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On the Blow Up and Condensation of Supercritical Solutions of the Nordheim Equation for Bosons

Abstract

In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons with bounded initial data blow up in finite time in the L norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contain a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.

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References

  1. 1

    Balescu R.: Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, New York (1974)

    Google Scholar 

  2. 2

    Carleman T.: Sur la théorie de l’équation intégro-differentielle de Boltzmann. Acta Math. 60, 91–146 (1933)

    Article  MathSciNet  Google Scholar 

  3. 3

    Escobedo, M., Velázquez, J.J.L.: Finite time blow-up and condensation for the bosonic Nordheim equation. Preprint arXiv:1206.5410

  4. 4

    Escobedo M., Mischler S., Velázquez J.J.L.: On the fundamental solution of a linearized Uehling Uhlenbeck equation. Arch. Rat. Mech. Anal. 186(2), 309–349 (2007)

    Article  MATH  Google Scholar 

  5. 5

    Escobedo M., Mischler S., Velázquez J.J.L.: Singular solutions for the Uehling Uhlenbeck equation. Proc. Royal Soc. Edinburgh. 138A, 67–107 (2008)

    Google Scholar 

  6. 6

    Huang K.: Statistical Mechanics. Wiley, New York (1963)

    Google Scholar 

  7. 7

    Josserand C., Pomeau Y., Rica S.: Self-similar singularities in the kinetics of condensation. J. Low Temp. Phys. 145, 231–265 (2006)

    ADS  Article  Google Scholar 

  8. 8

    Lacaze R., Lallemand P., Pomeau Y., Rica S.: Dynamical formation of a Bose–Einstein condensate. Physica D 152(153), 779–786 (2001)

    ADS  Article  MathSciNet  Google Scholar 

  9. 9

    Lu X.: On isotropic distributional solutions to the Boltzmann equation for Bose–Einstein particles. J. Stat. Phys. 116, 1597–1649 (2004)

    ADS  Article  MATH  Google Scholar 

  10. 10

    Lu X.: The Boltzmann equation for Bose–Einstein particles: velocity concentration and convergence to equilibrium. J. Stat. Phys. 119, 1027–1067 (2005)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  11. 11

    Lu X.: The Boltzmann equation for Bose–Einstein particles: condensation in finite time. J. Stat. Phys. 150, 11381176 (2013)

    Article  Google Scholar 

  12. 12

    Nordheim L.W.: On the kinetic method in the new statistics and its application in the electron theory of conductivity. Proc. R. Soc. Lond. A 119, 689–698 (1928)

    ADS  Article  MATH  Google Scholar 

  13. 13

    Semikov D.V., Tkachev I.I.: Kinetics of Bose condensation. Phys. Rev. Lett. 74, 3093–3097 (1995)

    ADS  Article  Google Scholar 

  14. 14

    Semikov D.V., Tkachev I.I.: Condensation of Bosons in the kinetic regime. Phys. Rev. D 55(2), 489–502 (1997)

    ADS  Article  Google Scholar 

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Correspondence to M. Escobedo.

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Communicated by H. Spohn

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Escobedo, M., Velázquez, J.J.L. On the Blow Up and Condensation of Supercritical Solutions of the Nordheim Equation for Bosons. Commun. Math. Phys. 330, 331–365 (2014). https://doi.org/10.1007/s00220-014-2034-9

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Keywords

  • Initial Data
  • Weak Solution
  • Mild Solution
  • Einstein Condensate
  • Dirac Mass