Derivation of nonlinear Gibbs measures from many-body quantum mechanics

Mathieu Lewin; Phan Thành Nam; Nicolas Rougerie

doi:10.5802/jep.18

Derivation of nonlinear Gibbs measures from many-body quantum mechanics
[Dérivation de mesures de Gibbs non linéaires comme limites d’un modèle de mécanique quantique à

N

corps]

Mathieu Lewin ¹ ; Phan Thành Nam ² ; Nicolas Rougerie ³

¹ CNRS & Université Paris-Dauphine, CEREMADE (UMR 7534) Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France
² IST Austria Am Campus 1, 3400 Klosterneuburg, Austria
³ Université Grenoble 1 & CNRS, LPMMC (UMR 5493) B.P. 166, F-38042 Grenoble, France

Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 65-115.

Résumé
Abstract

Nous prouvons que certaines mesures de Gibbs non linéaires peuvent être obtenues à partir des états de Gibbs grand-canoniques du problème à $N$ corps, dans une limite de champ moyen où la température $T$ diverge et la constante de couplage se comporte comme $1 / T$ . Nous commençons par caractériser les états de Gibbs en présence d’interactions comme minimiseurs d’une fonctionnelle comptant l’énergie libre relativement au cas sans interaction. Nous procédons ensuite à un analogue en dimension infinie d’une analyse semi-classique, en utilisant des propriétés fines de l’entropie relative quantique, le lien entre mesures de de Finetti et symboles supérieurs/inférieurs dans une base d’états cohérents, ainsi que des inégalités de type Berezin-Lieb. Nos résultats couvrent la mesure construite à partir de la fonctionnelle de Schrödinger non linéaire défocalisante sur un intervalle fini, ainsi que le cas d’interactions plus régulières en dimension supérieure.

We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature $T$ diverges and the interaction strength behaves as $1 / T$ . We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dimensional analogue of phase-space semiclassical analysis, using fine properties of the quantum relative entropy, the link between quantum de Finetti measures and upper/lower symbols in a coherent state basis, as well as Berezin-Lieb type inequalities. Our results cover the measure built on the defocusing nonlinear Schrödinger functional on a finite interval, as well as smoother interactions in dimensions $d \geq 2$ .

Reçu le : 2015-01-05
Accepté le : 2015-05-03

MR Zbl

DOI : 10.5802/jep.18

Classification : 81V70, 35Q40
Keywords: Many-body quantum mechanics, Bose-Einstein condensation, mean-field limit, non-linear Schrödinger equation, non-linear Gibbs measure, quantum de Finetti theorem
Mot clés : Mécanique quantique à $N$ corps, condensation de Bose-Einstein, limite de champ moyen, équation de Schrödinger non linéaire, mesure de Gibbs non linéaire, théorème de de Finetti quantique

Affiliations des auteurs :

Mathieu Lewin ¹ ; Phan Thành Nam ² ; Nicolas Rougerie ³

¹ CNRS & Université Paris-Dauphine, CEREMADE (UMR 7534) Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France
² IST Austria Am Campus 1, 3400 Klosterneuburg, Austria
³ Université Grenoble 1 & CNRS, LPMMC (UMR 5493) B.P. 166, F-38042 Grenoble, France

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Mathieu Lewin and Phan Th\`anh Nam and Nicolas Rougerie},
     title = {Derivation of nonlinear {Gibbs} measures from many-body quantum mechanics},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     publisher = {\'Ecole polytechnique},
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Mathieu Lewin; Phan Thành Nam; Nicolas Rougerie. Derivation of nonlinear Gibbs measures from many-body quantum mechanics. Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 65-115. doi : 10.5802/jep.18. https://jep.centre-mersenne.org/articles/10.5802/jep.18/

Bibliographie
Cité par

[1] S. Albeverio & R. Høegh-Krohn - “The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time”, J. Functional Analysis 16 (1974), p. 39-82 | DOI | MR | Zbl

