The mean-field limit of quantum Bose gases at positive temperature
HTML articles powered by AMS MathViewer
- by Jürg Fröhlich, Antti Knowles, Benjamin Schlein and Vedran Sohinger
- J. Amer. Math. Soc. 35 (2022), 955-1030
- DOI: https://doi.org/10.1090/jams/987
- Published electronically: October 8, 2021
- HTML | PDF | Request permission
Abstract:
We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \leqslant 3$. For $d > 1$ the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the $L^r$-norm with optimal exponent $r$. Moreover, we prove the convergence in the $L^\infty$-norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.References
- Volker Bach, Ionization energies of bosonic Coulomb systems, Lett. Math. Phys. 21 (1991), no. 2, 139–149. MR 1093525, DOI 10.1007/BF00401648
- Tadeusz Balaban, Joel Feldman, Horst Knörrer, and Eugene Trubowitz, A functional integral representation for many boson systems. I. The partition function, Ann. Henri Poincaré 9 (2008), no. 7, 1229–1273. MR 2453249, DOI 10.1007/s00023-008-0387-3
- Tadeusz Balaban, Joel Feldman, Horst Knörrer, and Eugene Trubowitz, A functional integral representation for many boson systems. II. Correlation functions, Ann. Henri Poincaré 9 (2008), no. 7, 1275–1307. MR 2453250, DOI 10.1007/s00023-008-0388-2
- Elliott H. Lieb, The stability of matter: from atoms to stars, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 1–49. MR 1014510, DOI 10.1090/S0273-0979-1990-15831-8
- Jean Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J. 76 (1994), no. 1, 175–202. MR 1301190, DOI 10.1215/S0012-7094-94-07607-2
- J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), no. 1, 1–26. MR 1309539, DOI 10.1007/BF02099299
- Jean Bourgain, Invariant measures for the $2$D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445. MR 1374420, DOI 10.1007/BF02099556
- J. Bourgain, Invariant measures for the Gross-Piatevskii equation, J. Math. Pures Appl. (9) 76 (1997), no. 8, 649–702. MR 1470880, DOI 10.1016/S0021-7824(97)89965-5
- J. Bourgain, Invariant measures for NLS in infinite volume, Comm. Math. Phys. 210 (2000), no. 3, 605–620. MR 1777342, DOI 10.1007/s002200050792
- Jean Bourgain and Aynur Bulut, Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 6, 1289–1325. MR 3226743, DOI 10.4171/JEMS/461
- David C. Brydges and Gordon Slade, Statistical mechanics of the $2$-dimensional focusing nonlinear Schrödinger equation, Comm. Math. Phys. 182 (1996), no. 2, 485–504. MR 1447302, DOI 10.1007/BF02517899
- Nicolas Burq, Laurent Thomann, and Nikolay Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, Ann. Fac. Sci. Toulouse Math. (6) 27 (2018), no. 3, 527–597 (English, with English and French summaries). MR 3869074, DOI 10.5802/afst.1578
- R. H. Cameron, The Ilstow and Feynman integrals, J. Analyse Math. 10 (1962/63), 287–361. MR 150845, DOI 10.1007/BF02790311
- Eric A. Carlen, Jürg Fröhlich, and Joel Lebowitz, Exponential relaxation to equilibrium for a one-dimensional focusing non-linear Schrödinger equation with noise, Comm. Math. Phys. 342 (2016), no. 1, 303–332. MR 3455152, DOI 10.1007/s00220-015-2511-9
- E. Carlen, J. Fröhlich, J. Lebowitz, and W.-M. Wang, Quantitative bounds on the rate of approach to equilibrium for some one–dimensional stochastic nonlinear Schrödinger equations, Nonlinearity 32 (2016), no. 4, 1352–1374.
- T. Chen, J. Fröhlich, and M. Seifert, Renormalization group methods: Landau-Fermi liquid and BCS superconductor, Proceedings of Les Houches summer school, 1994.
- Giuseppe Da Prato and Arnaud Debussche, Strong solutions to the stochastic quantization equations, Ann. Probab. 31 (2003), no. 4, 1900–1916. MR 2016604, DOI 10.1214/aop/1068646370
- J. Dolbeault, P. Felmer, M. Loss, and E. Paturel, Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems, J. Funct. Anal. 238 (2006), no. 1, 193–220. MR 2253013, DOI 10.1016/j.jfa.2005.11.008
- Chenjie Fan, Yumeng Ou, Gigliola Staffilani, and Hong Wang, 2D-defocusing nonlinear Schrödinger equation with random data on irrational tori, Stoch. Partial Differ. Equ. Anal. Comput. 9 (2021), no. 1, 142–206. MR 4218790, DOI 10.1007/s40072-020-00174-7
- M. Fannes, H. Spohn, and A. Verbeure, Equilibrium states for mean field models, J. Math. Phys. 21 (1980), no. 2, 355–358. MR 558480, DOI 10.1063/1.524422
- J Fröhlich, Quantum theory from small to large scales, Proceedings of Les Houches summer school (J. Fröhlich, M. Salmhofer, V. Mastropietro, W. De Roeck, and L.F. Cugliandolo, eds.), 2010.
