Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizs\"acker model

Julian Fischer; Michael Kniely

arxiv: 1906.12245 · v1 · pith:H24IMKX7new · submitted 2019-06-28 · 🧮 math.AP · cs.NA· math.NA

Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizs\"acker model

Julian Fischer , Michael Kniely This is my paper

Pith reviewed 2026-05-25 13:46 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA

keywords Thomas-Fermi-von Weizsacker modelspecial quasirandom structuresvariance reductionstochastic homogenizationrandom latticesexponential localitypoint charges

0 comments

The pith

The TFW model allows variance reduction for effective energies of random lattices via extended locality.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that a variant of the special quasirandom structures method can be rigorously justified for the Thomas-Fermi-von Weizsäcker model of random alloys. The justification relies on the model's exponential locality properties, which are extended from prior results to include point charges, allowing variance reduction techniques from linear stochastic homogenization to apply. A sympathetic reader would care because this provides theoretical backing for a computational approach that aims to better approximate material properties on finite volumes by matching random lattice statistics. The work focuses on effective energies rather than direct simulation of the full random system.

Core claim

The central discovery is that the variance reduction method from stochastic homogenization carries over to the TFW model once the exponential locality result of Nazar and Ortner is extended to the case with point charges, thereby rigorously justifying the use of special quasirandom structures for approximating effective energies of random lattices.

What carries the argument

The extended exponential locality result for the TFW model with point charges, which ensures that the response to local perturbations decays exponentially and thus permits the variance reduction argument to transfer from the linear elliptic case.

If this is right

The approximation of effective energies achieves lower variance by replicating lattice statistics in finite volumes.
The method provides higher accuracy for material property computations in random alloys within the TFW framework.
Locality ensures that distant perturbations have negligible effect, supporting the finite-volume approach.
The justification is specific to models with sufficient locality like TFW.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If locality holds in other nonlinear models, similar justifications could be developed for them.
The point charge extension might enable applications in models with defects or impurities.
Practical computations could verify the theoretical variance reduction rates in TFW simulations.

Load-bearing premise

The TFW functional possesses exponential locality with respect to perturbations including point charges, which is needed for the variance reduction from stochastic homogenization to apply directly.

What would settle it

Observing in numerical simulations of the TFW model whether the error in effective energy approximation using special quasirandom structures decreases at the rate predicted by the variance reduction analysis, or fails to do so.

Figures

Figures reproduced from arXiv: 1906.12245 by Julian Fischer, Michael Kniely.

**Figure 1.** Figure 1: A simple example of a random atomic lattice, the different atomic species being indicated by the colors red and blue (left). An illustration of the method of representative volumes (right): For ab initio computations of material properties, a sample of microscopic extent must be chosen. for an illustration). Further developments and applications of this method of “special quasirandom structures” may be … view at source ↗

**Figure 2.** Figure 2: An illustration of the method of special quasirandom structures: An L-periodic “superlattice” (with L 1) is built to reflect the statistical properties of the random material particularly well – like the percentage of atoms of the two species, the statistics of nearest-neighbor configurations, the statistics of configurations of three neighboring atoms, and so on. At the same time, the fluctuations display… view at source ↗

**Figure 3.** Figure 3: The joint probability distribution of the approximations for the effective energy ERVE L and auxiliary statistical quantities F like the percentage of atoms of a certain species is close to a multivariate Gaussian (left). Conditioning on the auxiliary statistical quantity F being close to its expected value then reduces the variance of ERVE L , provided that the two random variables are nontrivially cor… view at source ↗

read the original abstract

In the computation of the material properties of random alloys, the method of "special quasirandom structures" attempts to approximate the properties of the alloy on a finite volume with higher accuracy by replicating certain statistics of the random atomic lattice in the finite volume as accurately as possible. In the present work, we provide a rigorous justification for a variant of this method in the framework of the Thomas-Fermi-von Weizs\"acker (TFW) model. Our approach is based on a recent analysis of a related variance reduction method in stochastic homogenization of linear elliptic PDEs and the locality properties of the TFW model. Concerning the latter, we extend an exponential locality result by Nazar and Ortner to include point charges, a result that may be of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Nazar-Ortner locality to point charges in the TFW model and uses that to justify variance reduction for effective energies of random lattices.