[2] Z. Ammari - “Systèmes hamiltoniens en théorie quantique des champs : dynamique asymptotique et limite classique” (2013), Habilitation à diriger des recherches, Université de Rennes I

[3] Z. Ammari & F. Nier - “Mean field limit for bosons and infinite dimensional phase-space analysis”, Ann. Henri Poincaré 9 (2008), p. 1503-1574 | DOI | MR | Zbl

[4] Z. Ammari & F. Nier - “Mean field limit for bosons and propagation of Wigner measures”, J. Math. Phys. 50 (2009) no. 4, 042107 pages | DOI | MR | Zbl

[5] Z. Ammari & F. Nier - “Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states”, J. Math. Pures Appl. 95 (2011) no. 6, p. 585-626 | DOI | MR | Zbl

[6] 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 | DOI | MR | Zbl

[7] R. Benguria & E. H. Lieb - “Proof of the stability of highly negative ions in the absence of the Pauli principle”, Phys. Rev. Lett. 50 (1983), p. 1771-1774 | DOI

[8] F. A. Berezin - “Convex functions of operators”, Mat. Sb. (N.S.) 88(130) (1972), p. 268-276 | MR

[9] V. I. Bogachev - Gaussian measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998 | MR

[10] J. Bourgain - “Periodic nonlinear Schrödinger equation and invariant measures”, Comm. Math. Phys. 166 (1994) no. 1, p. 1-26 | Zbl

[11] J. Bourgain - “Invariant measures for the 2D-defocusing nonlinear Schrödinger equation”, Comm. Math. Phys. 176 (1996), p. 421-445 | DOI | Zbl

[12] J. Bourgain - “Invariant measures for the Gross-Piatevskii equation”, J. Math. Pures Appl. 76 (1997) no. 8, p. 649-02 | DOI | MR | Zbl

[13] J. Bourgain - “Invariant measures for NLS in infinite volume”, Comm. Math. Phys. 210 (2000) no. 3, p. 605-620 | DOI | MR | Zbl

[14] N. Burq, L. Thomann & N. Tzvetkov - “Long time dynamics for the one dimensional non linear Schrödinger equation”, Ann. Inst. Fourier (Grenoble) 63 (2013) no. 6, p. 2137-2198 | DOI | Numdam | Zbl

[15] N. Burq & N. Tzvetkov - “Random data Cauchy theory for supercritical wave equations. I. Local theory”, Invent. Math. 173 (2008) no. 3, p. 449-475 | DOI | MR | Zbl

[16] E. Carlen - “Trace inequalities and quantum entropy: an introductory course”, in Entropy and the Quantum (R. Sims & D. Ueltschi, eds.), Contemp. Math., vol. 529, American Mathematical Society, Providence, RI, 2010, p. 73-140 | DOI | MR

[17] E. Carlen, J. Fröhlich & J. L. Lebowitz - “Exponential relaxation to equilibrium for a one-dimensional focusing non-linear Schrödinger equation with noise” (2014), arXiv:1409.2327

[18] G. Chiribella - “On quantum estimation, quantum cloning and finite quantum de Finetti theorems”, in Theory of Quantum Computation, Communication, and Cryptography, Lect. Notes in Computer Science, vol. 6519, Springer, 2011 | DOI | MR | Zbl

[19] M. Christandl, R. König, G. Mitchison & R. Renner - “One-and-a-half quantum de Finetti theorems”, Comm. Math. Phys. 273 (2007) no. 2, p. 473-498 | DOI | MR | Zbl

[20] J. Colliander & T. Oh - “Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^{2} (𝕋)$ ”, Duke Math. J. 161 (2012) no. 3, p. 367-414 | DOI | Zbl

[21] M. Combescure & D. Robert - Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics, Springer, Dordrecht, 2012 | DOI | Zbl