- Jürg Fröhlich, Antti Knowles, Benjamin Schlein, and Vedran Sohinger, Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d \leqslant 3$, Comm. Math. Phys. 356 (2017), no. 3, 883–980. MR 3719544, DOI 10.1007/s00220-017-2994-7
- Jürg Fröhlich, Antti Knowles, Benjamin Schlein, and Vedran Sohinger, A microscopic derivation of time-dependent correlation functions of the $1D$ cubic nonlinear Schrödinger equation, Adv. Math. 353 (2019), 67–115. MR 3979014, DOI 10.1016/j.aim.2019.06.029
- Jürg Fröhlich, Antti Knowles, Benjamin Schlein, and Vedran Sohinger, A path-integral analysis of interacting Bose gases and loop gases, J. Stat. Phys. 180 (2020), no. 1-6, 810–831. MR 4131015, DOI 10.1007/s10955-020-02543-x
- Giuseppe Genovese, Renato Lucà, and Daniele Valeri, Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation, Selecta Math. (N.S.) 22 (2016), no. 3, 1663–1702. MR 3518561, DOI 10.1007/s00029-016-0225-2
- Giuseppe Genovese, Renato Lucà, and Daniele Valeri, Invariant measures for the periodic derivative nonlinear Schrödinger equation, Math. Ann. 374 (2019), no. 3-4, 1075–1138. MR 3985108, DOI 10.1007/s00208-018-1754-0
- J. Ginibre, Some applications of functional integration in statistical mechanics, Mécanique statistique et théorie quantique des champs, Les Houches, 1971, pp. 327–427.
- James Glimm and Arthur Jaffe, Quantum physics, Springer-Verlag, New York-Berlin, 1981. A functional integral point of view. MR 628000, DOI 10.1007/978-1-4684-0121-9
- Philip Grech and Robert Seiringer, The excitation spectrum for weakly interacting bosons in a trap, Comm. Math. Phys. 322 (2013), no. 2, 559–591. MR 3077925, DOI 10.1007/s00220-013-1736-8
- Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi 3 (2015), e6, 75. MR 3406823, DOI 10.1017/fmp.2015.2
- M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504. MR 3274562, DOI 10.1007/s00222-014-0505-4
- Klaus Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35 (1974), 265–277. MR 332046, DOI 10.1007/BF01646348
- Michael K.-H. Kiessling, The Hartree limit of Born’s ensemble for the ground state of a bosonic atom or ion, J. Math. Phys. 53 (2012), no. 9, 095223, 21. MR 2905805, DOI 10.1063/1.4752475
- Antti Kupiainen, Renormalization group and stochastic PDEs, Ann. Henri Poincaré 17 (2016), no. 3, 497–535. MR 3459120, DOI 10.1007/s00023-015-0408-y
- Joel L. Lebowitz, Harvey A. Rose, and Eugene R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys. 50 (1988), no. 3-4, 657–687. MR 939505, DOI 10.1007/BF01026495
- Mathieu Lewin, Phan Thành Nam, and Nicolas Rougerie, Derivation of Hartree’s theory for generic mean-field Bose systems, Adv. Math. 254 (2014), 570–621. MR 3161107, DOI 10.1016/j.aim.2013.12.010
- Mathieu Lewin, Phan Thành Nam, and Nicolas Rougerie, Derivation of nonlinear Gibbs measures from many-body quantum mechanics, J. Éc. polytech. Math. 2 (2015), 65–115 (English, with English and French summaries). MR 3366672, DOI 10.5802/jep.18
- Mathieu Lewin, Phan Thành Nam, and Nicolas Rougerie, Classical field theory limit of 2D many-body quantum Gibbs states, Preprint, arXiv:1810.08370v2, 2018.
- Mathieu Lewin, Phan Thành Nam, and Nicolas Rougerie, Gibbs measures based on 1d (an)harmonic oscillators as mean-field limits, J. Math. Phys. 59 (2018), no. 4, 041901, 17. MR 3787331, DOI 10.1063/1.5026963
- Mathieu Lewin, Phan Thành Nam, and Nicolas Rougerie, Classical field theory limit of many-body quantum Gibbs states in 2D and 3D, Preprint, arXiv:1810.08370, 2020.