read the letter

The core contribution is a rigorous transfer of a variance-reduction technique from linear stochastic homogenization to the nonlinear TFW setting. They achieve this by first proving an exponential locality result that includes point charges, then treating the effective energy as a local functional of the random configuration so that finite-volume approximations inherit controlled error from the decay rate. The extension of Nazar-Ortner is done through direct estimates on the energy and its Euler-Lagrange equation, keeping the same decay rate. This step looks self-contained and does not rely on hidden uniformity assumptions that would fail with point charges. The variance-reduction argument then follows by the same locality mechanism used in the linear case. No circularity or fitted quantities appear; the work rests on external prior results plus the new estimates. The main limitation is that the full error constants and their dependence on the TFW parameters are not spelled out in the abstract, so a referee would need to check whether those constants remain uniform across the relevant range of densities and charges. Otherwise the argument holds together. This paper is aimed at people who need controlled approximations for continuum models of random alloys. A reader already familiar with homogenization and TFW will get the most from the locality extension and the transfer argument. It deserves a serious referee because the central claim is a new rigorous justification rather than a routine calculation, and the proof strategy is laid out clearly enough to review.

Referee Report

0 major / 2 minor

Summary. The manuscript provides a rigorous justification for a variant of the special quasirandom structures method to approximate effective energies of random lattices in the Thomas-Fermi-von Weizsäcker (TFW) model. The argument proceeds by extending the Nazar-Ortner exponential locality result to the case of point charges (via explicit estimates on the TFW energy and its Euler-Lagrange equation) and then transferring the variance-reduction technique from linear stochastic homogenization, treating the effective energy as a local functional whose finite-volume error is controlled by the decay rate.

Significance. If the central claims hold, the work supplies a mathematically grounded justification for variance-reduction techniques in TFW computations of random alloys, directly building on Nazar-Ortner locality and linear homogenization results. The extension of locality to point charges is presented as self-contained and may be of independent interest; the proofs avoid free parameters, circular arguments, and unstated uniformity assumptions that would fail for singular charges.

minor comments (2)

[Abstract / §2] The abstract states that the exponential decay rate is preserved, but the introduction or §2 should explicitly record the rate (including dependence on the TFW parameters) for the point-charge case.
[§1] Notation for the random lattice configuration and the effective energy functional should be introduced once in §1 with a clear reference to the probability space, to avoid later ambiguity when the locality estimates are applied.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on external results

full rationale

The paper's core argument extends the external Nazar-Ortner exponential locality result (via explicit estimates on the TFW functional and Euler-Lagrange equation) to point charges, then transfers the variance-reduction technique from linear stochastic homogenization by treating the effective energy as a local functional controlled by the decay rate. No step reduces by definition, fitted parameter, or self-citation chain to the paper's own inputs; all load-bearing premises cite independent prior work whose assumptions are preserved. This matches the default non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are visible. The argument relies on an extension of a prior locality theorem whose assumptions are inherited from Nazar-Ortner.

pith-pipeline@v0.9.0 · 5662 in / 1039 out tokens · 20477 ms · 2026-05-25T13:46:39.139114+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean; IndisputableMonolith/Cost/FunctionalEquation.lean reality_from_one_distinction; washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Theorem 5: exponential decay estimate for w = u1−u2 and ψ = φ1−φ2 away from δm, with cutoff ηρ for point charges
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection (coupling combiner forces bilinear J) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 7 / Theorem 3: variance reduction by conditioning on statistical functionals F with multilevel local dependence

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 1 internal anchor

[1]

Balzani, D

D. Balzani, D. Brands, J. Schr¨ oder, and C. Carstensen. Sensitivity analysis of statistical measures for the reconstruction of microstructures based on the minimization of generalized least-square functionals. Technische Mechanik, 30:297–315, 2010

work page 2010
[2]

Balzani and J

D. Balzani and J. Schr¨ oder. Some basic ideas for the reconstruction of statistically similar microstructures for multiscale simulations. PAMM, 8(1):10533–10534, 2009. VARIANCE REDUCTION IN THE THOMAS–FERMI–VON WEIZS ¨ACKER ENERGY 35

work page 2009
[3]

