Quantum kinetic theory: modelling and numerics for Bose-Einstein condensation

  • Weizhu Bao
  • Lorenzo Pareschi
  • Peter A. Markowich
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


We review some modelling and numerical aspects in quantum kinetic theory for a gas of interacting bosons and we try to explain what makes Bose-Einstein condensation in a dilute gas mathematically interesting and numerically challenging. Particular care is devoted to the development of efficient numerical schemes for the quantum Boltzmann equation that preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and generalized Bose-Einstein distributions as steady states. These properties are essential in order to develop numerical methods that are able to capture the challenging phenomenon of bosons condensation. We also show that the resulting schemes can be evaluated with the use of fast algorithms. In order to study the evolution of the condensate wave function the Gross-Pitaevskii equation is presented together with some schemes for its efficient numerical solution.


Boltzmann Equation Collision Operator Trapping Potential Ground State Solution Nonlinear Schrodinger Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    A. Aftalion, and Q. Du, Vortices in a rotating Bose-Einstein condensate: Critical angular velocities and energy diagrams in the Thomas-Fermi regime, Phys. Rev. A, 64, 063603, (2001).CrossRefGoogle Scholar
  2. 2.
    W. Bao, Ground states and dynamics of multi-component Bose-Einstein condensates, SI AM Multiscale Modeling and Simulation, to appear (arXiv: cond-mat/0305 309).Google Scholar
  3. 3.
    W. Bao, D. Jaksch, An explicit unconditionally stable numerical method for solving damped nonlinear Schrodinger equations with a focusing nonlinearity, SIAM J. Numer. Anal., 41,1406–1426,(2003).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    W. Bao, Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comp., to appear (arXiv: cond-mat/0303241).Google Scholar
  5. 5.
    W. Bao, D. Jaksch, P. Markowich, Numerical solution of the Gross-Pitaevskii Equation for Bose-Einstein condensation, J. Comput. Phys., 187, 318–342, (2003).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    W. Bao, D. Jaksch, P. Markowich, Three Dimensional Simulation of Jet Formation in Collapsing Condensates, J. Phys. B: At. Mol. Opt. Phys., 37, 329–343, (2004).CrossRefGoogle Scholar
  7. 7.
    W. Bao, S. Jin, P.A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175,487–524, (2002).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    W. Bao, S. Jin, P.A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semi-clasical regimes, SIAM J. Sci. Comp., 25, 27–64, (2003).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    W. Bao, W. Tang, Ground state solution of trapped interacting Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187, 230–254, (2003).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti, Some Considerations on the deriva-tion of the nonlinear Quantum Boltzmann Equation. Mathematical Physics Archive, Uni-versity of Texas, 03–19, (2003).Google Scholar
  11. 11.
    S.N. Bose, Plancks Gesetz and Lichtquantenhypothese,Z Phys., 26, 178–181, (1924).MATHCrossRefGoogle Scholar
  12. 12.
    M.L. Chiofalo, S. Succi, M.P. Tosi, Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62,7438–7444, (2000).CrossRefGoogle Scholar
  13. 13.
    C. Buet, S. Cordier, Numerical method for the Compton scattering operator, in: Lecture Notes on the discretization of the Boltzmann equation, ed. N. Bellomo, World Scientific, (2002).Google Scholar
  14. 14.
    C. Buet, S. Cordier, P. Degond, M. Lemou, Fast algorithms for numerical, conservative, and entropy approximations of the Fokker-Planck equation, J. Comp. Phys., 133,310–322, (1997).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    S. Chapman, T. G. C78owling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, 1970. Third edition.Google Scholar
  16. 16.
    E. A. Cornell, J. R. Ensher, C. E. Wieman, Experiments in dilute atomic Bose-Einstein condensation in: Bose-Einstein Condensation in Atomic Gases, Proceedings of the Inter-national School of Physics Enrico Fermi Course CXL, M. Inguscio, S. Stringari and C. E. Wieman, Eds., Italian Physical Society, 1999), pp. 15–66 (cond-mat/9903109).Google Scholar
  17. 17.
    E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, C. E. Wieman, Dynamics of collapsing and exploding Bose-Einstein condensates, Nature, 412,295–299, 2001.CrossRefGoogle Scholar
  18. 18.
    A. Einstein, Quantentheorie des einatomingen idealen gases, Stiz. Presussische Akademie der Wissenshaften Phys-math. Klasse, Sitzungsberichte, 23,1–14, (1925).Google Scholar
  19. 19.
    A. Einstein, Zur quantentheorie des idealen gases, Stiz. Presussische Akademie der Wis-senshaften Phys-math. Klasse, Sitzungsberichte, 23,18–25, (1925).Google Scholar
  20. 20.
    L. Erdös, M. Salmhofer, H. Yau, On the quantum Boltzmann equation, preprint 2003.Google Scholar
  21. 21.
    M. Escobedo, S. Mischler, Equation de Boltzmann quantique homogene: existence et comportement asymptotique, C R. Acad. Sci. Paris 329 Serie I, 593–598 (1999).Google Scholar
  22. 22.
