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On the dynamics of finite temperature trapped Bose gases. (English) Zbl 1386.82035

The paper deals with the dynamics of the Bose-Einstein condensate which is described by two equations which are respectively a nonlinear Schrödinger equation and a quantum Boltzmann equation. The main contribution of the paper is based on the fact that one cannot merely extend classical techniques to solve these equations. The authors simplify the problem in a model referred to as toy model and they focus on it. The model is centered on the quantum Boltzmann equation of which an approximate solution is derived.

MSC:

82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
82C22 Interacting particle systems in time-dependent statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35Q20 Boltzmann equations
35B44 Blow-up in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
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References:

[1] Allemand, T., Derivation of a two-fluids model for a Bose gas from a quantum kinetic system, Kinet. Relat. Models, 2, 2, 379-402 (2009) · Zbl 1352.82015
[2] R. Alonso, V. Bagland, Y. Cheng, B. Lods, One dimensional dissipative Boltzmann equation: measure solutions, cooling rate and self-similar profile, submitted for publication, 2016.; R. Alonso, V. Bagland, Y. Cheng, B. Lods, One dimensional dissipative Boltzmann equation: measure solutions, cooling rate and self-similar profile, submitted for publication, 2016. · Zbl 1383.82024
[3] Alonso, R.; Gamba, I. M.; Tran, M.-B., The Cauchy problem for the quantum Boltzmann equation for bosons at very low temperature (2016), arXiv preprint
[4] Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A., Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269, 5221, 198-201 (1995)
[5] Andrews, M. R.; Townsend, C. G.; Miesner, H.-J.; Durfee, D. S.; Kurn, D. M.; Ketterle, W., Observation of interference between two Bose condensates, Science, 275, 5300, 637-641 (1997)
[6] Anglin, J. R.; Ketterle, W., Bose-Einstein condensation of atomic gases, Nature, 416, 6877, 211-218 (2002)
[7] Arkeryd, L., On the Boltzmann equation. I. Existence, Arch. Ration. Mech. Anal., 45, 1-16 (1972) · Zbl 0245.76059
[8] Arkeryd, L.; Nouri, A., Bose condensates in interaction with excitations: a kinetic model, Comm. Math. Phys., 310, 3, 765-788 (2012) · Zbl 1276.82016
[9] Arkeryd, L.; Nouri, A., A Milne problem from a Bose condensate with excitations, Kinet. Relat. Models, 6, 4, 671-686 (2013) · Zbl 1285.82031
[10] Arkeryd, L.; Nouri, A., Bose condensates in interaction with excitations: a two-component space-dependent model close to equilibrium, J. Stat. Phys., 160, 1, 209-238 (2015) · Zbl 1323.82025
[11] Bach, V.; Breteaux, S.; Chen, T.; Fröhlich, J.; Sigal, I. M., The time-dependent Hartree-Fock-Bogoliubov equations for bosons (2016), arXiv preprint
[12] Barbara Goss, L., Cornell, Ketterle, and Wieman share Nobel Prize for Bose-Einstein Condensates, Search and Discovery. Physics Today (2001), online
[13] Ben Arous, G.; Kirkpatrick, K.; Schlein, B., A central limit theorem in many-body quantum dynamics, Comm. Math. Phys., 321, 2, 371-417 (2013) · Zbl 1280.81157
[14] Bennemann, K. H.; Bennemann, J. B., The Physics of Liquid and Solid Helium, Interscience Monographs and Texts in Physics And Astronomy, vol. 1 (1976), Wiley: Wiley New York, Wiley, New York
[15] Bijlsma, M. J.; Zaremba, E.; Stoof, H. T.C., Condensate growth in trapped Bose gases, Phys. Rev. A, 62, 6, Article 063609 pp. (2000)
[16] Bressan, A., Notes on the Boltzmann Equation, Lecture Notes for a Summer Course (2005), S.I.S.S.A: S.I.S.S.A Trieste
[17] Briant, M.; Einav, A., On the Cauchy problem for the homogeneous Boltzmann-Nordheim equation for bosons: local existence, uniqueness and creation of moments, J. Stat. Phys., 163, 5, 1108-1156 (2016) · Zbl 1342.35203
