From bosonic grand-canonical ensembles to nonlinear Gibbs measures
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 5, 17 p.

In a recent paper, in collaboration with Mathieu Lewin and Phan Thành Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like functionals could be derived from many-body quantum mechanics, in a mean-field limit. This text summarizes these findings. It focuses on the simplest, but most physically relevant, case we could treat so far, namely that of the defocusing cubic NLS functional on a 1D interval. The measure obtained in the limit, which lives over H 1/2-ϵ , has been previously shown to be invariant under the NLS flow by Bourgain.

@article{SLSEDP_2014-2015____A5_0,
     author = {Rougerie, Nicolas},
     title = {From bosonic grand-canonical ensembles to nonlinear~Gibbs~measures},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:5},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2014-2015},
     doi = {10.5802/slsedp.71},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/slsedp.71/}
}
Rougerie, Nicolas. From bosonic grand-canonical ensembles to nonlinear Gibbs measures. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 5, 17 p. doi : 10.5802/slsedp.71. http://archive.numdam.org/articles/10.5802/slsedp.71/

[1] Ammari, Z. Systèmes hamiltoniens en théorie quantique des champs : dynamique asymptotique et limite classique. Habilitation à Diriger des Recherches, University of Rennes I, February 2013.

[2] Ammari, Z., and Nier, F. Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. Henri Poincaré 9 (2008), 1503–1574. | MR 2465733 | Zbl 1171.81014

[3] Ammari, Z., and Nier, F. Mean field limit for bosons and propagation of Wigner measures. J. Math. Phys. 50, 4 (2009), 042107. | MR 2513969 | Zbl 1214.81089

[4] Ammari, Z., and Nier, F. Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential. Ann. Sc. Norm. Sup. Pisa. (2015).

[5] Benedikter, N., Porta, M., and Schlein, B. Effective Evolution Equations from Quantum Dynamics, arXiv:1502.02498.

[6] Benguria, R., and Lieb, E. H. Proof of the Stability of Highly Negative Ions in the Absence of the Pauli Principle. Phys. Rev. Lett. 50 (May 1983), 1771–1774.

[7] Berezin, F. A. Convex functions of operators. Mat. Sb. (N.S.) 88(130) (1972), 268–276. | MR 300121 | Zbl 0271.47011

[8] Bogachev, V. I. Gaussian measures. Mathematical Surveys and Monographs No. 62. American Mathematical Soc., 1998. | MR 1642391 | Zbl 0913.60035

[9] Bourgain, J. Periodic nonlinear Schrödinger equation and invariant measures. Comm. Math. Phys. 166, 1 (1994), 1–26. | MR 1309539 | Zbl 0822.35126

[10] Bourgain, J. Invariant measures for the 2d-defocusing nonlinear Schrödinger equation. Comm. Math. Phys. 176 (1996), 421–445. | MR 1374420 | Zbl 0852.35131

[11] Bourgain, J. Invariant measures for the Gross-Piatevskii equation. Journal de Mathématiques Pures et Appliquées 76, 8 (1997), 649–02. | MR 1470880 | Zbl 0906.35095

[12] Burq, N., Thomann, L., and Tzvetkov, N. Long time dynamics for the one dimensional non linear Schrödinger equation. Ann. Inst. Fourier. 63 (2013), 2137–2198. | Numdam | MR 3237443

[13] Burq, N., and Tzvetkov, N. Random data Cauchy theory for supercritical wave equations. I. Local theory. Invent. Math. 173, 3 (2008), 449–475. | MR 2425133 | Zbl 1156.35062

[14] Cacciafesta, F., and de Suzzoni, A.-S. Invariant measure for the Schrödinger equation on the real line, arXiv:1405.5107.

[15] Carlen, E. Trace inequalities and quantum entropy: an introductory course. In Entropy and the Quantum (2010), R. Sims and D. Ueltschi, Eds., vol. 529 of Contemporary Mathematics, American Mathematical Society, pp. 73–140. Arizona School of Analysis with Applications, March 16-20, 2009, University of Arizona. | MR 2681769 | Zbl 1218.81023

[16] Dereziński, J., and Gérard, C. Mathematics of Quantization and Quantum Fields. Cambridge University Press, Cambridge, 2013. | Zbl 1271.81004

[17] Erdős, L., Schlein, B., and Yau, H.-T. Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential. J. Amer. Math. Soc. 22, 4 (2009), 1099–1156. | MR 2525781 | Zbl 1207.82031

[18] Fannes, M., Spohn, H., and Verbeure, A. Equilibrium states for mean field models. J. Math. Phys. 21, 2 (1980), 355–358. | MR 558480 | Zbl 0445.46049

[19] Fröhlich, J., Knowles, A., and Schwarz, S. On the mean-field limit of bosons with Coulomb two-body interaction. Commun. Math. Phys. 288, 3 (2009), 1023–1059. | MR 2504864 | Zbl 1177.82016

[20] Ginibre, J., and Velo, G. The classical field limit of scattering theory for nonrelativistic many-boson systems. I. Commun. Math. Phys. 66, 1 (1979), 37–76. | MR 530915 | Zbl 0443.35067

[21] Glimm, J., and Jaffe, A. Quantum Physics: A Functional Integral Point of View. Springer-Verlag, 1987. | MR 887102 | Zbl 0461.46051

[22] Golse, F. On the Dynamics of Large Particle Systems in the Mean Field Limit. Lecture notes for a course at the NDNS+ Applied Dynamical Systems Summer School “Macroscopic and large scale phenomena”, Universiteit Twente, Enschede (The Netherlands), arXiv:1301.5494. | MR 2050595

