Statistical mechanics of the uniform electron gas
[Mécanique statistique pour le gaz uniforme d’électrons]
Journal de l’École polytechnique — Mathématiques, Tome 5 (2018) , pp. 79-116.

Dans cet article nous définissons et étudions le gaz uniforme d’électrons, un système comprenant une infinité de particules arrangées de sorte que la densité moyenne soit constante dans tout l’espace. Ceci est en principe différent du Jellium, qui comprend une charge uniforme positive sans aucune contrainte sur la densité des électrons. Nous démontrons que le gaz uniforme d’électrons s’obtient en théorie de la fonctionnelle de la densité, dans la limite où la densité du système varie lentement. Nous construisons également le gaz uniforme quantique et montrons la convergence vers le gaz classique dans le régime de faible densité.

In this paper we define and study the classical Uniform Electron Gas (UEG), a system of infinitely many electrons whose density is constant everywhere in space. The UEG is defined differently from Jellium, which has a positive constant background but no constraint on the density. We prove that the UEG arises in Density Functional Theory in the limit of a slowly varying density, minimizing the indirect Coulomb energy. We also construct the quantum UEG and compare it to the classical UEG at low density.

Reçu le : 2017-06-12
Accepté le : 2017-11-07
Publié le : 2017-11-29
DOI : https://doi.org/10.5802/jep.64
Classification : 82B03,  81V70,  49K21
Mots clés: Gaz uniforme d’électrons, théorie de la fonctionnelle de la densité, limite thermodynamique, mécanique statistique, limites de champ moyen, transport optimal
@article{JEP_2018__5__79_0,
     author = {Mathieu Lewin and Elliott H. Lieb and Robert Seiringer},
     title = {Statistical mechanics of the~uniform~electron~gas},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {5},
     year = {2018},
     pages = {79-116},
     doi = {10.5802/jep.64},
     zbl = {06988574},
     mrnumber = {3732693},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2018__5__79_0/}
}
Lewin, Mathieu; Lieb, Elliott H.; Seiringer, Robert. Statistical mechanics of the uniform electron gas. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018) , pp. 79-116. doi : 10.5802/jep.64. https://jep.centre-mersenne.org/item/JEP_2018__5__79_0/

[1] M. Aizenman & P. A. Martin - “Structure of Gibbs states of one dimensional Coulomb systems”, Comm. Math. Phys. 78 (1980) no. 1, p. 99-116 | MR 597033

[2] V. Bach - “Error bound for the Hartree-Fock energy of atoms and molecules”, Comm. Math. Phys. 147 (1992) no. 3, p. 527-548 | MR 1175492 | Zbl 0771.46038

[3] V. Bach, E. H. Lieb & J. P. Solovej - “Generalized Hartree-Fock theory and the Hubbard model”, J. Statist. Phys. 76 (1994) no. 1-2, p. 3-89 | Article | MR 1297873 | Zbl 0839.60095

[4] A. D. Becke - “Density-functional thermochemistry. III. The role of exact exchange”, J. Chem. Phys. 98 (1993) no. 7, p. 5648-5652 | Article

[5] U. Bindini & L. De Pascale - “Optimal transport with Coulomb cost and the semiclassical limit of density functional theory”, J. Éc. polytech. Math. 4 (2017), p. 909-934 | Article | MR 3714366

[6] R. F. Bishop & K. H. Lührmann - “Electron correlations. II. Ground-state results at low and metallic densities”, Phys. Rev. B 26 (1982), p. 5523-5557 | Article

[7] X. Blanc & M. Lewin - “Existence of the thermodynamic limit for disordered quantum Coulomb systems”, J. Math. Phys. 53 (2012), article no. 095209 | Article | MR 2905791 | Zbl 1278.82031

[8] D. Borwein, J. M. Borwein & R. Shail - “Analysis of certain lattice sums”, J. Math. Anal. Appl. 143 (1989) no. 1, p. 126-137 | Article | MR 1019453 | Zbl 0682.10028

