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From a Non-Markovian System to the Landau Equation

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Abstract

In this paper, we prove that in macroscopic times of order one, the solutions to the truncated BBGKY hierarchy (to second order) converge in the weak coupling limit to the solution of the nonlinear spatially homogeneous Landau equation. The truncated problem describes the formal leading order behavior of the underlying particle dynamics, and can be reformulated as a non-Markovian hyperbolic equation that converges to the Markovian evolution described by the parabolic Landau equation. The analysis in this paper is motivated by Bogolyubov’s derivation of the kinetic equation by means of a multiple time scale analysis of the BBGKY hierarchy.

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References

  1. Alexandre R., Villani C.: On the Landau approximation in plasma physics. Ann. Inst. Henri Poincaré Anal. Nonlinéaire 21(1), 61–95 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  2. Balescu R.: Statistical Mechanics of Charged Particles. Monographs in Statistical Physics and Thermodynamics, vol. 4. Wiley, London (1963)

    MATH  Google Scholar 

  3. Balescu R.: Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, London (1975)

    MATH  Google Scholar 

  4. Basile G., Nota A., Pulvirenti M.: A diffusion limit for a test particle in a random distribution of scatterers. J. Stat. Phys. 155(6), 1087–1111 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bobylev A., Pulvirenti M., Saffirio C.: From particle systems to the Landau equation: a consistency result. Commun. Math. Phys. 319(3), 683–702 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bogoliubov, N.: Problems of a dynamical theory in statistical physics. In: Studies in Statistical Mechanics, vol. I, pp. 1–118. Interscience, New York (1962)

  7. Boldrighini C., Bunimovich L.A., Sinai Y.: On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32(3), 477–501 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Desvillettes L.: Entropy dissipation estimates for the Landau equation in the Coulomb case and applications. J. Funct. Anal. 269(5), 1359–1403 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Desvillettes L., Pulvirenti M.: The linear Boltzmann equation for long-range forces: a derivation from particle systems. Math. Models Methods Appl. Sci. 9(8), 1123–1145 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Desvillettes L., Ricci V.: A rigorous derivation of a linear kinetic equation of Fokker–Planck type in the limit of grazing collisions. J. Stat. Phys. 104(5–6), 1173–1189 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Desvillettes L., Villani C.: On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. Commun. Partial Differ. Equ. 25(1–2), 179–259 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Desvillettes L., Villani C.: On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications. Commun. Partial Differ. Equ. 25(1–2), 261–298 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Desvillettes L., Villani C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. DiPerna R., Lions P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130(2), 321–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gallagher I., Saint-Raymond L., Texier B.: From Newton to Boltzmann: Hard Spheres and Short-Range Potentials. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2013)

    MATH  Google Scholar 

  16. Gallavotti, G.: Grad–Boltzmann limit and Lorentz’s Gas. In: Statistical Mechanics: A Short Treatise (1999)

  17. Guo Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231(3), 391–434 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. John. F.: Partial Differential Equations, Applied Mathematical Sciences, vol. 1, 4th edn. Springer, New York (1991)

    Google Scholar 

  19. Landau L.: Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung. Phys. Z. Sow. Union 10, 154 (1936)

    MATH  Google Scholar 

  20. Lanford, O.: Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Physics, vol. 38, pp. 1–111 (1975)

  21. Lenard A.: On Bogoliubov’s kinetic equation for a spatially homogeneous plasma. Ann. Phys. 10, 390–400 (1960)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Lifshitz E., Pitaevskii L.: Course of Theoretical Physics. Pergamon Press, Oxford (1981)

    Google Scholar 

  23. Majda A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53. Springer, New York (1984)

    Book  Google Scholar 

  24. Pulvirenti M., Saffirio C., Simonella S.: On the validity of the Boltzmann equation for short range potentials. Rev. Math. Phys. 26(2), 64 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pulvirenti M., Simonella S.: The Boltzmann–Grad limit of a hard sphere system: analysis of the correlation error. Invent. Math. 207(3), 1135–1237 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Silvestre L.: Upper bounds for parabolic equations and the Landau equation. J. Differ. Equ. 262(3), 3034–3055 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Spohn H.: The Lorentz process converges to a random flight process. Commun. Math. Phys. 60(3), 277–290 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Spohn H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52(3), 569–615 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  29. Spohn H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (2012)

    MATH  Google Scholar 

  30. Strain R., Guo Y.: Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187(2), 287–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Toscani G., Villani C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203(3), 667–706 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Villani C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143(3), 273–307 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Villani C.: On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci. 8(6), 957–983 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Villani, C.: A Review of Mathematical Topics in Collisional Kinetic Theory. Handbook of Mathematical Fluid Dynamics, vol. 1. North-Holland, Amsterdam (2002)

  35. Wennberg B.: Stability and exponential convergence in L p for the spatially homogeneous Boltzmann equation. Nonlinear Anal. 20(8), 935–964 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Juan J. L. Velázquez & Raphael Winter

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  1. Juan J. L. Velázquez
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  2. Raphael Winter
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Correspondence to Raphael Winter.

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Communicated by H. Spohn

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Velázquez, J.J.L., Winter, R. From a Non-Markovian System to the Landau Equation. Commun. Math. Phys. 361, 239–287 (2018). https://doi.org/10.1007/s00220-018-3092-1

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