[23] Gottlieb, A. D. Examples of bosonic de Finetti states over finite dimensional Hilbert spaces. J. Stat. Phys. 121, 3-4 (2005), 497–509. | MR 2185337 | Zbl 1149.82308

[24] Hepp, K. The classical limit for quantum mechanical correlation functions. Comm. Math. Phys. 35, 4 (1974), 265–277. | MR 332046

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

[26] Lebowitz, J. L., Rose, H. A., and Speer, E. R. Statistical mechanics of the nonlinear Schrödinger equation. J. Statist. Phys. 50, 3-4 (1988), 657–687. | MR 939505 | Zbl 1084.82506

[27] Lewin, M., Nam, P. T., and Rougerie, N. Derivation of nonlinear Gibbs measures from many-body quantum mechanics. J. Éc. polytech. Math. 2 (2015), 65–115, arXiv:1410.0335. | MR 3335056

[28] Lewin, M., Nam, P. T., and Rougerie, N. Derivation of Hartree’s theory for generic mean-field Bose gases. Adv. Math. 254 (March 2014), 570–621, 1303.0981. | MR 3161107

[29] Lieb, E. H. The classical limit of quantum spin systems. Comm. Math. Phys. 31 (1973), 327–340. | MR 349181 | Zbl 1125.82305

[30] Lieb, E. H., and Ruskai, M. B. A fundamental property of quantum-mechanical entropy. Phys. Rev. Lett. 30 (1973), 434–436. | MR 373508

[31] Lieb, E. H., and Ruskai, M. B. Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14 (1973), 1938–1941. With an appendix by B. Simon. | MR 345558

[32] Lieb, E. H., and Seiringer, R. Derivation of the Gross-Pitaevskii equation for rotating Bose gases. Commun. Math. Phys. 264, 2 (2006), 505–537. | MR 2215615 | Zbl 1233.82004

[33] Lieb, E. H., Seiringer, R., Solovej, J. P., and Yngvason, J. The mathematics of the Bose gas and its condensation. Oberwolfach Seminars. Birkhäuser, 2005. | MR 2143817 | Zbl 1104.82012

[34] Lieb, E. H., and Yau, H.-T. The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 1 (1987), 147–174. | MR 904142 | Zbl 0641.35065

[35] Lörinczi, J., Hiroshima, F., and Betz, V. Feynman-Kac-Type Theorems and Gibbs Measures on Path Space: With Applications to Rigorous Quantum Field Theory. de Gruyter Studies in Mathematics. Walter de Gruyter GmbH & Company KG, 2011. | MR 2848339 | Zbl 1236.81003

[36] Nam, P. T., Rougerie, N., and Seiringer, R. Ground states of large Bose systems: The Gross-Pitaevskii limit revisited, arXiv:1503.07061.

[37] Ohya, M., and Petz, D. Quantum entropy and its use. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1993. | MR 1230389 | Zbl 0891.94008

[38] Pickl, P. A simple derivation of mean-field limits for quantum systems. Lett. Math. Phys. 97, 2 (2011), 151–164. | MR 2821235 | Zbl 1242.81150

[39] Raggio, G. A., and Werner, R. F. Quantum statistical mechanics of general mean field systems. Helv. Phys. Acta 62, 8 (1989), 980–1003. | MR 1034151 | Zbl 0938.82501

[40] Rodnianski, I., and Schlein, B. Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291, 1 (2009), 31–61. | MR 2530155 | Zbl 1186.82051

[41] Rougerie, N. De Finetti theorems, mean-field limits and Bose-Einstein condensation. Lecture Notes for a course at LMU, Munich, arXiv:1506.05263, 2014.

[42] Rougerie, N. Théorèmes de de Finetti, limites de champ moyen et condensation de Bose-Einstein. Lecture notes for a cours Peccot, 2014.

[43] Schlein, B. Derivation of effective evolution equations from microscopic quantum dynamics. Lecture Notes for a course at ETH Zurich, arXiv:0807.4307. | MR 3098647 | Zbl 1298.35203

[44] Simon, B. The P(φ) 2 Euclidean (quantum) field theory. Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics. | MR 489552 | Zbl 1175.81146

[45] Simon, B. The classical limit of quantum partition functions. Comm. Math. Phys. 71, 3 (1980), 247–276. | MR 565281 | Zbl 0436.22012

[46] Skorokhod, A. Integration in Hilbert space. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1974. | MR 466482 | Zbl 0307.28010

[47] Spohn, H. Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Modern Phys. 52, 3 (1980), 569–615. | MR 578142 | Zbl 0399.60082

[48] Summers, S. J. A Perspective on Constructive Quantum Field Theory, arXiv:1203.3991.

[49] Thomann, L., and Tzvetkov, N. Gibbs measure for the periodic derivative nonlinear Schrödinger equation. Nonlinearity 23, 11 (2010), 2771. | MR 2727169 | Zbl 1204.35154

[50] Tzvetkov, N. Invariant measures for the defocusing nonlinear Schrödinger equation. Ann. Inst. Fourier (Grenoble) 58, 7 (2008), 2543–2604. | Numdam | MR 2498359 | Zbl 1171.35116

[51] Velo, G., and Wightman, A., Eds. Constructive quantum field theory: The 1973 Ettore Majorana international school of mathematical physics. Lecture notes in physics. Springer-Verlag, 1973. | MR 395513 | Zbl 0325.00006

[52] Wehrl, A. General properties of entropy. Rev. Modern Phys. 50, 2 (1978), 221–260. | MR 496300