[9] D. Borwein, J. M. Borwein, R. Shail & I. J. Zucker - “Energy of static electron lattices”, J. Phys. A 21 (1988) no. 7, p. 1519-1531 | Article | MR 951042 | Zbl 0675.33010

[10] D. Borwein, J. M. Borwein & A. Straub - “On lattice sums and Wigner limits”, J. Math. Anal. Appl. 414 (2014) no. 2, p. 489-513 | Article | MR 3167976 | Zbl 1369.11046

[11] H. J. Brascamp & E. H. Lieb - “Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma”, in Functional Integration and Its Applications (A. Arthurs, ed.), Clarendon Press, Oxford, 1975 | Zbl 0348.26011

[12] D. C. Brydges & P. A. Martin - “Coulomb systems at low density: a review”, J. Statist. Phys. 96 (1999) no. 5-6, p. 1163-1330 | Article | MR 1722991 | Zbl 1080.82503

[13] G. Buttazzo, T. Champion & L. De Pascale - “Continuity and estimates for multimarginal optimal transportation problems with singular costs”, Appl. Math. Optim. (2017), doi:10.1007/s00245-017-9403-7 | Zbl 1400.49051

[14] P. Choquard, P. Favre & C. Gruber - “On the equation of state of classical one-component systems with long-range forces”, J. Statist. Phys. 23 (1980), p. 405-442 | Article | MR 596350

[15] M. Colombo, L. De Pascale & S. Di Marino - “Multimarginal optimal transport maps for one-dimensional repulsive costs”, Canad. J. Math. 67 (2015), p. 350-368 | Article | MR 3314838 | Zbl 1312.49052

[16] J. G. Conlon, E. H. Lieb & H.-T. Yau - “The N 7/5 law for charged bosons”, Comm. Math. Phys. 116 (1988) no. 3, p. 417-448 | MR 937769

[17] C. Cotar, G. Friesecke & C. Klüppelberg - “Density functional theory and optimal transportation with Coulomb cost”, Comm. Pure Appl. Math. 66 (2013) no. 4, p. 548-599 | Article | MR 3020313 | Zbl 1266.82057

[18] C. Cotar, G. Friesecke & B. Pass - “Infinite-body optimal transport with Coulomb cost”, Calc. Var. Partial Differential Equations 54 (2015) no. 1, p. 717-742 | MR 3385178 | Zbl 1322.49073

[19] S. Di Marino, 2017, in preparation

[20] S. Di Marino, A. Gerolin & L. Nenna - “Optimal transportation theory with repulsive costs”, in Topological optimization and optimal transport in the applied sciences (F. Santambrogio, T. Champion, G. Carlier, M. Rumpf, É. Oudet & M. Bergounioux, eds.), Radon series on computational and applied mathematics, vol. 17, De Gruyter, 2017, p. 204-256 | MR 3729378

[21] N. D. Drummond, Z. Radnai, J. R. Trail, M. D. Towler & R. J. Needs - “Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals”, Phys. Rev. B (2004), article no. 085116 | Article

[22] M. E. Fisher - “The free energy of a macroscopic system”, Arch. Rational Mech. Anal. 17 (1964), p. 377-410 | Article | MR 172644

[23] S. Fournais, M. Lewin & J. P. Solovej - “The semi-classical limit of large fermionic systems” (2015), arXiv:1510.01124

[24] J. Fröhlich & Y. M. Park - “Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems”, Comm. Math. Phys. 59 (1978) no. 3, p. 235-266 | MR 496191

[25] P. Gori-Giorgi & M. Seidl - “Density functional theory for strongly-interacting electrons: perspectives for physics and chemistry”, Phys. Chem. Chem. Phys. 12 (2010), p. 14405-14419 | Article

[26] G. M. Graf & D. Schenker - “On the molecular limit of Coulomb gases”, Comm. Math. Phys. 174 (1995) no. 1, p. 215-227 | MR 1372807 | Zbl 0837.58048