    M. Escobedo, S. Mischler, M. A. Valle, Homogeneous Boltzmann equation in quantum relativistic kinetic theory, Electronic Journal of Differential Equations, Monograph 04, 2003, 85 pages, ( or or Scholar
  23. 23.
    M. Escobedo, S. Mischler, On a quantum Boltzmann equation for a gas of photons, J. Math. PuresAppl, 9:80, 471–515, (2001).MathSciNetCrossRefGoogle Scholar
  24. 24.
    C.W. Gardiner, D. Jaksch, P. Zoller, Quantum Kinetic Theory II: Simulation of the Quantum Boltzmann Master Equation, Phys. Rev. A, 56, 575, (1997).CrossRefGoogle Scholar
  25. 25.
    C.W. Gardiner, P. Zoller, Quantum Kinetic Theory I: A Quantum Kinetic Master Equation for Condensation of a weakly interacting Bose gas without a trapping potential, Phys. Rev. A, 55, 2902, (1997).CrossRefGoogle Scholar
  26. 26.
    C.W. Gardiner, P. Zoller, Quantum Kinetic Theory III: Quantum kinetic master equation for strongly condensed trapped systems, Phys. Rev. A, 58, 536, (1998).CrossRefGoogle Scholar
  27. 27.
    V.L. Ginzburg, L.P. Pitaevskii, On the theory of superfluidity, Zh. Eksp. Teor.Fiz., 34,1240 (1958) [Sov. Phys. JETP 7, 858 (1958)].MathSciNetGoogle Scholar
  28. 28.
    E. P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4, 195 (1963).CrossRefGoogle Scholar
  29. 29.
    E.P. Gross, Structure of a quantized vortex in boson systems, Nuovo. Cimento., 20, 454, (1961).MATHCrossRefGoogle Scholar
  30. 30.
    D. Jaksch, P. Markowich, L. Pareschi, M. Wenin, P. Zoller, Increasing phase-space density by varying the trap potential, work in progress.Google Scholar
  31. 31.
    B. Jackson, E. Zaremba, Dynamical simulations of trapped Bose gases at finite tempera-tures, Laser Phys., 12, 93–105, (2002).Google Scholar
  32. 32.
    W. Ketterle, D. S. Durfee, D.M. Stamper-Kurn, Making, probing and understanding Bose- Einstein condensates, in: Bose-Einstein condensation in atomic gases, Proceedings of the International School of Physics Enrico Fermi, Course CXL, M. Inguscio, S. Stringari, and CE. Wieman, eds., (IOS Press, Amsterdam, 1999), pp. 67–176 (cond-mat/9904034).Google Scholar
  33. 33.
    L. Landau, E. Lifschitz, Quantum Mechanics: Non-relativistic Theory, Pergamon Press, New York, (1977).Google Scholar
  34. 34.
    P. Leboeuf, N. Pavloff, Phys. Rev. A 64, 033602 (2001); V. Dunjko, V. Lorent, and M. 01- shanii, Phys. Rev. Lett. 86, 5413 (2001).Google Scholar
  35. 35.
    M.Lemou, Multipole expansions for the Fokker-Planck-Landau operator, Numerische Mathematik, 78, 597–618, (1998).MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    E.H. Lieb, R. Seiringer, J. Yngvason, Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional, Phys. Rev. A, 61, 3602, (2000).CrossRefGoogle Scholar
  37. 37.
    X. Lu, On spatially homogeneous solutions of a modified Boltzmann equation for Fermi- Dirac particles, J. Statist. Phys., 105, 353–388, (2001).MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    X. Lu, A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long-time behavior, J. Statist. Phys., 98, 1335–1394, (2000).MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    P. Markowich, L.Pareschi, Fast, conservative and entropic numerical methods for the boson Boltzmann equation, preprint 2002.Google Scholar
  40. 40.
    L. Pareschi, Computational methods and fast algorithms for Boltzmann equations. Lecture Notes on the discretization of the Boltzmann equation, ed. N. Bellomo, World Scientific, (2002).Google Scholar
  41. 41.
    L. Pareschi, G. Russo, G. Toscani, Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comp. Phys, 165, 1–21, (2000).MathSciNetCrossRefGoogle Scholar
  42. 42.
    L. Pareschi, G.Toscani, C Villani, Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit, Numerische Mathematik, (to appear).Google Scholar
  43. 43.
    L.P. Pitaevskii, Zh. Eksp. Teor. Fiz., 40, 646, (1961) (Sov. Phys. JETP, 13, 451, (1961)).Google Scholar
  44. 44.
    D.V. Semikoz, I.I. Tkachev, Kinetics of Bose condensation, Physical Review Letters, 74, 3093–3097, (1995).CrossRefGoogle Scholar
  45. 45.
    D.V. Semikoz, I.I. Tkachev, Condensation of bosons in the kinetic regime, Physical Review D, 55,489–502, (1997).CrossRefGoogle Scholar
  46. 46.
    D.M. Stamper-Kurn, H.J. Miesner, A.R Cdikkatur, S. Inouye, J. Stenger, W Ketterle, Phys. Rev, Lett., 81, p. 2194, (1988).CrossRefGoogle Scholar
  47. 47.
    G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5, (1968) pp. 506.MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    J.T.M. Walraven, Quantum dyanmics of simple systems, edited by G.L.Oppo, S.L.Burnett, E.Riis and M.Wilkinson, Bristol 1996. (SUSSP Proceedings, vol. 44).Google Scholar
  49. 49.
    H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150, 262–268, (1990).MathSciNetCrossRefGoogle Scholar
  50. 50.
    E. Zaremba, T. Nikuni, A. Griffin, J. Low Temp. Phys., 116, p. 277, (1999).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Weizhu Bao
    • 1
  • Lorenzo Pareschi
    • 2
  • Peter A. Markowich
    • 3
  1. 1.Department of Computational ScienceNational University of SingaporeSingapore
  2. 2.Department of MathematicsUniversity of FerraraItaly
  3. 3.Department of MathematicsUniversity of ViennaAustria

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