Local existence of solutions to randomized Gross-Pitaevskii hierarchies
HTML articles powered by AMS MathViewer
- by Vedran Sohinger PDF
- Trans. Amer. Math. Soc. 368 (2016), 1759-1835 Request permission
Abstract:
In this paper, we study the local-in-time existence of solutions to randomized forms of the Gross-Pitaevskii hierarchy on periodic domains. In particular, we study the independently randomized Gross-Pitaevskii hierarchy and the dependently randomized Gross-Pitaevskii hierarchy, which were first introduced in the author’s joint work with Staffilani (2013). For these hierarchies, we construct local-in-time low-regularity solutions in spaces which contain a random component. The constructed density matrices will solve the full randomized hierarchies, thus extending the results from the author and Staffilani’s joint work, where solutions solving arbitrarily long subhierarchies were given.
Our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of T. Chen and Pavlović (2013). The presence of randomization in the problem adds additional difficulties, most notably to estimating the Duhamel expansions that are crucial in the truncation argument. These difficulties are overcome by a detailed analysis of the Duhamel expansions. In the independently randomized case, we need to keep track of which randomization parameters appear in the Duhamel terms, whereas in the dependently randomized case, we express the Duhamel terms directly in terms of the initial data. In both cases, we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data.
References
- Riccardo Adami, Claude Bardos, François Golse, and Alessandro Teta, Towards a rigorous derivation of the cubic NLSE in dimension one, Asymptot. Anal. 40 (2004), no. 2, 93–108. MR 2104130
- Riccardo Adami, François Golse, and Alessandro Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 (2007), no. 6, 1193–1220. MR 2331036, DOI 10.1007/s10955-006-9271-z
- Stefan Adams, Andrea Collevecchio, and Wolfgang König, A variational formula for the free energy of an interacting many-particle system, Ann. Probab. 39 (2011), no. 2, 683–728. MR 2789510, DOI 10.1214/10-AOP565
- M. Aizenman, E. H. Lieb, R. Seiringer, J. P. Solovej, J. Yngvason, Bose-Einstein quantum phase transition in an optical lattice model, Phys. Rev. A 70 (2004), 023612.
- M. Aizenman, E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason, Bose-Einstein condensation as a quantum phase transition in an optical lattice, Mathematical physics of quantum mechanics, Lecture Notes in Phys., vol. 690, Springer, Berlin, 2006, pp. 199–215. MR 2234912, DOI 10.1007/3-540-34273-7_{1}6
- Zied Ammari and Francis Nier, Mean field limit for bosons and infinite dimensional phase-space analysis, Ann. Henri Poincaré 9 (2008), no. 8, 1503–1574. MR 2465733, DOI 10.1007/s00023-008-0393-5
- Z. Ammari and F. Nier, Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states, J. Math. Pures Appl. (9) 95 (2011), no. 6, 585–626 (English, with English and French summaries). MR 2802894, DOI 10.1016/j.matpur.2010.12.004
- Ioannis Anapolitanos, Rate of convergence towards the Hartree-von Neumann limit in the mean-field regime, Lett. Math. Phys. 98 (2011), no. 1, 1–31. MR 2836427, DOI 10.1007/s11005-011-0477-x
- M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observations of Bose-Einstein condensation in a dilute atomic vapor, Science 269 (1995), 198–201.
- Antoine Ayache and Nikolay Tzvetkov, $L^p$ properties for Gaussian random series, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4425–4439. MR 2395179, DOI 10.1090/S0002-9947-08-04456-5
- J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic theory of superconductivity, Phys. Rev. (2) 106 (1957), 162–164. MR 106739
- J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. (2) 108 (1957), 1175–1204. MR 95694
- Claude Bardos, François Golse, and Norbert J. Mauser, Weak coupling limit of the $N$-particle Schrödinger equation, Methods Appl. Anal. 7 (2000), no. 2, 275–293. Cathleen Morawetz: a great mathematician. MR 1869286, DOI 10.4310/MAA.2000.v7.n2.a2
- William Beckner, Multilinear embedding estimates for the fractional Laplacian, Math. Res. Lett. 19 (2012), no. 1, 175–189. MR 2923184, DOI 10.4310/MRL.2012.v19.n1.a14
- W. Beckner, Convolution estimates and the Gross-Pitaevskii hierarchy, preprint (2011), http://arxiv.org/abs/1111.3857.