[27] G. M. Graf & J. P. Solovej - “A correlation estimate with applications to quantum systems with Coulomb interactions”, Rev. Math. Phys. 06 (1994) no. 05a, p. 977-997 | Article | MR 1301362

[28] C. Gruber, J. L. Lebowitz & P. A. Martin - “Sum rules for inhomogeneous Coulomb systems”, J. Chem. Phys. 75 (1981) no. 2, p. 944-954 | MR 621569

[29] C. Gruber, C. Lugrin & P. A. Martin - “Equilibrium equations for classical systems with long range forces and application to the one dimensional Coulomb gas”, Helv. Phys. Acta 51 (1978) no. 5-6, p. 829-866 | MR 542803

[30] C. Gruber, C. Lugrin & P. A. Martin - “Equilibrium properties of classical systems with long-range forces. BBGKY equation, neutrality, screening, and sum rules”, J. Statist. Phys. 22 (1980), p. 193-236 | Article | MR 560555

[31] C. Gruber & P. A. Martin - “Translation invariance in statistical mechanics of classical continuous systems”, Ann. Physics 131 (1981) no. 1, p. 56 -72 | MR 608086

[32] C. Hainzl, M. Lewin & J. P. Solovej - “The thermodynamic limit of quantum Coulomb systems. Part I. General theory”, Advances in Math. 221 (2009), p. 454-487 | Article | Zbl 1165.81041

[33] C. Hainzl, M. Lewin & J. P. Solovej - “The thermodynamic limit of quantum Coulomb systems. Part II. Applications”, Advances in Math. 221 (2009), p. 488-546 | Article | Zbl 1165.81042

[34] J. E. Harriman - “Orthonormal orbitals for the representation of an arbitrary density”, Phys. Rev. A (3) 24 (1981) no. 2, p. 680-682 | Article

[35] M. Hoffmann-Ostenhof & T. Hoffmann-Ostenhof - “Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules”, Phys. Rev. A (3) 16 (1977) no. 5, p. 1782-1785 | Article

[36] P. Hohenberg & W. Kohn - “Inhomogeneous electron gas”, Phys. Rev. 136 (1964) no. 3B, p. B864-B871 | Article | MR 180312

[37] J. Z. Imbrie - “Debye screening for jellium and other Coulomb systems”, Comm. Math. Phys. 87 (1982) no. 4, p. 515-565 | MR 691043

[38] G. Kin-Lic Chan & N. C. Handy - “Optimized Lieb-Oxford bound for the exchange-correlation energy”, Phys. Rev. A (3) 59 (1999) no. 4, p. 3075-3077 | Article

[39] W. Kohn & L. J. Sham - “Self-consistent equations including exchange and correlation effects”, Phys. Rev. (2) 140 (1965), p. A1133-A1138 | Article | MR 189732

[40] H. Kunz - “The one-dimensional classical electron gas”, Ann. Physics 85 (1974) no. 2, p. 303 -335 | MR 426742

[41] O. Lazarev & E. H. Lieb - “A smooth, complex generalization of the Hobby-Rice theorem”, Indiana Univ. Math. J. 62 (2013) no. 4, p. 1133-1141 | Article | MR 3179686 | Zbl 1295.41025

[42] T. Leblé & S. Serfaty - “Large deviation principle for empirical fields of Log and Riesz gases”, Invent. Math. (2017), doi:10.1007/s00222-017-0738-0 | MR 3735628 | Zbl 1397.82007

[43] M. Lewin - “Geometric methods for nonlinear many-body quantum systems”, J. Funct. Anal. 260 (2011), p. 3535-3595 | Article | MR 2781970 | Zbl 1216.81180

[44] M. Lewin & E. H. Lieb - “Improved Lieb-Oxford exchange-correlation inequality with gradient correction”, Phys. Rev. A (3) 91 (2015) no. 2, article no. 022507 | MR 3405770

