of a cube $Q$; this is the weakest convexity condition under which such integral functionals are weakly lower semicontinuous), together with an asymptotic cell formula for it. Identify the hypotheses on the point process (stationarity, ergodicity, admissibility, moment conditions) and on the electronic model (including the boundary conditions of the electronic problem) under which the limit exists, is almost surely deterministic, and is finite.

Structural objective (symmetry survival). Determine which invariances the limit density inherits: frame indifference $W(RF)=W(F)$ for all $R\in SO(d)$, the group of rotations of $\mathbb{R}^d$; and, for point processes whose statistics are invariant under a subgroup of the lattice-invariance group $GL_d(\mathbb{Z})$ of integer matrices with determinant $\pm1$ (for instance independent identically distributed perturbations of $\mathbb{Z}^d$), which subgroup leaves $W$ invariant. The source note observes that the $GL_d(\mathbb{Z})$-invariance expected from the Cauchy–Born rule fails for the density produced by its stochastic formula except in very special cases, such as a deterministic periodic lattice. The question is to be answered at least for the classical models where the limit is known and, within the same framework, for the quantum limit once objective (a) or (b) is established; the objective includes the quantum limit, not only the classical case.

Related observation (non-minimality of the reference configuration). The source note remarks that, even when the mean position of the atoms is a minimum of the microscopic energy, the macroscopic energy is not minimized at the identity deformation, so the random reference configuration used implicitly is not an energy minimum. The note poses no further question on this point; the problems it suggests — natural states (stress-free configurations) of the limit functional, stability, and the linearization of the energy (linearized elastic coefficients) around such non-minimal reference states — remain open beyond the existing deterministic atomistic theory, but they lie outside the question posed here.

Scope boundary. The Γ-convergence of classical pair and volumetric interactions on admissible stochastic lattices, the increment-stationarity theory, the periodic multi-body theory, and the thermodynamic limits of disordered quantum systems are settled and form the background of the question. The open target is the quantum (electronic-structure) macroscopic elastic limit on stochastic lattices, together with the symmetry-survival question above, at the level of generality of the source.

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