- Gérard Ben Arous, Kay Kirkpatrick, and Benjamin Schlein, A central limit theorem in many-body quantum dynamics, Comm. Math. Phys. 321 (2013), no. 2, 371–417. MR 3063915, DOI 10.1007/s00220-013-1722-1
- Niels Benedikter, Marcello Porta, and Benjamin Schlein, Mean-field evolution of fermionic systems, Comm. Math. Phys. 331 (2014), no. 3, 1087–1131. MR 3248060, DOI 10.1007/s00220-014-2031-z
- Niels Benedikter, Marcello Porta, and Benjamin Schlein, Mean-field dynamics of fermions with relativistic dispersion, J. Math. Phys. 55 (2014), no. 2, 021901, 10. MR 3202863, DOI 10.1063/1.4863349
- N. Benedikter, G. de Oliveira, and B. Schlein, Quantitative derivation of the Gross-Pitaevskii equation, preprint (2012), http://arxiv.org/abs/1208.0373.
- A. Benyi, T. Oh, and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, preprint (2014), http://arxiv.org/abs/1405.7326, to appear in Excursions in Harmonic Analysis.
- A. Benyi, T. Oh, and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb {R}^d$, $d \geq 3$, Trans. Amer. Math. Soc. Ser. B 2 (2015), 1–50, DOI 10.1090/btran/6.
- S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik 26 (1924), 178.
- J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), no. 1, 1–26. MR 1309539
- 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
- Jean Bourgain, Invariant measures for the $2$D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445. MR 1374420
- 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, Refinements of Strichartz’ inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices 5 (1998), 253–283. MR 1616917, DOI 10.1155/S1073792898000191
- J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), no. 1, 145–171. MR 1626257, DOI 10.1090/S0894-0347-99-00283-0
- Jean Bourgain and Aynur Bulut, Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball, C. R. Math. Acad. Sci. Paris 350 (2012), no. 11-12, 571–575 (English, with English and French summaries). MR 2956145, DOI 10.1016/j.crma.2012.05.006
- 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
- Jean Bourgain and Aynur Bulut, Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball, J. Funct. Anal. 266 (2014), no. 4, 2319–2340. MR 3150162, DOI 10.1016/j.jfa.2013.06.002
- N. Burq, L. Thomann, and N. Tzvetkov, Global infinite energy solutions for the cubic wave equation, preprint (2012), http://arxiv.org/pdf/1210.2086.pdf.
- Nicolas Burq and Nikolay Tzvetkov, Invariant measure for a three dimensional nonlinear wave equation, Int. Math. Res. Not. IMRN 22 (2007), Art. ID rnm108, 26. MR 2376217, DOI 10.1093/imrn/rnm108
- Nicolas Burq and Nikolay Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475. MR 2425133, DOI 10.1007/s00222-008-0124-z
- Nicolas Burq and Nikolay Tzvetkov, Random data Cauchy theory for supercritical wave equations. II. A global existence result, Invent. Math. 173 (2008), no. 3, 477–496. MR 2425134, DOI 10.1007/s00222-008-0123-0
- Nicolas Burq and Nikolay Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 1, 1–30. MR 3141727, DOI 10.4171/JEMS/426
- F. Cacciafesta and A.-S. de Suzzoni, Invariant measure for the Schrödinger equation on the real line, preprint (2014), http://arxiv.org/abs/1405.5107.
- Sourav Chatterjee and Persi Diaconis, Fluctuations of the Bose-Einstein condensate, J. Phys. A 47 (2014), no. 8, 085201, 23. MR 3165088, DOI 10.1088/1751-8113/47/8/085201
- Li Chen, Ji Oon Lee, and Benjamin Schlein, Rate of convergence towards Hartree dynamics, J. Stat. Phys. 144 (2011), no. 4, 872–903. MR 2826623, DOI 10.1007/s10955-011-0283-y
- Thomas Chen, Christian Hainzl, Nataša Pavlović, and Robert Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, preprint (2013), http://arxiv.org/abs/1307.3168.
