Journal of Statistical Physics

, Volume 119, Issue 5–6, pp 1027–1067 | Cite as

The Boltzmann Equation for Bose–Einstein Particles: Velocity Concentration and Convergence to Equilibrium

  • Xuguang Lu


Long time behavior of solutions of the spatially homogeneous Boltzmann equation for Bose–Einstein particles is studied for hard potentials with certain cutoffs and for the hard sphere model. It is proved that in the cutoff case solutions as time \(t\rightarrow\infty\) converge to the Bose–Einstein distribution in L1 topology with the weighted measure \((\varrho +|v|^2)dv\), where \(\varrho=1\) for temperature \(T\geq T_c\) and \(\varrho=0\) for T<T c . In particular this implies that if T<T c then the solutions in the velocity regions \(\{v\in{\bf R}^3|\,\,|v|\leq \delta (t)\}\) (with \(\delta(t)\rightarrow 0\)) converge to a unique Dirac delta function (velocity concentration). All these convergence are uniform with respect to the cutoff constants. For the hard sphere model, these results hold also for weak or distributional solutions. Our methods are based on entropy inequalities and an observation that the convergence to Bose–Einstein distributions can be reduced to the convergence to Maxwell distributions.


Bose–Einstein particles temperature entropy velocity concentration convergence to equilibrium 


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  1. 1.
    Ball, J. M., Murat, F. 1989Remarks on Chacon’s biting lemmaProc. Am. Math. Soc.107655663Google Scholar
  2. 2.
    Bouchut, F., Desvillettes, L. 1998A proof of the smoothing properties of the positive part of Boltzmann’s kernelRev. Mat. Iberoamericana144761Google Scholar
  3. 3.
    Caflisch, R. E., Levermore, C. D. 1986Equilibrium for radiation in a homogeneous plasmaPhys. Fluids29748752CrossRefGoogle Scholar
  4. 4.
    Carlen, E. A., Carvalho, M. C. 1992Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equationJ. Stat. Phys.67575608CrossRefGoogle Scholar
  5. 5.
    Carlen, E.A., Carvalho, M.C. 1994Entropy production estimates for Boltzmann equations with physically realistic collision kernelsJ. Stat. Phys.74743782Google Scholar
  6. 6.
    Cercignani, C. 1988The Boltzmann Equation and its ApplicationsSpringer-VerlagNew YorkGoogle Scholar
  7. 7.
    Cercignani, C., Illner, R., Pulvirenti, M. 1994The Mathematical Theory of Dilute GasesSpringer-VerlagNew YorkGoogle Scholar
  8. 8.
    Cercignani, C. 1982H-theorem and trend to equilibrium in the kinetic theory of gasesArch. Mech.34231241Google Scholar
  9. 9.
    S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd ed. (Cambridge University Press, 1970).Google Scholar
  10. 10.
    Desvillettes, L. 1989Entropy dissipation rate and convergence in kinetic equationsCommun. Math. Phys.123687702CrossRefGoogle Scholar
  11. 11.
    Escobedo, M., Herrero, M. A., Velazquez, J. J. L. 1998A nonlinear Fokker–Planck equation modelling the approach to thermal equilibrium in a homogeneous plasmaTrans. Am. Math. Soc.35038373901CrossRefGoogle Scholar
  12. 12.
    Escobedo, M., Mischler, S. 2001On a quantum Boltzmann equation for a gas of photonsJ. Math. Pures Appl.80471515CrossRefGoogle Scholar
  13. 13.
    M. Escobedo, Boltzmann equation for quantum particles and Fokker Planck approximation, Elliptic and Parabolic Problems (Rolduc/Gaeta, 2001), (World Sci. Publishing, River Edge, NJ, 2002), pp. 75–87.Google Scholar
  14. 14.
    Escobedo, M., Mischler, S., Valle, M.A. 2003Homogeneous Boltzmann equation in quantum relativistic kinetic theory, Electronic Journal of Differential Equations MonographSouthwest Texas State UniversitySan Marcos, TXVol. 4Google Scholar
  15. 15.
    Escobedo, M., Mischler, S., Velazquez, J. J. L. 2004Asymptotic description of Dirac mass formation in kinetic equations for quantum particlesJ. Differ. Equ.202208230CrossRefMathSciNetGoogle Scholar
  16. 16.
    Huang, K. 1963Statistical MechanicsJohn Wiley & Sons Inc.New York, LondonGoogle Scholar
  17. 17.
    L. D.Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. Part 1 (Pergamon Press, 1980).Google Scholar
  18. 18.
    Lions, P. L. 1994Compactness in Boltzmann’s equation via Fourier integral operators and applications, IJ. Math. Kyoto Univ.34391427Google Scholar
  19. 19.
    Lu, X. G. 1998A direct method for the regularity of the gain term in the Boltzmann equationJ. Math. Anal. Appl.228409435CrossRefGoogle Scholar
  20. 20.
    Lu, X. G. 2000A modified Boltzmann equation for Bose–Einstein particles: isotropic solutions and long-time behaviorJ. Stat. Phys.9813351394CrossRefGoogle Scholar
  21. 21.
    Lu, X. G. 2001On spatially homogeneous solutions of a modified Boltzmann equation for Fermi–Dirac particlesJ. Stat. Phys.105353388CrossRefGoogle Scholar
  22. 22.
    Lu, X. G. 2004On isotropic distributional solutions to the Boltzmann equation for Bose–Einstein particlesJ. Stat. Phys.11615971649CrossRefMathSciNetGoogle Scholar
  23. 23.
    Nordheim, L. W. 1928On the kinetic methods in the new statistics and its applications in the electron theory of conductivityProc. Roy. Soc. London Ser. A119689Google Scholar
  24. 24.
    R. K. Pathria, Statistical Mechanics (Pergamon Press, 1972).Google Scholar
  25. 25.
    Rudin, W. 1974Real and Complex AnalysisMcGraw-HillNew YorkGoogle Scholar
  26. 26.
    Saint-Raymond, L. 2004Kinetic models for superfluids: a review of mathematical resultsC.R. Phys.56575Google Scholar
  27. 27.
    Semikov, D. V., Tkachev, I. I. 1995Kinetics of Bose condensationPhys. Rev. Lett.7430933097CrossRefPubMedGoogle Scholar
  28. 28.
    Semikov, D. V., Tkachev, I. I. 1997Condensation of Bose in the kinetic regimePhys. Rev. D55489502CrossRefGoogle Scholar
  29. 29.
    Toscani, G., Villani, C. 1999Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equationCommun. Math. Phys.203667706CrossRefGoogle Scholar
  30. 30.
    Truesdell, C., Muncaster, R.G. 1980Fundamentals Maxwell’s Kinetic Theory of a Simple Monoatomic GasAcademic PressNew YorkGoogle Scholar
  31. 31.
    Uehling, E. A., Uhlenbeck, G.E. 1933Transport phenomena in Einstein–Bose and Fermi–Dirac gases, IPhys. Rev.43552561CrossRefGoogle Scholar
  32. 32.
    Villani, C. 2002A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid DynamicsNorth-HollandAmsterdam71305Vol. IGoogle Scholar
  33. 33.
    Villani, C. 2003Cercignani’s conjecture is sometimes true and always almost trueCommun. Math. Phys.234455490CrossRefGoogle Scholar
  34. 34.
    Wennberg, B. 1994Regularity in the Boltzmann equation and the Radon transformCommun. Partial Differ. Equ.1920572074Google Scholar
  35. 35.
    Wennberg, B. 1997The geometry of binary collisions and generalized Radon transformsArch. Rational Mech. Anal.139291302CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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