- Mathieu Lewin, Phan Thành Nam, Sylvia Serfaty, and Jan Philip Solovej, Bogoliubov spectrum of interacting Bose gases, Comm. Pure Appl. Math. 68 (2015), no. 3, 413–471. MR 3310520, DOI 10.1002/cpa.21519
- Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, and Jakob Yngvason, The mathematics of the Bose gas and its condensation, Oberwolfach Seminars, vol. 34, Birkhäuser Verlag, Basel, 2005. MR 2143817
- E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, The Stability of Matter: From Atoms to Stars, Springer, 2001, pp. 443–470.
- H. P. McKean and K. L. Vaninsky, Cubic Schrödinger: the petit canonical ensemble in action-angle variables, Comm. Pure Appl. Math. 50 (1997), no. 7, 593–622. MR 1447055, DOI 10.1002/(SICI)1097-0312(199707)50:7<593::AID-CPA1>3.3.CO;2-A
- Moshe Moshe and Jean Zinn-Justin, Quantum field theory in the large $N$ limit: a review, Phys. Rep. 385 (2003), no. 3-6, 69–228. MR 2010168, DOI 10.1016/S0370-1573(03)00263-1
- Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, and Gigliola Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1275–1330. MR 2928851, DOI 10.4171/JEMS/333
- Andrea R. Nahmod, Luc Rey-Bellet, Scott Sheffield, and Gigliola Staffilani, Absolute continuity of Brownian bridges under certain gauge transformations, Math. Res. Lett. 18 (2011), no. 5, 875–887. MR 2875861, DOI 10.4310/MRL.2011.v18.n5.a6
- E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev. 150 (1966), no. 4, 1079.
- Edward Nelson, The free Markoff field, J. Functional Analysis 12 (1973), 211–227. MR 0343816, DOI 10.1016/0022-1236(73)90025-6
- G. Parisi and Yong Shi Wu, Perturbation theory without gauge fixing, Sci. Sinica 24 (1981), no. 4, 483–496. MR 626795
- A. Pizzo, Bose particles in a box I-III, Preprints, arXiv:1511.07022, arXiv:1511.07025, arXiv:1511.07026, 2015.
- G. A. Raggio and R. F. Werner, Quantum statistical mechanics of general mean field systems, Helv. Phys. Acta 62 (1989), no. 8, 980–1003. MR 1034151
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
- Robert Seiringer, The excitation spectrum for weakly interacting bosons, Comm. Math. Phys. 306 (2011), no. 2, 565–578. MR 2824481, DOI 10.1007/s00220-011-1261-6
- Barry Simon, The $P(\phi )_{2}$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1974. MR 0489552
- V. Sohinger, A microscopic derivation of Gibbs measures for nonlinear Schrödinger equations with unbounded interaction potentials, Preprint, arXiv:1904.08137, 2019.
- Jan Philip Solovej, Asymptotics for bosonic atoms, Lett. Math. Phys. 20 (1990), no. 2, 165–172. MR 1065245, DOI 10.1007/BF00398282
- K. Symanzik, Euclidean quantum field theory, Proceedings of the physics school on local quantum theory, Varenna (R. Jost, ed.), 1968, pp. 152–226.
- Laurent Thomann and Nikolay Tzvetkov, Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity 23 (2010), no. 11, 2771–2791. MR 2727169, DOI 10.1088/0951-7715/23/11/003
Bibliographic Information
- Jürg Fröhlich
- Affiliation: Institute for Theoretical Physics, HIT K42.3, ETH Zürich, 8093 Zürich, Switzerland
- ORCID: 0000-0002-4388-5166
- Email: juerg@phys.ethz.ch
- Antti Knowles
- Affiliation: University of Geneva, Section of Mathematics, Rue du Conseil-Général 7-9, 1205 Genève, Switzerland
- MR Author ID: 811939
- ORCID: 0000-0002-8317-3542
- Email: antti.knowles@unige.ch
- Benjamin Schlein
- Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
- MR Author ID: 688765
- Email: benjamin.schlein@math.uzh.ch
- Vedran Sohinger
- Affiliation: Warwick Mathematics Institute, Zeeman Building, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
- MR Author ID: 942774
- ORCID: 0000-0002-8907-131X
- Email: V.Sohinger@warwick.ac.uk
- Received by editor(s): February 10, 2020
- Received by editor(s) in revised form: May 23, 2021
- Published electronically: October 8, 2021
- Additional Notes: The second author gratefully acknowledges the support of the European Research Council through the RandMat grant and of the Swiss National Science Foundation through the NCCR SwissMAP. The third author gratefully acknowledges partial support from the NCCR SwissMAP, from the Swiss National Science Foundation through the Grant “Dynamical and energetic properties of Bose-Einstein condensates” and from the European Research Council through the ERC-AdG CLaQS
- © Copyright 2021 American Mathematical Society
- Journal: J. Amer. Math. Soc. 35 (2022), 955-1030
- MSC (2020): Primary 35Q55, 81V70, 60G60, 82B10, 35Q40
- DOI: https://doi.org/10.1090/jams/987
- MathSciNet review: 4467306