- Thomas Chen, Christian Hainzl, Nataša Pavlović, and Robert Seiringer, On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti, Lett. Math. Phys. 104 (2014), no. 7, 871–891. MR 3210237, DOI 10.1007/s11005-014-0693-2
- Thomas Chen and Nataša Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discrete Contin. Dyn. Syst. 27 (2010), no. 2, 715–739. MR 2600687, DOI 10.3934/dcds.2010.27.715
- Thomas Chen and Nataša Pavlović, Recent results on the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Math. Model. Nat. Phenom. 5 (2010), no. 4, 54–72. MR 2662450, DOI 10.1051/mmnp/20105403
- Thomas Chen and Nataša Pavlović, The quintic NLS as the mean field limit of a boson gas with three-body interactions, J. Funct. Anal. 260 (2011), no. 4, 959–997. MR 2747009, DOI 10.1016/j.jfa.2010.11.003
- Thomas Chen and Nataša Pavlović, A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies, Proc. Amer. Math. Soc. 141 (2013), no. 1, 279–293. MR 2988730, DOI 10.1090/S0002-9939-2012-11308-5
- Thomas Chen and Nataša Pavlović, Higher order energy conservation and global well-posedness of solutions for Gross-Pitaevskii hierarchies, Comm. Partial Differential Equations 39 (2014), no. 9, 1597–1634. MR 3246036, DOI 10.1080/03605302.2013.816858
- Thomas Chen and Nataša Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d=3$ based on spacetime norms, Ann. Henri Poincaré 15 (2014), no. 3, 543–588. MR 3165917, DOI 10.1007/s00023-013-0248-6
- Thomas Chen, Nataša Pavlović, and Nikolaos Tzirakis, Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 5, 1271–1290. MR 2683760, DOI 10.1016/j.anihpc.2010.06.003
- T. Chen, N. Pavlović, and N. Tzirakis, Multilinear Morawetz identities for the Gross-Pitaevskii hierarchy, Recent advances in harmonic analysis and partial differential equations, Contemp. Math., vol. 581, Amer. Math. Soc., Providence, RI, 2012, pp. 39–62. MR 3013052, DOI 10.1090/conm/581/11491
- T. Chen and K. Taliaferro, Positive semidefiniteness and global well-posedness of solutions to the Gross-Pitaevskii hierarchy, preprint (2013), http://arxiv.org/abs/1305.1404.
- Xuwen Chen, Second order corrections to mean field evolution for weakly interacting bosons in the case of three-body interactions, Arch. Ration. Mech. Anal. 203 (2012), no. 2, 455–497. MR 2885567, DOI 10.1007/s00205-011-0453-8
- Xuwen Chen, Second order corrections to mean field evolution for weakly interacting bosons in the case of three-body interactions, Arch. Ration. Mech. Anal. 203 (2012), no. 2, 455–497. MR 2885567, DOI 10.1007/s00205-011-0453-8
- Xuwen Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps, J. Math. Pures Appl. (9) 98 (2012), no. 4, 450–478 (English, with English and French summaries). MR 2968164, DOI 10.1016/j.matpur.2012.02.003
- Xuwen Chen, On the rigorous derivation of the 3D cubic nonlinear Schrödinger equation with a quadratic trap, Arch. Ration. Mech. Anal. 210 (2013), no. 2, 365–408. MR 3101788, DOI 10.1007/s00205-013-0645-5
- Xuwen Chen and Justin Holmer, On the rigorous derivation of the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics, Arch. Ration. Mech. Anal. 210 (2013), no. 3, 909–954. MR 3116008, DOI 10.1007/s00205-013-0667-z
- X. Chen, J. Holmer, On the Klainerman-Machedon conjecture of the quantum BBGKY hierarchy with self-interaction, preprint (2013), http://arxiv.org/abs/1303.5385.
- X. Chen, J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation, preprint (2013), http://arxiv.org/abs/1308.3895.