[45] M. Lewin, P. T. Nam, S. Serfaty & J. P. Solovej - “Bogoliubov spectrum of interacting Bose gases”, Comm. Pure Appl. Math. 68 (2015) no. 3, p. 413-471 | Article | MR 3310520 | Zbl 1318.82030

[46] E. H. Lieb - “A lower bound for Coulomb energies”, Phys. Lett. A 70 (1979), p. 444-446 | Article | MR 588128

[47] E. H. Lieb - “Density functionals for Coulomb systems”, Int. J. Quantum Chem. 24 (1983), p. 243-277 | Article

[48] E. H. Lieb & M. Loss - Analysis, Graduate Studies in Math., vol. 14, American Mathematical Society, Providence, RI, 2001

[49] E. H. Lieb & H. Narnhofer - “The thermodynamic limit for jellium”, J. Statist. Phys. 12 (1975) no. 4, p. 291-310 | Article | MR 401029 | Zbl 0973.82500

[50] E. H. Lieb & S. Oxford - “Improved lower bound on the indirect Coulomb energy”, Int. J. Quantum Chem. 19 (1980) no. 3, p. 427-439 | Article

[51] E. H. Lieb & R. Schrader - “Current densities in density-functional theory”, Phys. Rev. A (3) 88 (2013) no. 3, article no. 032516

[52] E. H. Lieb & R. Seiringer - The stability of matter in quantum mechanics, Cambridge Univ. Press, 2010

[53] E. H. Lieb, J. P. Solovej & J. Yngvason - “Ground states of large quantum dots in magnetic fields”, Phys. Rev. B 51 (1995), p. 10646-10665 | Article

[54] D. Lundholm, P. T. Nam & F. Portmann - “Fractional Hardy-Lieb-Thirring and related inequalities for interacting systems”, Arch. Rational Mech. Anal. 219 (2016) no. 3, p. 1343-1382 | Article | MR 3448930 | Zbl 1332.81292

[55] P. A. Martin & T. Yalcin - “The charge fluctuations in classical Coulomb systems”, J. Statist. Phys. 22 (1980), p. 435-463 | Article | MR 574007

[56] S. A. Mikhailov & K. Ziegler - “Floating Wigner molecules and possible phase transitions in quantum dots”, European Phys. J. B 28 (2002) no. 1, p. 117-120 | Article

[57] M. Navet, E. Jamin & M. R. Feix - “«Virial» pressure of the classical one-component plasma”, J. Physique Lett. 41 (1980) no. 3, p. 69-73 | Article

[58] M. M. Odashima & K. Capelle - “How tight is the Lieb-Oxford bound?”, J. Chem. Phys. 127 (2007) no. 5, 054106 pages

[59] J. P. Perdew - “Unified theory of exchange and correlation beyond the local density approximation”, in Electronic Structure of Solids ’91 (P. Ziesche & H. Eschrig, eds.), Akademie Verlag, Berlin, 1991, p. 11-20

[60] J. P. Perdew, K. Burke & M. Ernzerhof - “Generalized gradient approximation made simple”, Phys. Rev. Lett. 77 (1996), p. 3865-3868 | Article

[61] J. P. Perdew & S. Kurth - “Density functionals for non-relativistic Coulomb systems in the new century”, in A primer in density functional theory (C. Fiolhais, F. Nogueira & M. A. L. Marques, eds.), Springer, Berlin, Heidelberg, 2003, p. 1-55

[62] J. P. Perdew & Y. Wang - “Accurate and simple analytic representation of the electron-gas correlation energy”, Phys. Rev. B 45 (1992), p. 13244-13249 | Article

[63] M. Petrache & S. Serfaty - “Next order asymptotics and renormalized energy for Riesz interactions”, J. Inst. Math. Jussieu 16 (2015) no. 3, p. 1-69 | MR 3646281 | Zbl 1373.82013

