Skip to main content

Advertisement

SpringerLink
Log in

Uniform in Time Lower Bound for Solutions to a Quantum Boltzmann Equation of Bosons

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this paper, we consider a quantum Boltzmann equation, which describes the interaction between excited atoms and a condensate. The collision integrals are taken–over energy manifolds, having the full form of the Bogoliubov dispersion law for particle energy. We prove that nonnegative radially symmetric solutions of the quantum Boltzmann equation are bounded from below by a Gaussian distribution, uniformly in time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Allemand, Thibaut: Derivation of a two-fluids model for a Bose gas from a quantum kinetic system. Kinet. Relat. Models. 2(2), 379–402 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allemand, Thibaut: Modèles mathématiques pour les gaz quantiques, Department of Mathematics and their Applications, Ecole Normale Supérieure. PhD Thesis under the supervision of Laure Saint-Raymond 2010

  3. Alonso, R., Gamba, I. M., Tran, M.-B.: The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation system for bosons at very low temperature. arXiv preprint arXiv:1609.07467, 2016

  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)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  6. Arkeryd, L., Nouri, A.: Bose condensate in interaction with excitations - a two-component space-dependent model close to equilibrium. ArXiv e-prints, July 2013

  7. Arkeryd, Leif., Nouri, Anne.: Bose condensates in interaction with excitations: a kinetic model. Comm. Math. Phys. 310(3), 765–788 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Arkeryd, Leif., Nouri, Anne.: A Milne problem from a Bose condensate with excitations. Kinet. Relat. Models. 6(4), 671–686 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Arkeryd, Leif., Nouri, Anne.: Bose condensates in interaction with excitations: a two-component space-dependent model close to equilibrium. J. Stat. Phys. 160(1), 209–238 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Carleman, Torsten: Sur la théorie de l'équation intégrodifférentielle de Boltzmann. Acta Math. 60(1), 91–146 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  11. Craciun, G., Tran, M.-B.: A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases. arXiv preprint arXiv:1608.05438, 2016

  12. Eckern, U.: Relaxation processes in a condensed Bose gas. J. Low Temp. Phys. 54, 333–359 (1984)

    Article  ADS  Google Scholar 

  13. Escobedo, M., Velázquez, J.J.L.: Finite time blow-up and condensation for the bosonic Nordheim equation. Invent. Math. 200(3), 761–847 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Escobedo, M., Velázquez, J. J. L.: On the theory of weak turbulence for the nonlinear Schrödinger equation. Mem. Amer. Math. Soc. 238(1124), v+107 2015

  15. Escobedo, Miguel, Mischler, Stéphane, Valle, Manuel A.: Homogeneous Boltzmann equation in quantum relativistic kinetic theory, volume 4 of Electronic Journal of Differential Equations. Monograph. Southwest Texas State University, San Marcos, TX, 2003

  16. Escobedo, Miguel., Pezzotti, Federica., Valle, Manuel.: Analytical approach to relaxation dynamics of condensed Bose gases. Ann. Physics. 326(4), 808–827 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Escobedo, Miguel., Tran, Minh-Binh.: Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic and Related Models. 8(3), 493–531 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gamba, I. M., Smith, L. M., Tran, M.-B.: On the wave turbulence theory for stratified flows in the ocean. arXiv preprint arXiv:1709.08266, 2017

  19. Gardiner, C., Zoller, P.: Quantum kinetic theory. A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential, volume 55 of Phys. Rev. A. 1997

  20. Gardiner, C., Zoller, P.: Quantum kinetic theory. III. Quantum kinetic master equation for strongly condensed trapped systems, volume 58 of Phys. Rev. A. 1998

  21. Germain, P., Ionescu, A. D., Tran, M.-B.: Optimal local well-posedness theory for the kinetic wave equation. arXiv preprint arXiv:1711.05587, 2017

  22. Griffin, Allan,. Nikuni, Tetsuro., Zaremba, Eugene.: Bose-condensed gases at finite temperatures. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  23. Gust, E. D., Reichl, L. E.: Collision integrals in the kinetic equations ofdilute Bose–Einstein condensates. arXiv:1202.3418, 2012

  24. Gust, E.D., Reichl, L.E.: Relaxation rates and collision integrals for Bose-Einstein condensates. Phys. Rev. A. 170, 43–59 (2013)

    Google Scholar 

  25. Imamovic-Tomasovic, M., Griffin, A.: Quasiparticle kinetic equation in a trapped Bose gas at low temperatures. J. Low Temp. Phys. 122, 617–655 (2001)

    Article  ADS  Google Scholar 

  26. Jaksch, D., Gardiner, C., Zoller, P.: Quantum kinetic theory. II. Simulation of the quantum Boltzmann master equation, volume 56 of Phys. Rev. A. 1997

  27. Jin, S., Tran, M.-B.: Quantum hydrodynamic approximations to the finite temperature trapped Bose gases. Physica D: Nonlinear Phenomena accepted, arXiv preprint arXiv:1703.00825, 2017