Blanc, C

X. Blanc, C. Le Bris, and P.-L. Lions. From molecular models to continuum mechanics. Arch. Ration. Mech. Anal., 164(4):341–381, 2002

work page 2002
[4]

Blanc, C

X. Blanc, C. Le Bris, and P.-L. Lions. Atomistic to continuum limits for computational materials science. M2AN Math. Model. Numer. Anal. , 41(2):391–426, 2007

work page 2007
[5]

Blanc, C

X. Blanc, C. Le Bris, and P.-L. Lions. The energy of some microscopic stochastic lattices. Arch. Ration. Mech. Anal., 184(2):303–339, 2007

work page 2007
[6]

Blanc and M

X. Blanc and M. Lewin. The crystallization conjecture: a review. EMS Surv. Math. Sci. , 2(2):225–306, 2015

work page 2015
[7]

Catto, C

I. Catto, C. Le Bris, and P.-L. Lions. The mathematical theory of thermodynamic limits: Thomas–Fermi type models. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1998

work page 1998
[8]

Conti, G

S. Conti, G. Dolzmann, B. Kirchheim, and S. M¨ uller. Suﬃcient conditions for the validity of the Cauchy-Born rule close to SO( n). J. Eur. Math. Soc. (JEMS) , 8(3):515–530, 2006

work page 2006
[9]

M. Fago, R. L. Hayes, E. A. Carter, and M. Ortiz. Density-functional-theory-based local quasicontinuum method: Prediction of dislocation nucleation. Phys. Rev. B, 70:100102, 2004

work page 2004
[10]

J. Fischer. Quantitative normal approximation for sums of random variables with multilevel local dependence. Preprint, 2018. arXiv:1905.10273

work page internal anchor Pith review Pith/arXiv arXiv 2018
[11]

J. Fischer. The choice of representative volumes in the approximation of eﬀective properties of random materials. to appear in Arch. Ration. Mech. Anal. , 2019. arXiv:1807.00834

work page arXiv 2019
[12]

Friesecke and F

G. Friesecke and F. Theil. Validity and failure of the Cauchy-Born hypothesis in a two- dimensional mass-spring lattice. J. Nonlinear Sci. , 12(5):445–478, 2002

work page 2002
[13]

Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial diﬀerential equations of second order . Springer-Verlag, Berlin – Heidelberg, 2001

work page 2001
[14]

Hohenberg and W

P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev. (2) , 136:B864–B871, 1964

work page 1964
[15]

Kohn and L

W. Kohn and L. J. Sham. Self-consistent equations including exchange and correlation eﬀects. Phys. Rev. (2) , 140:A1133–A1138, 1965

work page 1965
[16]

Le Bris, F

C. Le Bris, F. Legoll, and W. Minvielle. Special quasirandom structures: a selection approach for stochastic homogenization. Monte Carlo Methods Appl. , 22(1):25–54, 2016

work page 2016
[17]

E. H. Lieb. Thomas–Fermi and related theories of atoms and molecules. Rev. Mod. Phys. , 53:603–641, 1981

work page 1981
[18]

E. H. Lieb and B. Simon. The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math., 23:22–116, 1977

work page 1977
[19]

G. Lu, E. B. Tadmor, and E. Kaxiras. From electrons to ﬁnite elements: A concurrent multiscale approach for metals. Phys. Rev. B , 73:024108, 2006

work page 2006
[20]

F. Q. Nazar and C. Ortner. Locality of the Thomas–Fermi–von Weizs¨ acker equations. Arch. Ration. Mech. Anal., 224:817–870, 2017

work page 2017
[21]

R. G. Parr and W. Yang. Density-functional theory of the electronic structure of molecules. Annual Review of Physical Chemistry , 46(1):701–728, 1995

work page 1995
[22]

Rodney, L

D. Rodney, L. Ventelon, E. Clouet, L. Pizzagalli, and F. Willaime. Ab initio modeling of dislocation core properties in metals and semiconductors. Acta Materialia , 124:633 – 659, 2017

work page 2017
[23]

