Derivation of nonlinear Gibbs measures from many-body quantum mechanics

Lewin, Mathieu; Nam, Phan Thành; Rougerie, Nicolas

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

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 $d\ge 2$.

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

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

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

[4] Ammari, Z.; Nier, F. Mean field limit for bosons and propagation of Wigner measures, J. Math. Phys., Volume 50 (2009) no. 4, pp. 042107 | MR | Zbl

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

[6] Bach, V.; Lieb, E. H.; Solovej, J. P. Generalized Hartree-Fock theory and the Hubbard model, J. Statist. Phys., Volume 76 (1994) no. 1-2, pp. 3-89 | MR | Zbl

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

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

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

[10] Bourgain, J. Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., Volume 166 (1994) no. 1, pp. 1-26 http://projecteuclid.org/getRecord?id=euclid.cmp/1104271501 | MR | Zbl

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

[12] Bourgain, J. Invariant measures for the Gross-Piatevskii equation, J. Math. Pures Appl., Volume 76 (1997) no. 8, p. 649-02 http://www.sciencedirect.com/science/article/pii/S0021782497899655 | DOI | MR | Zbl

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

[14] Burq, N.; Thomann, L.; Tzvetkov, N. Long time dynamics for the one dimensional non linear Schrödinger equation, Ann. Inst. Fourier (Grenoble), Volume 63 (2013) no. 6, pp. 2137-2198 | Numdam | MR

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

[16] Carlen, E.; Sims, R.; Ueltschi, D. Trace inequalities and quantum entropy: an introductory course, Entropy and the Quantum (Contemp. Math.), Volume 529, American Mathematical Society, Providence, RI, 2010, pp. 73-140 | MR | Zbl

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

[18] Chiribella, G. On quantum estimation, quantum cloning and finite quantum de Finetti theorems, Theory of Quantum Computation, Communication, and Cryptography (Lect. Notes in Computer Science), Volume 6519, Springer, 2011 | MR

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

[20] Colliander, J.; Oh, T. Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2\left(𝕋\right)$, Duke Math. J., Volume 161 (2012) no. 3, pp. 367-414 | DOI | MR | Zbl

[21] Combescure, M.; Robert, D. Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics, Springer, Dordrecht, 2012, pp. xiv+415 | DOI | MR | Zbl

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

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

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

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

[26] Gottlieb, A. D.; Schumm, T. Quantum noise thermometry for bosonic Josephson junctions in the mean-field regime, Phys. Rev. A, Volume 79 (2009), pp. 063601 http://link.aps.org/doi/10.1103/PhysRevA.79.063601 | DOI

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

[28] Guerra, F.; Rosen, L.; Simon, B. The $\mathbf{P}{\left(\phi \right)}_2$ Euclidean quantum field theory as classical statistical mechanics. I, II, Ann. of Math. (2), Volume 101 (1975), p. 111-189; ibid. (2) 101 (1975), 191–259 | MR

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

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

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

[32] Juliá-Díaz, B.; Gottlieb, A. D.; Martorell, J.; Polls, A. Quantum and thermal fluctuations in bosonic Josephson junctions, Phys. Rev. A, Volume 88 (2013), pp. 033601 http://link.aps.org/doi/10.1103/PhysRevA.88.033601 | DOI

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

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

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

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

[37] Lewin, M.; Nam, P. T.; Rougerie, N. 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] Lewin, M.; Nam, P. T.; Rougerie, N. Remarks on the quantum de Finetti theorem for bosonic systems, Appl. Math. Res. Express. AMRX, Volume 1 (2015), pp. 48-63 | DOI | MR

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

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

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

[42] Lieb, E. H.; Ruskai, M. B. A fundamental property of quantum-mechanical entropy, Phys. Rev. Lett., Volume 30 (1973), pp. 434-436 | MR

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

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

[45] Lieb, E. H.; Seiringer, R.; Yngvason, J. Justification of $c$-number substitutions in bosonic Hamiltonians, Phys. Rev. Lett., Volume 94 (2005), pp. 080401 http://link.aps.org/doi/10.1103/PhysRevLett.94.080401 | DOI

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

[47] Lörinczi, J.; Hiroshima, F.; Betz, V. 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 http://books.google.fr/books?id=5va_PAAACAAJ | MR | Zbl

[48] Nelson, E. Construction of quantum fields from Markoff fields, J. Funct. Anal., Volume 12 (1973) no. 1, pp. 97 -112 http://www.sciencedirect.com/science/article/pii/0022123673900918 | DOI | MR | Zbl

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

[50] Oh, T.; Quastel, J. On invariant Gibbs measures conditioned on mass and momentum, J. Math. Soc. Japan, Volume 65 (2013) no. 1, pp. 13-35 http://jlc.jst.go.jp/DN/JST.JSTAGE/jmath/65.13 | MR | Zbl

[51] Ohya, M.; Petz, D. Quantum entropy and its use, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1993, pp. viii+335 | DOI | MR | Zbl

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

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

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

[55] Simon, B. The $P{\left(\phi \right)}_2$ Euclidean (quantum) field theory, Princeton University Press, Princeton, N.J., 1974, pp. xx+392 (Princeton Series in Physics) | MR | Zbl

[56] Simon, B. Trace ideals and their applications, LMS Lecture Note Series, 35, Cambridge University Press, Cambridge, 1979, pp. viii+134 | MR | Zbl

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

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

[59] Stinespring, W. F. Positive functions on $C^{*}$-algebras, Proc. Amer. Math. Soc., Volume 6 (1955) no. 2, pp. 211-216 http://www.jstor.org/stable/2032342 | MR | Zbl

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

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

[62] Thomann, L.; Tzvetkov, N. Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, Volume 23 (2010) no. 11, pp. 2771 http://stacks.iop.org/0951-7715/23/i=11/a=003 | MR | Zbl

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

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

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