- Zeqian Chen and Chuangye Liu, On the Cauchy problem for Gross-Pitaevskii hierarchies, J. Math. Phys. 52 (2011), no. 3, 032103, 13. MR 2814691, DOI 10.1063/1.3567168
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal. 33 (2001), no. 3, 649–669. MR 1871414, DOI 10.1137/S0036141001384387
- James Colliander and Tadahiro Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\Bbb T)$, Duke Math. J. 161 (2012), no. 3, 367–414. MR 2881226, DOI 10.1215/00127094-1507400
- L. N. Cooper, Bound electron pairs in a degenerate Fermi gas, Phys. Rev. 104 (1956), 1189–1190
- M. Cramer and J. Eisert, A quantum central limit theorem for non-equilibrium systems: exact relaxation of correlated states, New. J. Phys. 12 (2009), 055020.
- C. D. Cushen and R. L. Hudson, A quantum-mechanical central limit theorem, J. Appl. Probability 8 (1971), 454–469. MR 289082, DOI 10.2307/3212170
- K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75 (1995), no. 22, 3969–3973.
- B. de Finetti, Funzione caratteristica di un fenomeno aleatorio, Atti R. Accad. Naz. Lincei, Ser. 6, Mem. Cl. Sci. Fis. Mat. Natur. (1931).
- Bruno de Finetti, La prévision : ses lois logiques, ses sources subjectives, Ann. Inst. H. Poincaré 7 (1937), no. 1, 1–68 (French). MR 1508036
- Chao Deng and Shangbin Cui, Random-data Cauchy problem for the Navier-Stokes equations on $\Bbb T^3$, J. Differential Equations 251 (2011), no. 4-5, 902–917. MR 2812576, DOI 10.1016/j.jde.2011.05.002
- Chao Deng and Shangbin Cui, Random-data Cauchy problem for the periodic Navier-Stokes equations with initial data in negative-order Sobolev spaces, preprint (2011), http://arxiv.org/abs/1103.6170.
- Yu Deng, Two-dimensional nonlinear Schrödinger equation with random radial data, Anal. PDE 5 (2012), no. 5, 913–960. MR 3022846, DOI 10.2140/apde.2012.5.913
- Y. Deng, Invariance of the Gibbs measure for the Benjamin-Ono equation, preprint (2012), http://arxiv.org/abs/1210.1542.
- Y. Deng, N. Tzvetkov, N. Visciglia, Invariant measures and long time behaviour for the Benjamin-Ono equation III, preprint (2014), http://arxiv.org/abs/1405.4954.
- P. Diaconis and D. Freedman, Finite exchangeable sequences, Ann. Probab. 8 (1980), no. 4, 745–764. MR 577313
- J. L. Doob, Stochastic processes with an integral-valued parameter, Trans. Amer. Math. Soc. 44 (1938), no. 1, 87–150. MR 1501964, DOI 10.1090/S0002-9947-1938-1501964-2
- E. B. Dynkin, Classes of equivalent random quantities, Uspehi Matem. Nauk (N.S.) 8 (1953), no. 2(54), 125–130 (Russian). MR 0055601
- A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften 1: 3. (1925).
- Alexander Elgart, László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal. 179 (2006), no. 2, 265–283. MR 2209131, DOI 10.1007/s00205-005-0388-z
- Alexander Elgart and Benjamin Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math. 60 (2007), no. 4, 500–545. MR 2290709, DOI 10.1002/cpa.20134
- László Erdős and Benjamin Schlein, Quantum dynamics with mean field interactions: a new approach, J. Stat. Phys. 134 (2009), no. 5-6, 859–870. MR 2518972, DOI 10.1007/s10955-008-9570-7
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Comm. Pure Appl. Math. 59 (2006), no. 12, 1659–1741. MR 2257859, DOI 10.1002/cpa.20123
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), no. 3, 515–614. MR 2276262, DOI 10.1007/s00222-006-0022-1
- L. Erdős, B. Schlein, and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation, Phys. Rev. Lett. 98 (2007), no. 4, 040404.
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc. 22 (2009), no. 4, 1099–1156. MR 2525781, DOI 10.1090/S0894-0347-09-00635-3
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math. (2) 172 (2010), no. 1, 291–370. MR 2680421, DOI 10.4007/annals.2010.172.291
- László Erdős and Horng-Tzer Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169–1205. MR 1926667, DOI 10.4310/ATMP.2001.v5.n6.a6
- P. Federbush, A partially alternate derivation of a result of Nelson, J. Math. Phys. 10 (1969), 50-52.