Schr¨ oder, D

J. Schr¨ oder, D. Balzani, and D. Brands. Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions. Archive of Applied Mechanics, 81(7):975–997, Jul 2011

work page 2011
[24]

J. Solovej. Universality in the Thomas–Fermi–von Weizs¨ acker model of atoms and molecules. Commun. Math. Phys. , 129:561–598, 1990

work page 1990
[25]

Suryanarayana, K

P. Suryanarayana, K. Bhattacharya, and M. Ortiz. Coarse-graining Kohn-Sham density func- tional theory. Journal of the Mechanics and Physics of Solids , 61(1):38 – 60, 2013

work page 2013
[26]

N. S. Trudinger. Linear elliptic operators with measurable coeﬃcients. Ann. Scuola Norm. – Sci., 27:265–308, 1973

work page 1973
[27]

von Pezold, A

J. von Pezold, A. Dick, M. Fri´ ak, and J. Neugebauer. Generation and performance of special quasirandom structures for studying the elastic properties of random alloys: Application to Al-Ti. Phys. Rev. B , 81:094203, 2010

work page 2010
[28]

S.-H. Wei, L. G. Ferreira, J. E. Bernard, and A. Zunger. Electronic properties of random alloys: Special quasirandom structures. Phys. Rev. B , 42:9622–9649, Nov 1990

work page 1990
[29]

Zunger, S.-H

A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard. Special quasirandom structures. Phys. Rev. Lett., 65:353–356, Jul 1990. 36 JULIAN FISCHER AND MICHAEL KNIELY Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria, E-Mail: julian.fischer@ist.ac.at Institute of Science and Technology Austria (IST Austria...

work page 1990

[1] [1]

Balzani, D

D. Balzani, D. Brands, J. Schr¨ oder, and C. Carstensen. Sensitivity analysis of statistical measures for the reconstruction of microstructures based on the minimization of generalized least-square functionals. Technische Mechanik, 30:297–315, 2010

work page 2010

[2] [2]

Balzani and J

D. Balzani and J. Schr¨ oder. Some basic ideas for the reconstruction of statistically similar microstructures for multiscale simulations. PAMM, 8(1):10533–10534, 2009. VARIANCE REDUCTION IN THE THOMAS–FERMI–VON WEIZS ¨ACKER ENERGY 35

work page 2009

[3] [3]

Blanc, C

X. Blanc, C. Le Bris, and P.-L. Lions. From molecular models to continuum mechanics. Arch. Ration. Mech. Anal., 164(4):341–381, 2002

work page 2002

[4] [4]

Blanc, C

X. Blanc, C. Le Bris, and P.-L. Lions. Atomistic to continuum limits for computational materials science. M2AN Math. Model. Numer. Anal. , 41(2):391–426, 2007

work page 2007

[5] [5]

Blanc, C

X. Blanc, C. Le Bris, and P.-L. Lions. The energy of some microscopic stochastic lattices. Arch. Ration. Mech. Anal., 184(2):303–339, 2007

work page 2007

[6] [6]

Blanc and M

X. Blanc and M. Lewin. The crystallization conjecture: a review. EMS Surv. Math. Sci. , 2(2):225–306, 2015

work page 2015

[7] [7]

Catto, C

I. Catto, C. Le Bris, and P.-L. Lions. The mathematical theory of thermodynamic limits: Thomas–Fermi type models. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1998

work page 1998

[8] [8]

Conti, G

S. Conti, G. Dolzmann, B. Kirchheim, and S. M¨ uller. Suﬃcient conditions for the validity of the Cauchy-Born rule close to SO( n). J. Eur. Math. Soc. (JEMS) , 8(3):515–530, 2006

work page 2006

[9] [9]

M. Fago, R. L. Hayes, E. A. Carter, and M. Ortiz. Density-functional-theory-based local quasicontinuum method: Prediction of dislocation nucleation. Phys. Rev. B, 70:100102, 2004

work page 2004

[10] [10]

J. Fischer. Quantitative normal approximation for sums of random variables with multilevel local dependence. Preprint, 2018. arXiv:1905.10273

work page internal anchor Pith review Pith/arXiv arXiv 2018

[11] [11]