[18] Buckmaster, T.; Germain, P.; Hani, Z.; Shatah, J., Effective dynamics of the nonlinear Schrödinger equation on large domains (2016), arXiv preprint · Zbl 1394.35464
[19] Buckmaster, T.; Germain, P.; Hani, Z.; Shatah, J., Analysis of the (CR) equation in higher dimensions, Int. Math. Res. Not. IMRN (2017), in press
[20] Carleman, T., Sur la théorie de l’équation intégrodifférentielle de Boltzmann, Acta Math., 60, 1, 91-146 (1933) · JFM 59.0404.02
[21] Cercignani, C., Theory and Application of the Boltzmann Equation (1975), Elsevier: Elsevier New York · Zbl 0403.76065
[22] Cercignani, C., The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, vol. 67 (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0646.76001
[23] Cercignani, C.; Illner, R.; Pulvirenti, M., The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106 (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0813.76001
[24] Craciun, G.; Tran, M.-B., A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases (2016), arXiv preprint
[25] Deckert, D-A.; Fröhlich, J.; Pickl, P.; Pizzo, A., Dynamics of sound waves in an interacting Bose gas, Adv. Math., 293, 275-323 (2016) · Zbl 1334.81140
[26] Eckern, U., Relaxation processes in a condensed Bose gas, J. Low Temp. Phys., 54, 333-359 (1984)
[27] Escobedo, M.; Pezzotti, F.; Valle, M., Analytical approach to relaxation dynamics of condensed Bose gases, Ann. Physics, 326, 4, 808-827 (2011) · Zbl 1214.82099
[28] Escobedo, M.; Tran, M.-B., Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature, Kinet. Relat. Models, 8, 3, 493-531 (2015) · Zbl 1328.35144
[29] Escobedo, M.; Velázquez, J. J.L., Finite time blow-up and condensation for the bosonic Nordheim equation, Invent. Math., 200, 3, 761-847 (2015) · Zbl 1317.35256
[30] Escobedo, M.; Velázquez, J. J.L., On the theory of weak turbulence for the nonlinear Schrödinger equation, Mem. Amer. Math. Soc., 238, 1124 (2015), v+107 · Zbl 1329.35278
[31] Faou, E.; Germain, P.; Hani, Z., The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation, J. Amer. Math. Soc., 29, 4, 915-982 (2016) · Zbl 1364.35332
[32] Gamba, I. M.; Smith, L. M.; Tran, M.-B., On the wave turbulence theory for stratified flows in the ocean (2017), arXiv preprint
[33] Gardiner, C. W.; Lee, M. D.; Ballagh, R. J.; Davis, M. J.; Zoller, P., Quantum kinetic theory of condensate growth: comparison of experiment and theory, Phys. Rev. Lett., 81, 5266 (1998)
[34] Gardiner, C.; Zoller, P., Quantum kinetic theory. A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential, Phys. Rev. A, 55, 2902 (1997)
[35] Gardiner, C.; Zoller, P., Quantum kinetic theory. III. Quantum kinetic master equation for strongly condensed trapped systems, Phys. Rev. A, 58, 536 (1998)
[36] Gardiner, C.; Zoller, P., Quantum kinetic theory. V. Quantum kinetic master equation for mutual interaction of condensate and noncondensate, Phys. Rev. A, 61, Article 033601 pp. (2000)
[37] Gardiner, C.; Zoller, P.; Ballagh, R. J.; Davis, M. J., Kinetics of Bose-Einstein condensation in a trap, Phys. Rev. Lett., 79, 1793 (1997)
[38] Germain, P.; Hani, Z.; Thomann, L., On the continuous resonant equation for NLS, II. Statistical study, Anal. PDE, 8, 7, 1733-1756 (2015) · Zbl 1326.35344
[39] Germain, P.; Hani, Z.; Thomann, L., On the continuous resonant equation for NLS, I. Deterministic analysis, J. Math. Pures Appl., 105, 1, 131-163 (2016) · Zbl 1344.35133
[40] Germain, P.; Ionescu, A. D.; Tran, M.-B., Optimal local well-posedness theory for the kinetic wave equation (2017), arXiv preprint
[41] Germain, P.; Thomann, L., On the high frequency limit of the lll equation (2015), arXiv preprint