[22] A.-S. de Suzzoni - “Invariant measure for the cubic wave equation on the unit ball of $ℝ^{3}$ ”, Dyn. Partial Differ. Equ. 8 (2011) no. 2, p. 127-147 | MR | Zbl

[23] J. Dolbeault, P. Felmer, M. Loss & E. Paturel - “Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems”, J. Funct. Anal. 238 (2006) no. 1, p. 193-220 | DOI | MR | Zbl

[24] J. Glimm & A. Jaffe - Quantum physics: a functional integral point of view, Springer-Verlag, 1987

[25] A. D. Gottlieb - “Examples of bosonic de Finetti states over finite dimensional Hilbert spaces”, J. Statist. Phys. 121 (2005) no. 3-4, p. 497-509 | DOI | MR | Zbl

[26] A. D. Gottlieb & T. Schumm - “Quantum noise thermometry for bosonic Josephson junctions in the mean-field regime”, Phys. Rev. A 79 (2009), 063601 pages | DOI

[27] P. Grech & R. Seiringer - “The excitation spectrum for weakly interacting bosons in a trap”, Comm. Math. Phys. 322 (2013) no. 2, p. 559-591 | DOI | MR | Zbl

[28] F. Guerra, L. Rosen & B. Simon - “The $P {(φ)}_{2}$ Euclidean quantum field theory as classical statistical mechanics. I, II”, Ann. of Math. (2) 101 (1975), p. 111-189; ibid. (2) 101 (1975), 191–259 | DOI | MR

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

[30] A. Harrow - “The church of the symmetric subspace” (2013), arXiv:1308.6595

[31] R. L. Hudson & G. R. Moody - “Locally normal symmetric states and an analogue of de Finetti’s theorem”, Z. Wahrsch. Verw. Gebiete 33 (1975/76) no. 4, p. 343-351 | DOI | MR

[32] B. Juliá-Díaz, A. D. Gottlieb, J. Martorell & A. Polls - “Quantum and thermal fluctuations in bosonic Josephson junctions”, Phys. Rev. A 88 (2013), 033601 pages | DOI

[33] A. Knowles - “Limiting dynamics in large quantum systems”, Doctoral thesis, ETH Zürich, 2009

[34] J. L. Lebowitz, H. A. Rose & E. R. Speer - “Statistical mechanics of the nonlinear Schrödinger equation”, J. Statist. Phys. 50 (1988) no. 3-4, p. 657-687 | DOI | Zbl

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

[36] M. Lewin, P. T. Nam & N. Rougerie - “Derivation of Hartree’s theory for generic mean-field Bose systems”, Adv. Math. 254 (2014), p. 570-621 | DOI | MR | Zbl

[37] M. Lewin, P. T. Nam & N. Rougerie - “The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases” (2014), to appear in TRAAM, http://arxiv.org/abs/1405.3220

[38] M. Lewin, P. T. Nam & N. Rougerie - “Remarks on the quantum de Finetti theorem for bosonic systems”, Appl. Math. Res. Express. AMRX 1 (2015), p. 48-63 | DOI | MR | Zbl

[39] M. Lewin & J. Sabin - “A family of monotone quantum relative entropies”, Lett. Math. Phys. 104 (2014) no. 6, p. 691-705 | DOI | MR | Zbl

[40] E. H. Lieb - “The classical limit of quantum spin systems”, Comm. Math. Phys. 31 (1973), p. 327-340 | DOI | MR | Zbl

[41] E. H. Lieb - “Convex trace functions and the Wigner-Yanase-Dyson conjecture”, Adv. Math. 11 (1973), p. 267-288 | DOI | MR | Zbl

[42] E. H. Lieb & M. B. Ruskai - “A fundamental property of quantum-mechanical entropy”, Phys. Rev. Lett. 30 (1973), p. 434-436 | DOI | MR

[43] E. H. Lieb & M. B. Ruskai - “Proof of the strong subadditivity of quantum-mechanical entropy”, J. Math. Phys. 14 (1973), p. 1938-1941, With an appendix by B. Simon | DOI | MR