  28. Kirkpatrick, T. R., Dorfman, J. R.: Transport theory for a weakly interacting condensed Bose gas. Phys. Rev. A (3). 28(4), 2576–2579 1983

  29. Kirkpatrick, T.R., Dorfman, J.R.: Transport coefficients in a dilute but condensed Bose gas. J. Low Temp. Phys. 58, 399–415 (1985)

    Article  ADS  Google Scholar 

  30. 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)

    Article  ADS  Google Scholar 

  31. Lacaze, Robert, Lallemand, Pierre, Pomeau, Yves, Rica, Sergio: Dynamical formation of a Bose-Einstein condensate. Phys. D. 152/153, 779–786 2001 Advances in nonlinear mathematics and science

  32. Lukkarinen, Jani., Spohn, Herbert.: Weakly nonlinear Schrödinger equation with random initial data. Invent. Math. 183(1), 79–188 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Mouhot, Clément: Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions. Comm. Partial Differential Equations. 30(4-6), 881–917 2005

  34. Mouhot, Clément., Villani, Cédric.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Ration. Mech. Anal. 173(2), 169–212 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Nguyen, T.T., Tran, M.-B.: On the kinetic equation in Zakharov's wave turbulence theory for capillary waves. SIAM Journal on Mathematical Analysis. 50(2), 2020–2047 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nordheim, L.W.: Transport phenomena in Einstein-Bose and fermi-dirac gases. Proc. Roy. Soc. London A 119, 689 (1928)

    Article  ADS  Google Scholar 

  37. Peierls, R.: Zur kinetischen theorie der warmeleitung in kristallen. Annalen der Physik. 395(8), 1055–1101 (1929)

    Article  ADS  MATH  Google Scholar 

  38. Peierls, R. E.: Quantum theory of solids. In Theoretical physics in the twentieth century (Pauli memorial volume), pages 140–160. Interscience, New York, 1960

  39. Pulvirenti, Ada., Wennberg, Bernt.: A Maxwellian lower bound for solutions to the Boltzmann equation. Comm. Math. Phys. 183(1), 145–160 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Reichl, L.E., Gust, E.D.: Transport theory for a dilute Bose-Einstein condensate. J Low Temp Phys. 88, 053603 (2013)

    Google Scholar 

  41. Reichl, L. E., Tran, M.-B.: A kinetic model for very low temperature dilute Bose gases. arXiv preprint arXiv:1709.09982, 2017

  42. Soffer, A., Tran, M.-B.: On coupling kinetic and Schrodinger equations. Journal of Differential Equations. 265(5), 2243–2279 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Soffer, A., Tran, M.-B.: On the dynamics of finite temperature trapped Bose gases. Advances in Mathematics. 325, 533–607 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Spohn, Herbert: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124(2–4), 1041–1104 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Spohn, Herbert: Weakly nonlinear wave equations with random initial data. In Proceedings of the International Congress of Mathematicians. Volume III, pages 2128–2143. Hindustan Book Agency, New Delhi, 2010

  46. Uehling, E.A., Uhlenbeck, G.E.: On the kinetic methods in the new statistics and its applications in the electron theory of conductivity. I Phys. Rev. 43, 552–561 (1933)

    Article  ADS  MATH  Google Scholar 

  47. Villani, Cédric: A review of mathematical topics in collisional kinetic theory. In Handbook of mathematical fluid dynamics, Vol. I, pages 71–305. North-Holland, Amsterdam, 2002

  48. M'etens, S., Pomeau, Y., Brachet, M. A., Rica, S.: Théorie cinétique d'un gaz de Bose dilué avec condensat. C. R. Acad. Sci. Paris S'er. IIb M'ec. Phys. Astr. 327, 791–798 1999

  49. Zakharov, V. E.: editor. Nonlinear waves and weak turbulence, volume 182 of American Mathematical Society Translations, Series 2. American Mathematical Society, Providence, RI, 1998 Advances in the Mathematical Sciences, 36

Download references

Acknowledgements

The authors thank the referees for their constructive comments. M.-B Tran has been supported by NSF Grant DMS-1814149, NSF Grant RNMS (Ki-Net) 1107291. M.-B Tran would like to thank Professor Daniel Heinzen, Professor Linda Reichl, Professor Mark Raizen and Professor Robert Dorfman for fruitful discussions on the topic. The research was carried on while M.-B. Tran was visiting University of Texas at Austin and Penn State University. He would like to thank these institutions for their hospitality.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.

Author information

Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, State College, PA, 16802, USA

    Toan T. Nguyen

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Minh-Binh Tran

Authors
  1. Toan T. Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minh-Binh Tran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Toan T. Nguyen.

Additional information

Communicated by P.-L. Lions

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen, T.T., Tran, MB. Uniform in Time Lower Bound for Solutions to a Quantum Boltzmann Equation of Bosons. Arch Rational Mech Anal 231, 63–89 (2019). https://doi.org/10.1007/s00205-018-1271-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-018-1271-z

Search

Navigation