[42] Glassey, R. T., The Cauchy Problem in Kinetic Theory (1996), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0858.76001
[43] Griffin, A.; Nikuni, T.; Zaremba, E., Bose-Condensed Gases at Finite Temperatures (2009), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1177.82003
[44] Grillakis, M.; Machedon, M., Beyond mean field: on the role of pair excitations in the evolution of condensates, J. Fixed Point Theory Appl., 14, 1, 91-111 (2013) · Zbl 1304.81072
[45] Grillakis, M.; Machedon, M., Pair excitations and the mean field approximation of interacting bosons, I, Comm. Math. Phys., 324, 2, 601-636 (2013) · Zbl 1277.82034
[46] Grillakis, M.; Machedon, M.; Margetis, D., Second-order corrections to mean field evolution of weakly interacting bosons, II, Adv. Math., 228, 3, 1788-1815 (2011) · Zbl 1226.82033
[47] Gust, E. D.; Reichl, L. E., Collision integrals in the kinetic equations of dilute Bose-Einstein condensates (2012)
[48] Gust, E. D.; Reichl, L. E., Relaxation rates and collision integrals for Bose-Einstein condensates, Phys. Rev. A, 170, Article 43 pp. (2013)
[49] Gust, E. D.; Reichl, L. E., Transport coefficients from the boson Uehling-Uhlenbeck equation, Phys. Rev. E, 87, 4, Article 042109 pp. (2013)
[50] Imamovic-Tomasovic, M.; Griffin, A., Quasiparticle kinetic equation in a trapped Bose gas at low temperatures, J. Low Temp. Phys., 122, 617-655 (2001)
[51] Inguscio, M.; Stringari, S.; Wieman, C. E., Bose-Einstein Condensation in Atomic Gases, vol. 140 (1999), IOS Press: IOS Press Amsterdam
[52] Jaksch, D.; Gardiner, C.; Gheri, K. M.; Zoller, P., Quantum kinetic theory. IV. Intensity and amplitude fluctuations of a Bose-Einstein condensate at finite temperature including trap loss, Phys. Rev. A, 58, 1450 (1998)
[53] Jaksch, D.; Gardiner, C.; Zoller, P., Quantum kinetic theory. II. Simulation of the quantum Boltzmann master equation, Phys. Rev. A, 56, 575 (1997)
[54] Jin, S.; Tran, M.-B., Quantum hydrodynamic approximations to the finite temperature trapped Bose gases (2017), arXiv preprint
[55] Kagan, Y.; Svistunov, B. V., Evolution of correlation properties and appearance of broken symmetry in the process of Bose-Einstein condensation, Phys. Rev. Lett., 79, 18, 3331 (1997)
[56] Kirkpatrick, T. R.; Dorfman, J. R., Transport theory for a weakly interacting condensed Bose gas, Phys. Rev. A (3), 28, 4, 2576-2579 (1983)
[57] Kirkpatrick, T. R.; Dorfman, J. R., Transport in a dilute but condensed nonideal Bose gas: kinetic equations, J. Low Temp. Phys., 58, 301-331 (1985)
[58] Lieb, E. H.; Seiringer, R., Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett., 88, 17, Article 170409 pp. (2002)
[59] Lu, X., On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles, J. Stat. Phys., 116, 5-6, 1597-1649 (2004) · Zbl 1097.82023
[60] Lu, X., The Boltzmann equation for Bose-Einstein particles: velocity concentration and convergence to equilibrium, J. Stat. Phys., 119, 5-6, 1027-1067 (2005) · Zbl 1135.82029
[61] Lu, X., The Boltzmann equation for Bose-Einstein particles: condensation in finite time, J. Stat. Phys., 150, 6, 1138-1176 (2013) · Zbl 1277.82041
[62] Lukkarinen, J.; Spohn, H., Weakly nonlinear Schrödinger equation with random initial data, Invent. Math., 183, 1, 79-188 (2011) · Zbl 1235.76136
[63] Martin, R. H., Nonlinear Operators and Differential Equations in Banach Spaces, Pure Appl. Math. (1976), Wiley-Interscience
[64] Mitrouskas, D.; Petrat, S.; Pickl, P., Bogoliubov corrections and trace norm convergence for the Hartree dynamics (2016), arXiv preprint · Zbl 1435.35321
[65] Nazarenko, S., Wave Turbulence, Lecture Notes in Phys., vol. 825 (2011), Springer: Springer Heidelberg · Zbl 1220.76006
[66] Nepomnyashchii, Yu. A.; Nepomnyashchii, A. A., Infrared divergence in field theory of a base system with a condensate, Sov. Phys. JETP, 493, 48 (1978)