[64] E. Räsänen, S. Pittalis, K. Capelle & C. R. Proetto - “Lower bounds on the exchange-correlation energy in reduced dimensions”, Phys. Rev. Lett. 102 (2009) no. 20, article no. 206406 | Article

[65] E. Räsänen, M. Seidl & P. Gori-Giorgi - “Strictly correlated uniform electron droplets”, Phys. Rev. B 83 (2011) no. 19, article no. 195111 | Article

[66] S. Rota Nodari & S. Serfaty - “Renormalized energy equidistribution and local charge balance in 2d Coulomb system”, Internat. Math. Res. Notices (2015) no. 11, p. 3035-3093 | MR 3373044 | Zbl 1321.82029

[67] N. Rougerie & S. Serfaty - “Higher dimensional Coulomb gases and renormalized energy functionals”, Comm. Pure Appl. Math. 69 (2016) no. 3, p. 519-605 | Article | MR 3455593 | Zbl 1338.82043

[68] D. Ruelle - Statistical mechanics. Rigorous results, World Scientific & Imperial College Press, Singapore & London, 1999 | Zbl 1016.82500

[69] V. Rutherfoord - “On the Lazarev-Lieb extension of the Hobby-Rice theorem”, Adv. in Math. 244 (2013), p. 16-22 | Article | MR 3077864 | Zbl 1350.41035

[70] E. Sandier & S. Serfaty - “1D log gases and the renormalized energy: crystallization at vanishing temperature”, Probab. Theory Relat. Fields (2014), p. 1-52

[71] E. Sandier & S. Serfaty - “2D Coulomb gases and the renormalized energy”, Ann. Probability 43 (2015) no. 4, p. 2026-2083 | Article | MR 3353821 | Zbl 1328.82006

[72] M. Seidl - “Strong-interaction limit of density-functional theory”, Phys. Rev. A (3) 60 (1999) no. 6, p. 4387-4395 | Article

[73] M. Seidl, S. Di Marino, A. Gerolin, L. Nenna, K. J. H. Giesbertz & P. Gori-Giorgi - “The strictly-correlated electron functional for spherically symmetric systems revisited” (2017), arXiv:1702.05022

[74] M. Seidl, P. Gori-Giorgi & A. Savin - “Strictly correlated electrons in density-functional theory: a general formulation with applications to spherical densities”, Phys. Rev. A (3) 75 (2007), article no. 042511

[75] M. Seidl, J. P. Perdew & M. Levy - “Strictly correlated electrons in density-functional theory”, Phys. Rev. A (3) 59 (1999) no. 1, p. 51-54

[76] M. Seidl, S. Vuckovic & P. Gori-Giorgi - “Challenging the Lieb-Oxford bound in a systematic way”, Molecular Phys. 114 (2016) no. 7-8, p. 1076-1085 | Article

[77] S. Serfaty - “Ginzburg-Landau vortices, Coulomb gases, and renormalized energies”, J. Statist. Phys. 154 (2014) no. 3, p. 660-680 | Article | MR 3163544 | Zbl 1291.82142

[78] J. Sun, J. P. Perdew & A. Ruzsinszky - “Semilocal density functional obeying a strongly tightened bound for exchange”, Proc. Nat. Acad. Sci. U.S.A. 112 (2015), p. 685-689 | Article

[79] J. Sun, R. C. Remsing, Y. Zhang, Z. Sun, A. Ruzsinszky, H. Peng, Z. Yang, A. Paul, U. Waghmare, X. Wu, M. L. Klein & J. P. Perdew - “Accurate first-principles structures and energies of diversely bonded systems from an efficient density functional”, Nature Chemistry 8 (2016), 831–836 pages

[80] J. Sun, A. Ruzsinszky & J. P. Perdew - “Strongly Constrained and Appropriately Normed Semilocal Density Functional”, Phys. Rev. Lett. 115 (2015), article no. 036402