Inventiones mathematicae

, Volume 203, Issue 2, pp 493–553 | Cite as

The Brownian motion as the limit of a deterministic system of hard-spheres

  • Thierry Bodineau
  • Isabelle Gallagher
  • Laure Saint-Raymond


We provide a rigorous derivation of the Brownian motion as the limit of a deterministic system of hard-spheres as the number of particles \(N\) goes to infinity and their diameter \(\varepsilon \) simultaneously goes to \(0\), in the fast relaxation limit \(\alpha = N\varepsilon ^{d-1}\rightarrow \infty \) (with a suitable diffusive scaling of the observation time). As suggested by Hilbert in his sixth problem, we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions: we use indeed the linear Boltzmann equation to describe one tagged particle in a gas close to global equilibrium. Our proof is based on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.


  1. 1.
    Alexander, R.: The infinite hard sphere system, Ph.D. dissertation, Department of Mathematics, University of California, Berkeley (1975)Google Scholar
  2. 2.
    Alexander, R.: Time evolution for infinitely many hard spheres. Comm. Math. Phys. 49(3), 217–232 (1976)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bardos, C.: Problèmes aux limites pour les quations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport. Ann. Sci. École Norm. Sup. 4(3), 185–233 (1970)MathSciNetMATHGoogle Scholar
  4. 4.
    Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of the Boltzmann equation I. J. Stat. Phys. 63, 323–344 (1991)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of kinetic equations II: convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46, 667–753 (1993)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc 284(2), 617–649 (1984)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    van Beijeren, H., Lanford III, O.E., Lebowitz, J.L., Spohn, H.: Equilibrium time correlation functions in the low density limit. J. Stat. Phys. 22, 237–257 (1980)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Billingsley, P.: Probability and measure. Wiley series in probability and mathematical statistics. Wiley (1995)Google Scholar
  9. 9.
    Boldrighini, C., Bunimovich, L.A., Sinai, Y.: On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32(3), 477–501 (1983)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Bunimovich, L.A., Sinai, Y.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78, 479–497 (1980/81)Google Scholar
  11. 11.
    Caglioti, E., Golse, F.: On the distribution of free path lengths for the periodic Lorentz gas III. Commun. Math. Phys. 236, 199–221 (2003)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Caglioti, E., Golse, F.: On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas. J. Stat. Phys. 141, 264–317 (2010)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Springer, New York (1994)CrossRefMATHGoogle Scholar
  14. 14.
    De Roeck, W., Fröhlich, J.: Diffusion of a massive quantum particle coupled to a quasi-free thermal medium. Comm. Math. Phys. 303(3), 613–707 (2011)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Desvillettes, L., Golse, F.: A remark concerning the Chapman-Enskog asymptotics. In: Perthame, B. (ed.) Advances in kinetic theory and computing. Ser. Adv. Math. Appl. Sci., vol. 22, pp. 191–203. World Science Publishing, River Edge (1994)Google Scholar
  16. 16.
    Desvillettes, L., Pulvirenti, M.: The linear Boltzmann equation for long-range forces: a derivation from particles. Math. Meth. Mod. Appl. Sci. 9(8), 11231145 (1999)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    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)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Erdős, L.: Lecture notes on quantum Brownian motion, quantum theory from small to large scales: lecture notes of the Les Houches summer school, 95, Oxford University Press (2010)Google Scholar
  19. 19.
    Erdős, L., Salmhofer, M., Yau, H.T.: Quantum diffusion of the random Schrodinger evolution in the scaling limit. Acta Math. 200(2), 211–278 (2008)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Esposito, R., Marra, R., Yau, H.T.: Navier–Stokes equations for stochastic particle systems on the lattice. Comm. Math. Phys. 182, 395–456 (1996)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltzmann: the case of hard-spheres and short-range potentials, Zürich lectures in advanced mathematics 18. Erratum to Chapter 5 (2014)Google Scholar