- Karl-Heinz Fichtner, On the position distribution of the ideal Bose gas, Math. Nachr. 151 (1991), 59–67. MR 1121197, DOI 10.1002/mana.19911510105
- Jürg Fröhlich, Sandro Graffi, and Simon Schwarz, Mean-field- and classical limit of many-body Schrödinger dynamics for bosons, Comm. Math. Phys. 271 (2007), no. 3, 681–697. MR 2291792, DOI 10.1007/s00220-007-0207-5
- J. Fröhlich, A. Knowles, and A. Pizzo, Atomism and quantization, J. Phys. A 40 (2007), no. 12, 3033–3045. MR 2313859, DOI 10.1088/1751-8113/40/12/S09
- Jürg Fröhlich, Antti Knowles, and Simon Schwarz, On the mean-field limit of bosons with Coulomb two-body interaction, Comm. Math. Phys. 288 (2009), no. 3, 1023–1059. MR 2504864, DOI 10.1007/s00220-009-0754-z
- Jürg Fröhlich and Enno Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, Séminaire: Équations aux Dérivées Partielles. 2003–2004, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2004, pp. Exp. No. XIX, 26. MR 2117050
- Jürg Fröhlich, Tai-Peng Tsai, and Horng-Tzer Yau, On a classical limit of quantum theory and the non-linear Hartree equation, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 189–207. MR 1805891
- Jürg Fröhlich, Tai-Peng Tsai, and Horng-Tzer Yau, On a classical limit of quantum theory and the non-linear Hartree equation, Geom. Funct. Anal. Special Volume (2000), 57–78. GAFA 2000 (Tel Aviv, 1999). MR 1826249, DOI 10.1007/978-3-0346-0422-2_{3}
- Jürg Fröhlich, Tai-Peng Tsai, and Horng-Tzer Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys. 225 (2002), no. 2, 223–274. MR 1889225, DOI 10.1007/s002200100579
- J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems. I, Comm. Math. Phys. 66 (1979), no. 1, 37–76. MR 530915
- J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems. II, Comm. Math. Phys. 68 (1979), no. 1, 45–68. MR 539736
- James Glimm, Boson fields with non-linear self-interaction in two dimensions, Comm. Math. Phys. 8 (1968), 12–25.
- D. Goderis, A. Verbeure, and P. Vets, About the mathematical theory of quantum fluctuations, Mathematical methods in statistical mechanics (Leuven, 1988) Leuven Notes Math. Theoret. Phys. Ser. A Math. Phys., vol. 1, Leuven Univ. Press, Leuven, 1989, pp. 31–47. MR 1060771
- Philip Gressman, Vedran Sohinger, and Gigliola Staffilani, On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy, J. Funct. Anal. 266 (2014), no. 7, 4705–4764. MR 3170216, DOI 10.1016/j.jfa.2014.02.006
- M. Grillakis and M. Machedon, Pair excitations and the mean field approximation of interacting bosons, I, Comm. Math. Phys. 324 (2013), no. 2, 601–636. MR 3117522, DOI 10.1007/s00220-013-1818-7
- Manoussos G. Grillakis, Matei Machedon, and Dionisios Margetis, Second-order corrections to mean field evolution of weakly interacting bosons. I, Comm. Math. Phys. 294 (2010), no. 1, 273–301. MR 2575484, DOI 10.1007/s00220-009-0933-y
- M. Grillakis, M. Machedon, and D. Margetis, Second-order corrections to mean field evolution of weakly interacting bosons. II, Adv. Math. 228 (2011), no. 3, 1788–1815. MR 2824569, DOI 10.1016/j.aim.2011.06.028
- E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento (10) 20 (1961), 454–477 (English, with Italian summary). MR 128907
- Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688
- Leonard Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford Dirichlet form, Duke Math. J. 42 (1975), no. 3, 383–396. MR 372613
- M. Hayashi, Quantum estimation and the quantum central limit theorem, Science and Technology 227 (2006), 95.
- Klaus Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35 (1974), 265–277. MR 332046
- K. Hepp and E. H. Lieb, Phase transitions in reservoir-driven open systems with applications to lasers and superconductors, Helv. Phys. Acta 46 (1973), 573–603.
