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Annales Henri Poincaré

, Volume 19, Issue 6, pp 1869–1889 | Cite as

The Shape of the Emerging Condensate in Effective Models of Condensation

  • Volker Betz
  • Steffen Dereich
  • Peter Mörters
Article
  • 24 Downloads

Abstract

We consider effective models of condensation where the condensation occurs as time t goes to infinity. We provide natural conditions under which the buildup of the condensate occurs on a spatial scale of 1 / t and has the universal form of a Gamma density. The exponential parameter of this density is determined only by the equation and the total mass of the condensate, while the power law parameter may in addition depend on the decay properties of the initial condition near the condensation point. We apply our results to some examples, including simple models of Bose–Einstein condensation.

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References

  1. 1.
    Buffet, E., de Smedt, P., Pulé, J.V.: On the dynamics of Bose–Einstein condensation. Annales de l’Institut Henri Poincaré (C) Analyse non linéaire 1, 413–451 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buffet, E., de Smedt, P., Pulé, J.V.: The dynamics of the open bose gas. Ann. Phys. 155, 269–304 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chatterjee, S., Diaconis, P.: Fluctuations of the Bose–Einstein condensate. J. Phys. A Math. Theor. 47(8), 085201 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dereich, S.: Preferential attachment with fitness: unfolding the condensate. Electron. J. Probab. 21(3), 1–38 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dereich, S., Mörters, P.: Emergence of condensation in Kingmanâs model of selection and mutation. Acta Appl. Math. 127, 17–26 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dereich, S., Mailler, C., Mörters, P.: Nonextensive condensation in reinforced branching processes. Ann. Appl. Probab. 27, 2539–2568 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Escobedo, M., Mischler, S.: Equation de Boltzmann quantique homogene: existence et comportement asymptotique. C. R. Acad. Sci. Paris Serie I 329, 593–598 (1999)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Escobedo, M., Mischler, S.: On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl. 80, 471–515 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Escobedo, M., Mischler, S., Velázquez, J.J.L.: Asymptotic description of Dirac mass formation in kinetic equations for quantum particles. J. Differ. Equ. 202, 208–230 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Escobedo, M., Velázquez, J.J.L.: Finite time blow-up and condensation for the bosonic Nordheim equation. Invent. Math. 200, 761–847 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kingman, J.F.C.: A simple model for the balance between selection and mutation. J. Appl. Probab. 15, 1–12 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lu, X.: On isotropic distributional solutions to the Boltzmann equation for Bose–Einstein particles. J. Stat. Phys. 116, 1597–1649 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lu, X.: The Boltzmann equation for Bose–Einstein particles: velocity concentration and convergence to equilibrium. J. Stat. Phys. 119, 1027–1067 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Park, S.-C., Krug, J.: Evolution in random fitness landscapes: the infinite sites model. J. Stat. Mech. Theory Exp. 4, P04014 (2008)Google Scholar
  15. 15.
    Spohn, H.: Kinetics of the Bose–Einstein condensation. Phys. D Nonlinear Phenom. 239, 627–634 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yuan, L.: A generalization of Kingman’s model of selection and mutation and the Lenski experiment. Math. Biosci. 285, 61–67 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikTU DarmstadtDarmstadtGermany
  2. 2.Institut für mathemtaische StochastikUniversität MünsterMunsterGermany
  3. 3.Mathematisches InstitutUniversität zu KölnCologneGermany

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