  22. 22.
    Gallavotti, G.: Statistical mechanics. A short treatise. Texts and monographs in physics. Springer, Berlin (1999)MATHGoogle Scholar
  23. 23.
    Golse, F.: On the periodic Lorentz gas and the Lorentz kinetic equation. Ann. Fac. Sci. Toulouse Math. 17, 735–749 (2008)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Hilbert, D.: Begründung der kinetischen Gastheorie. (German). Math. Ann. 72(4), 562–577 (1912)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Kac, M.: Probability and related topics in physical sciences, Am. Math. Soc., p. 1 (1959)Google Scholar
  26. 26.
    King, F.: BBGKY hierarchy for positive potentials, Ph.D. dissertation, Dept. Mathematics, Univ. California, Berkeley (1975)Google Scholar
  27. 27.
    Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov processes. Time symmetry and martingale approximation. Grundlehren der Mathematischen Wissenschaften, vol. 345. Springer, Heidelberg (2012)CrossRefMATHGoogle Scholar
  28. 28.
    Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Lecture notes in physics, vol. 38, pp. 1–111, Springer (1975)Google Scholar
  29. 29.
    Lebowitz, J., Spohn, H.: Microscopic basis for Fick’s law for self-diffusion. J. Stat. Phys. 28, 539–556 (1982)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Lebowitz, J., Spohn, H.: Steady state self-diffusion at low density. J. Stat. Phys. 29, 39–55 (1982)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Lorentz, H.: Le mouvement des électrons dans les métaux. Arch. Neerl. 10, 336–371 (1905)MATHGoogle Scholar
  32. 32.
    Marklof, J.: Kinetic transport in crystals. In: Proceedings of the XVI International Congress on Mathematical Physics, Prague 2009, World Scientific, pp. 162–179 (2010)Google Scholar
  33. 33.
    Marklof, J., Strömbergsson, A.: The Boltzmann–Grad limit of the periodic Lorentz gas. Ann. Math. 174, 225–298 (2011)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Marklof, J., Toth, B.: Superdiffusion in the periodic Lorentz gas, arXiv:1403.6024, preprint (2014)
  35. 35.
    Pettersson, R.: On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation. J. Stat. Phys. 72, 355–380 (1993)CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    Olla, S., Varadhan, S., Yau, H.-T.: Hydrodynamical limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155, 523–560 (1993)CrossRefMathSciNetMATHGoogle Scholar
  37. 37.
    Pulvirenti, M., Saffirio, C., Simonella, S.: On the validity of the Boltzmann equation for short range potentials. Rev. Math. Phys. 26(2), 1450001 (2014)CrossRefMathSciNetMATHGoogle Scholar
  38. 38.
    Quastel, J., Yau, H.-T.: Lattice gases, large deviations, and the incompressible Navier–Stokes equations. Ann. Math. 148, 51–108 (1998)CrossRefMathSciNetMATHGoogle Scholar
  39. 39.
    Saint-Raymond, L.: Hydrodynamic limits of the Boltzmann equation. Lecture notes in mathematics, vol. 1971, Springer (2009)Google Scholar
  40. 40.
    Simonella, S.: Evolution of correlation functions in the hard sphere dynamics. J. Stat. Phys. 155(6), 1191–1221 (2014)CrossRefMathSciNetMATHGoogle Scholar
  41. 41.
    Spohn, H.: The Lorentz process converges to a random flight process. Commun. Math. Phys. 60, 277–290 (1978)CrossRefMathSciNetMATHGoogle Scholar
  42. 42.
    Spohn, H.: Large scale dynamics of interacting particles, vol. 174, Springer (1991)Google Scholar
  43. 43.
    Szasz, D., Toth, B.: Towards a unified dynamical theory of the Brownian particle in an ideal gas. Comm. Math. Phys. 111, 41–62 (1987)CrossRefMathSciNetMATHGoogle Scholar
  44. 44.
    Uchiyama, K.: Derivation of the Boltzmann equation from particle dynamics. Hiroshima Math. J. 18(2), 245–297 (1988)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Thierry Bodineau
    • 1
  • Isabelle Gallagher
    • 2
  • Laure Saint-Raymond
    • 3
  1. 1.CMAPEcole Polytechnique and CNRSPalaiseauFrance
  2. 2.IMJUniversité Paris DiderotParisFrance
  3. 3.DMAEcole Normale Supérieure and Université Pierre et Marie CurieParisFrance

Personalised recommendations