[44] E. H. Lieb, R. Seiringer, J. P. Solovej & J. Yngvason - The mathematics of the Bose gas and its condensation, Oberwolfach Seminars, Birkhäuser, 2005 | Zbl

[45] E. H. Lieb, R. Seiringer & J. Yngvason - “Justification of $c$ -number substitutions in bosonic Hamiltonians”, Phys. Rev. Lett. 94 (2005), 080401 pages | DOI

[46] E. H. Lieb & H.-T. Yau - “The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics”, Comm. Math. Phys. 112 (1987) no. 1, p. 147-174 | MR | Zbl

[47] J. Lörinczi, F. Hiroshima & V. Betz - Feynman-Kac-type theorems and Gibbs measures on path space: with applications to rigorous quantum field theory, de Gruyter Studies in Math., Walter de Gruyter, 2011 | DOI | Zbl

[48] E. Nelson - “Construction of quantum fields from Markoff fields”, J. Funct. Anal. 12 (1973) no. 1, p. 97 -112 | DOI | MR | Zbl

[49] E. Nelson - “The free Markoff field”, J. Funct. Anal. 12 (1973), p. 211-227 | DOI | MR | Zbl

[50] T. Oh & J. Quastel - “On invariant Gibbs measures conditioned on mass and momentum”, J. Math. Soc. Japan 65 (2013) no. 1, p. 13-35 | DOI | MR | Zbl

[51] M. Ohya & D. Petz - Quantum entropy and its use, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1993 | DOI | Zbl

[52] D. Petz - “Monotonicity of quantum relative entropy revisited”, Rev. Math. Phys. 15 (2003) no. 1, p. 79-91 | DOI | MR | Zbl

[53] N. Rougerie - “Théorèmes de de Finetti, limites de champ moyen et condensation de Bose-Einstein” (2014), Lecture notes

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

[55] B. Simon - The $P {(φ)}_{2}$ Euclidean (quantum) field theory, Princeton University Press, Princeton, N.J., 1974, Princeton Series in Physics | Zbl

[56] B. Simon - Trace ideals and their applications, LMS Lecture Note Series, vol. 35, Cambridge University Press, Cambridge, 1979 | MR | Zbl

[57] B. Simon - “The classical limit of quantum partition functions”, Comm. Math. Phys. 71 (1980) no. 3, p. 247-276 | DOI | MR | Zbl

[58] A. V. Skorokhod - Integration in Hilbert space, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1974 | Zbl

[59] W. F. Stinespring - “Positive functions on $C^{*}$ -algebras”, Proc. Amer. Math. Soc. 6 (1955) no. 2, p. 211-216 | MR | Zbl

[60] E. Størmer - “Symmetric states of infinite tensor products of $C^{*}$ -algebras”, J. Funct. Anal. 3 (1969), p. 48-68 | DOI | MR

[61] S. J. Summers - “A perspective on constructive quantum field theory” (2012), arXiv:1203.3991

[62] L. Thomann & N. Tzvetkov - “Gibbs measure for the periodic derivative nonlinear Schrödinger equation”, Nonlinearity 23 (2010) no. 11, 2771 pages | DOI | Zbl

[63] N. Tzvetkov - “Invariant measures for the defocusing nonlinear Schrödinger equation”, Ann. Inst. Fourier (Grenoble) 58 (2008) no. 7, p. 2543-2604 | DOI | Numdam | Zbl

[64] - Constructive quantum field theory: The 1973 Ettore Majorana international school of mathematical physics (G. Velo & A. S. Wightman, eds.), Lect. Notes in Physics, Springer-Verlag, 1973 | Zbl

[65] A. Wehrl - “General properties of entropy”, Rev. Modern Phys. 50 (1978) no. 2, p. 221-260 | MR

Cité par Sources :