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Derivation of an Ornstein–Uhlenbeck Process for a Massive Particle in a Rarified Gas of Particles

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Abstract

We consider the statistical motion of a convex rigid body in a gas of N smaller (spherical) atoms close to thermodynamic equilibrium. Because the rigid body is much bigger and heavier, it undergoes a lot of collisions leading to small deflections. We prove that its velocity is described, in a suitable limit, by an Ornstein–Uhlenbeck process. The strategy of proof relies on Lanford’s arguments (Lecture notes in physics, vol 38, Springer, New York, pp 1–111, 1975) together with the pruning procedure from Bodineau et al. (Invent Math 203(2):493–553, 2016) to reach diffusive times, much larger than the mean free time. Furthermore, we need to introduce a modified dynamics to avoid pathological collisions of atoms with the rigid body: these collisions, due to the geometry of the rigid body, require developing a new type of trajectory analysis.

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References

  1. Alexander, R.: The Infinite Hard Sphere System. Ph.D. dissertation, Dept. Mathematics, Univ. California, Berkeley (1975)

  2. Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  3. Bodineau, T., Gallagher, I., Saint-Raymond, L.: The Brownian motion as the limit of a deterministic system of hard-spheres. Invent Math. 203(2), 493–553 (2016)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  4. Bodineau, T., Gallagher, I., Saint-Raymond, L.: From hard sphere dynamics to the Stokes–Fourier equations: an \(L^2\) analysis of the Boltzmann-Grad limit. Ann. PDE 3, 2 (2017)

    Article  Google Scholar 

  5. Caprino, S., Marchioro, C., Pulvirenti, M.: Approach to equilibrium in a microscopic model of friction. Commun. Math. Phys. 264(1), 167–189 (2006)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  6. Cavallaro, G., Marchioro, C.: On the motion of an elastic body in a free gas. Rep. Math. Phys. 69(2), 251–264 (2012)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, New York (1994)

    Book  MATH  Google Scholar 

  8. Dobson, M., Legoll, F., Lelièvre, T., Stoltz, G.: Derivation of Langevin dynamics in a nonzero background flow field. ESAIM: M2AN 47(6), 1583–1626 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duplantier, B.: Brownian Motion, Diverse and Undulating. Birkhäuser, Basel (2005)

    Book  MATH  Google Scholar 

  10. Dürr, D., Goldstein, S., Lebowitz, J.L.: A mechanical model of Brownian motion. Commun. Math. Phys. 78(4), 507–530 (1981)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. Dürr, D., Goldstein, S., Lebowitz, J.L.: A mechanical model for the Brownian motion of a convex body. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 62(4), 427–448 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Durrett, R.: Stochastic Calculus. A Practical Introduction. Probability and Stochastics Series. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

  13. 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 (EMS), Zürich (2013)

  14. Holley, R.: The motion of a heavy particle in an infinite one dimensional gas of hard spheres. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 17, 181–219 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kusuoka, S., Liang, S.: A classical mechanical model of Brownian motion with plural particles. Rev. Math. Phys. 22, 733–838 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Lecture Notes in Physics, vol. 38, pp. 1–111. Springer, New York (1975)

    Google Scholar 

  17. Szász, D., Tóth, B.: Towards a unified dynamical theory of the Brownian particle in an ideal gas. Commun. Math. Phys. 111(1), 41–62 (1987)

    Article  MathSciNet  MATH  ADS  Google Scholar 

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Acknowledgements

We would like to thank S. Simonella for his very useful comments on the first version of this paper. We would like to thank F. Alouges for enlightening discussions on solid reflection laws. T.B. would like to thank the support of ANR-15-CE40-0020-02 Grant LSD.

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Authors and Affiliations

  1. CMAP, Ecole Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128, Palaiseau, France

    Thierry Bodineau

  2. DMA, École normale supérieure, CNRS, PSL Research University, 75005, Paris, France

    Isabelle Gallagher

  3. UFR de mathématiques, Université Paris-Diderot Sorbonne Paris-Cité, 75013, Paris, France

    Isabelle Gallagher

  4. UMPA, École normale supérieure de Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Laure Saint-Raymond

Authors
  1. Thierry Bodineau
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  2. Isabelle Gallagher
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  3. Laure Saint-Raymond
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Correspondence to Thierry Bodineau.

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Communicated by Christian Maes.

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Bodineau, T., Gallagher, I. & Saint-Raymond, L. Derivation of an Ornstein–Uhlenbeck Process for a Massive Particle in a Rarified Gas of Particles. Ann. Henri Poincaré 19, 1647–1709 (2018). https://doi.org/10.1007/s00023-018-0674-6

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  • DOI: https://doi.org/10.1007/s00023-018-0674-6

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