- Edwin Hewitt and Leonard J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955), 470–501. MR 76206, DOI 10.1090/S0002-9947-1955-0076206-8
- Y. Hong, K. Taliaferro, and Z. Xie, Unconditional uniqueness of the cubic Gross-Pitaevskii hierarchy with low regularity, preprint (2014), http://arxiv.org/abs/1402.5347.
- R. L. Hudson and G. R. Moody, Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), no. 4, 343–351. MR 397421, DOI 10.1007/BF00534784
- V. Jakšić, Y. Pautrat, and C.-A. Pillet, A quantum central limit theorem for sums of independent identically distributed random variables, J. Math. Phys. 51 (2010), no. 1, 015208, 8. MR 2605841, DOI 10.1063/1.3285287
- Shizuo Kakutani, Notes on infinite product measure spaces. I, Proc. Imp. Acad. Tokyo 19 (1943), 148–151. MR 14403
- Kay Kirkpatrick, Benjamin Schlein, and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math. 133 (2011), no. 1, 91–130. MR 2752936, DOI 10.1353/ajm.2011.0004
- S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268. MR 1231427, DOI 10.1002/cpa.3160460902
- Sergiu Klainerman and Matei Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys. 279 (2008), no. 1, 169–185. MR 2377632, DOI 10.1007/s00220-008-0426-4
- Antti Knowles and Peter Pickl, Mean-field dynamics: singular potentials and rate of convergence, Comm. Math. Phys. 298 (2010), no. 1, 101–138. MR 2657816, DOI 10.1007/s00220-010-1010-2
- A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1977 (German). Reprint of the 1933 original. MR 0494348
- M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335, 3, February 2012.
- Greg Kuperberg, A tracial quantum central limit theorem, Trans. Amer. Math. Soc. 357 (2005), no. 2, 459–471. MR 2095618, DOI 10.1090/S0002-9947-03-03449-4
- 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
- Ji Oon Lee, Rate of convergence towards semi-relativistic Hartree dynamics, Ann. Henri Poincaré 14 (2013), no. 2, 313–346. MR 3028041, DOI 10.1007/s00023-012-0188-6
- 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
- M. Lewin, P. T. Nam, and N. Rougerie, The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, preprint (2014), http://arxiv.org/ abs/1405.3220.
- Mathieu Lewin and Julien Sabin, The Hartree equation for infinitely many particles I. Well-posedness theory, Comm. Math. Phys. 334 (2015), no. 1, 117–170. MR 3304272, DOI 10.1007/s00220-014-2098-6
- Mathieu Lewin and Julien Sabin, The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D, Anal. PDE 7 (2014), no. 6, 1339–1363. MR 3270166, DOI 10.2140/apde.2014.7.1339
- E. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett. 88 (2002), 170409-1-4.
- 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. Lieb, R. Seiringer, and J. P. Yngvason, Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A 61 (2000), 043602.
- Elliott H. Lieb, Robert Seiringer, and Jakob Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys. 224 (2001), no. 1, 17–31. Dedicated to Joel L. Lebowitz. MR 1868990, DOI 10.1007/s002200100533
- Jonas Lührmann, Mean-field quantum dynamics with magnetic fields, J. Math. Phys. 53 (2012), no. 2, 022105, 19. MR 2920451, DOI 10.1063/1.3687024
- Jonas Lührmann and Dana Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on $\Bbb {R}^3$, Comm. Partial Differential Equations 39 (2014), no. 12, 2262–2283. MR 3259556, DOI 10.1080/03605302.2014.933239
- Józef Marcinkiewicz, Collected papers, Państwowe Wydawnictwo Naukowe, Warsaw, 1964. Edited by Antoni Zygmund. With the collaboration of Stanislaw Lojasiewicz, Julian Musielak, Kazimierz Urbanik and Antoni Wiweger. Instytut Matematyczny Polskiej Akademii Nauk. MR 0168434
- H. P. McKean and K. L. Vaninsky, Statistical mechanics of nonlinear wave equations, Trends and perspectives in applied mathematics, Appl. Math. Sci., vol. 100, Springer, New York, 1994, pp. 239–264. MR 1277197, DOI 10.1007/978-1-4612-0859-4_{8}
- H. P. McKean and K. L. Vaninsky, Action-angle variables for the cubic Schrödinger equation, Comm. Pure Appl. Math. 50 (1997), no. 6, 489–562. MR 1441912, DOI 10.1002/(SICI)1097-0312(199706)50:6<489::AID-CPA1>3.0.CO;2-4
- 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
- Alessandro Michelangeli and Benjamin Schlein, Dynamical collapse of boson stars, Comm. Math. Phys. 311 (2012), no. 3, 645–687. MR 2909759, DOI 10.1007/s00220-011-1341-7
- 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
- Andrea R. Nahmod, Nataša Pavlović, and Gigliola Staffilani, Almost sure existence of global weak solutions for supercritical Navier-Stokes equations, SIAM J. Math. Anal. 45 (2013), no. 6, 3431–3452. MR 3131480, DOI 10.1137/120882184
- A. Nahmod and G. Staffilani, Randomization in nonlinear PDE and the supercritical periodic quintic NLS in $3D$, preprint (2013), http://arxiv.org/abs/1308.1169.