[67] Nguyen, T. T.; Tran, M.-B., Uniform in time lower bound for solutions to a quantum Boltzmann equation of bosons (2016), arXiv preprint
[68] Nguyen, T. T.; Tran, M.-B., On the kinetic equation in Zakharov’s wave turbulence theory for capillary waves (2017), arXiv preprint
[69] Nordheim, L. W., On the kinetic methods in the new statistics and its applications in the electron theory of conductivity, Proc. R. Soc. Lond. Ser. A, 119, 689-698 (1928) · JFM 54.0988.05
[70] Pomeau, Y.; Brachet, M.-E.; Métens, S.; Rica, S., Théorie cinétique d’un gaz de Bose dilué avec condensat, C. R. Acad. Sci., Ser. IIB Mech. Phys. Astron., 327, 8, 791-798 (1999) · Zbl 1016.82030
[71] Popov, V. N.; Seredniakov, A. V., Low-frequency asymptotic form of the self-energy parts of a superfluid Bose system at T=0, Sov. Phys. JETP, 193, 50 (1979)
[72] Proukakis, N.; Gardiner, S.; Davis, M.; Szymanska, M., Cold Atoms: Volume 1, Quantum Gases Finite Temperature and Non-Equilibrium Dynamics (2013), Imperial College Press
[73] Proukakis, N. P.; Jackson, B., Finite-temperature models of Bose-Einstein condensation, J. Phys., B At. Mol. Opt. Phys., 41, 20, Article 203002 pp. (2008)
[74] Reichl, L. E.; Gust, E. D., Transport theory for a dilute Bose-Einstein condensate, J. Low Temp. Phys., 88, Article 053603 pp. (2013)
[75] Reichl, L. E.; Tran, M.-B., A kinetic model for very low temperature dilute Bose gases (2017), arXiv preprint
[76] Seiringer, R., The excitation spectrum for weakly interacting bosons, Comm. Math. Phys., 306, 2, 565-578 (2011) · Zbl 1226.82039
[77] Semikoz, D. V.; Tkachev, I. I., Kinetics of Bose condensation, Phys. Rev. Lett., 74, 16, 3093 (1995)
[78] Semikoz, D. V.; Tkachev, I. I., Condensation of bosons in the kinetic regime, Phys. Rev. D, 55, 2, 489 (1997)
[79] Soffer, A.; Tran, M.-B., On coupling kinetic and Schrodinger equations (2016), arXiv preprint
[80] Spohn, H., Kinetics of the Bose-Einstein condensation, Phys. D, 239, 627-634 (2010) · Zbl 1186.82052
[81] Spohn, H., Weakly nonlinear wave equations with random initial data, (Proceedings of the International Congress of Mathematicians. Volume III (2010), Hindustan Book Agency: Hindustan Book Agency New Delhi), 2128-2143 · Zbl 1230.82035
[82] Stoof, H., Coherent versus incoherent dynamics during Bose-Einstein condensation in atomic gases, J. Low Temp. Phys., 114, 11-108 (1999)
[83] Uhlenbeck, G. E.; Uehling, E. A., Transport phenomena in Einstein-Bose and Fermi-Dirac gases, Phys. Rev., 43, 552-561 (1933) · Zbl 0006.33404
[84] Villani, C., A review of mathematical topics in collisional kinetic theory, (Handbook of Mathematical Fluid Dynamics, vol. I (2002), North-Holland: North-Holland Amsterdam), 71-305 · Zbl 1170.82369
[85] Wennberg, B., Entropy dissipation and moment production for the Boltzmann equation, J. Stat. Phys., 86, 5-6, 1053-1066 (1997) · Zbl 0935.82035
[86] Williams, J. E.; Zaremba, E.; Jackson, B.; Nikuni, T.; Griffin, A., Dynamical instability of a condensate induced by a rotating thermal gas, Phys. Rev. Lett., 88, 7, Article 070401 pp. (2002)
[87] (Zakharov, V. E., Nonlinear Waves and Weak Turbulence. Nonlinear Waves and Weak Turbulence, American Mathematical Society Translations, Series 2, vol. 182 (1998), American Mathematical Society: American Mathematical Society Providence RI), Advances in the Mathematical Sciences, 36 · Zbl 0879.00029
[88] Zakharov, V. E.; L’vov, V. S.; Falkovich, G., Kolmogorov Spectra of Turbulence I: Wave Turbulence (2012), Springer Science & Business: Springer Science & Business Media · Zbl 0786.76002
[89] Zaremba, E.; Nikuni, T.; Griffin, A., Dynamics of trapped Bose gases at finite temperatures, J. Low Temp. Phys., 116, 277-345 (1999)
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