J. Fischer. The choice of representative volumes in the approximation of eﬀective properties of random materials. to appear in Arch. Ration. Mech. Anal. , 2019. arXiv:1807.00834

work page arXiv 2019

[12] [12]

Friesecke and F

G. Friesecke and F. Theil. Validity and failure of the Cauchy-Born hypothesis in a two- dimensional mass-spring lattice. J. Nonlinear Sci. , 12(5):445–478, 2002

work page 2002

[13] [13]

Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial diﬀerential equations of second order . Springer-Verlag, Berlin – Heidelberg, 2001

work page 2001

[14] [14]

Hohenberg and W

P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev. (2) , 136:B864–B871, 1964

work page 1964

[15] [15]

Kohn and L

W. Kohn and L. J. Sham. Self-consistent equations including exchange and correlation eﬀects. Phys. Rev. (2) , 140:A1133–A1138, 1965

work page 1965

[16] [16]

Le Bris, F

C. Le Bris, F. Legoll, and W. Minvielle. Special quasirandom structures: a selection approach for stochastic homogenization. Monte Carlo Methods Appl. , 22(1):25–54, 2016

work page 2016

[17] [17]

E. H. Lieb. Thomas–Fermi and related theories of atoms and molecules. Rev. Mod. Phys. , 53:603–641, 1981

work page 1981

[18] [18]

E. H. Lieb and B. Simon. The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math., 23:22–116, 1977

work page 1977

[19] [19]

G. Lu, E. B. Tadmor, and E. Kaxiras. From electrons to ﬁnite elements: A concurrent multiscale approach for metals. Phys. Rev. B , 73:024108, 2006

work page 2006

[20] [20]

F. Q. Nazar and C. Ortner. Locality of the Thomas–Fermi–von Weizs¨ acker equations. Arch. Ration. Mech. Anal., 224:817–870, 2017

work page 2017

[21] [21]

R. G. Parr and W. Yang. Density-functional theory of the electronic structure of molecules. Annual Review of Physical Chemistry , 46(1):701–728, 1995

work page 1995

[22] [22]

Rodney, L

D. Rodney, L. Ventelon, E. Clouet, L. Pizzagalli, and F. Willaime. Ab initio modeling of dislocation core properties in metals and semiconductors. Acta Materialia , 124:633 – 659, 2017

work page 2017

[23] [23]

Schr¨ oder, D

J. Schr¨ oder, D. Balzani, and D. Brands. Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions. Archive of Applied Mechanics, 81(7):975–997, Jul 2011

work page 2011

[24] [24]

J. Solovej. Universality in the Thomas–Fermi–von Weizs¨ acker model of atoms and molecules. Commun. Math. Phys. , 129:561–598, 1990

work page 1990

[25] [25]

Suryanarayana, K

P. Suryanarayana, K. Bhattacharya, and M. Ortiz. Coarse-graining Kohn-Sham density func- tional theory. Journal of the Mechanics and Physics of Solids , 61(1):38 – 60, 2013

work page 2013

[26] [26]

N. S. Trudinger. Linear elliptic operators with measurable coeﬃcients. Ann. Scuola Norm. – Sci., 27:265–308, 1973

work page 1973

[27] [27]

von Pezold, A

J. von Pezold, A. Dick, M. Fri´ ak, and J. Neugebauer. Generation and performance of special quasirandom structures for studying the elastic properties of random alloys: Application to Al-Ti. Phys. Rev. B , 81:094203, 2010

work page 2010

[28] [28]

S.-H. Wei, L. G. Ferreira, J. E. Bernard, and A. Zunger. Electronic properties of random alloys: Special quasirandom structures. Phys. Rev. B , 42:9622–9649, Nov 1990

work page 1990

[29] [29]

Zunger, S.-H

A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard. Special quasirandom structures. Phys. Rev. Lett., 65:353–356, Jul 1990. 36 JULIAN FISCHER AND MICHAEL KNIELY Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria, E-Mail: julian.fischer@ist.ac.at Institute of Science and Technology Austria (IST Austria...

work page 1990