- Edward Nelson, The free Markoff field, J. Functional Analysis 12 (1973), 211–227. MR 0343816, DOI 10.1016/0022-1236(73)90025-6
- Tadahiro Oh, Invariant Gibbs measures and a.s. global well posedness for coupled KdV systems, Differential Integral Equations 22 (2009), no. 7-8, 637–668. MR 2532115
- Tadahiro Oh, Invariance of the white noise for KdV, Comm. Math. Phys. 292 (2009), no. 1, 217–236. MR 2540076, DOI 10.1007/s00220-009-0856-7
- Tadahiro Oh, Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system, SIAM J. Math. Anal. 41 (2009/10), no. 6, 2207–2225. MR 2579711, DOI 10.1137/080738180
- Tadahiro Oh and Catherine Sulem, On the one-dimensional cubic nonlinear Schrödinger equation below $L^2$, Kyoto J. Math. 52 (2012), no. 1, 99–115. MR 2892769, DOI 10.1215/21562261-1503772
- R. E. A. C. Paley and A. Zygmund, On some series of functions 1, Proc. Camb. Phil. Soc. 26 (1930), 337–357.
- R. E. A. C. Paley and A. Zygmund, On some series of functions 2, Proc. Camb. Phil. Soc. 26 (1930), 458–474.
- R. E. A. C. Paley and A. Zygmund, On some series of functions 3, Proc. Camb. Phil. Soc. 28 (1932), 190–205.
- Peter Pickl, Derivation of the time dependent Gross-Pitaevskii equation with external fields, Rev. Math. Phys. 27 (2015), no. 1, 1550003, 45. MR 3317556, DOI 10.1142/S0129055X15500038
- Peter Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys. 97 (2011), no. 2, 151–164. MR 2821235, DOI 10.1007/s11005-011-0470-4
- L. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP 13 (1961), 451–454.
- N. Prokof’ev and B. Svistunov, Bold diagrammatic Monte Carlo technique: When the sign problem is welcome, Phys. Rev. Lett. 99 (2007), 250201.
- N. Prokof’ev and B. Svistunov, Bold diagrammatic Monte Carlo: A generic sign-problem tolerant technique for polaron models and possibly interacting many-body problems, Phys. Rev. B 77 (2008), 125101.
- Hans Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), no. 1-2, 112–138 (German). MR 1512104, DOI 10.1007/BF01458040
- M. Rafler, Gaussian Loop- and Pólya processes: A point process approach, Ph.D. thesis, Univ. Potsdam.
- G. Richards, Invariance of the Gibbs measure for the periodic quartic gKdV, preprint (2012), http://arxiv.org/abs/1209.4337.
- Igor Rodnianski and Benjamin Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys. 291 (2009), no. 1, 31–61. MR 2530155, DOI 10.1007/s00220-009-0867-4
- Benjamin Schlein, Derivation of effective evolution equations from microscopic quantum dynamics, Evolution equations, Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 511–572. MR 3098647
- V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbb {T}^3$ from the dynamics of many-body quantum systems, preprint (2014), http://arxiv.org/abs/1405.3003.
- V. Sohinger and G. Staffilani, Randomization and the Gross-Pitaevskii hierarchy, preprint (2013), http://arxiv.org/abs/1308.3714.
- Herbert Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52 (1980), no. 3, 569–615. MR 578142, DOI 10.1103/RevModPhys.52.569
- Erling Størmer, Symmetric states of infinite tensor products of $C^{\ast }$-algebras, J. Functional Analysis 3 (1969), 48–68. MR 0241992, DOI 10.1016/0022-1236(69)90050-0
- Anne-Sophie de Suzzoni, Invariant measure for the cubic wave equation on the unit ball of $\Bbb R^3$, Dyn. Partial Differ. Equ. 8 (2011), no. 2, 127–147. MR 2857361, DOI 10.4310/DPDE.2011.v8.n2.a4
- A.-S. de Suzzoni, On the use of normal forms in the propagation of random waves, preprint (2013), http://arxiv.org/abs/1307.0619.
- A.-S. de Suzzoni, Invariant measure for the Klein-Gordon equation in a non periodic setting, preprint (2014), http://arxiv.org/abs/1403.2274.
- Anne-Sophie de Suzzoni and Nikolay Tzvetkov, On the propagation of weakly nonlinear random dispersive waves, Arch. Ration. Mech. Anal. 212 (2014), no. 3, 849–874. MR 3187679, DOI 10.1007/s00205-014-0728-y
- Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
- Laurent Thomann, Random data Cauchy problem for supercritical Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 6, 2385–2402 (English, with English and French summaries). MR 2569900, DOI 10.1016/j.anihpc.2009.06.001
- 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
- Nikolay Tzvetkov, Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ. 3 (2006), no. 2, 111–160. MR 2227040, DOI 10.4310/DPDE.2006.v3.n2.a2
- Nikolay Tzvetkov, Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2543–2604 (English, with English and French summaries). MR 2498359
- N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Related Fields 146 (2010), no. 3-4, 481–514. MR 2574736, DOI 10.1007/s00440-008-0197-z
- K. Van Houcke, E. Kozik, N. Prokof’ev, and B. Svistunov, in Computer Simulation Studies in Condensed Matter Physics XXI (eds. D. P. Landau, S. P. Lewis, H. B. Schuttler), Springer, 2008.
- K. Van Houcke, F. Werner, E. Kozik, N. Prokof’ev, B. Svistunov, M. J. H. Ku, A. T. Summer, L. W. Cheuk, A. Schirotzek, and M. W. Zwierlein, Feynman diagrams versus Fermi-gas Feynman emulator, Nature Physics 8, May 2012.
- Thomas H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and a preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254, DOI 10.1090/ulect/029
- Zhihui Xie, Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity in $d=1,2$, Differential Integral Equations 28 (2015), no. 5-6, 455–504. MR 3328130
- S. Xu, Invariant Gibbs measure for 3D NLW in infinite volume, preprint (2014), http:// arxiv.org/abs/1405.3856.
- Ting Zhang and Daoyuan Fang, Random data Cauchy theory for the incompressible three dimensional Navier-Stokes equations, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2827–2837. MR 2801624, DOI 10.1090/S0002-9939-2011-10762-7
- P. E. Zhidkov, An invariant measure for the nonlinear Schrödinger equation, Dokl. Akad. Nauk SSSR 317 (1991), no. 3, 543–546 (Russian); English transl., Soviet Math. Dokl. 43 (1991), no. 2, 431–434. MR 1121337
- Peter E. Zhidkov, Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory, Lecture Notes in Mathematics, vol. 1756, Springer-Verlag, Berlin, 2001. MR 1831831
Additional Information
- Vedran Sohinger
- Affiliation: Department of Mathematics, David Rittenhouse Lab, University of Pennsylvania, Office 3N4B, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
- Address at time of publication: Eidgenössische Technische Hochschule Zürich, Departement Mathematik, Rämistrasse 101, 8092 Zürich, Switzerland
- Email: vedranso@math.upenn.edu, vedran.sohinger@math.ethz.ch
- Received by editor(s): January 4, 2014
- Published electronically: June 15, 2015
- Additional Notes: The author was supported by a Simons Postdoctoral Fellowship.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 1759-1835
- MSC (2010): Primary 35Q55, 70E55
- DOI: https://doi.org/10.1090/tran/6479
- MathSciNet review: 3449225