Inventiones mathematicae

, Volume 193, Issue 1, pp 1–147 | Cite as

Kac’s program in kinetic theory

  • Stéphane Mischler
  • Clément MouhotEmail author


This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguishable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean (Kac in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III, pp. 171–197, 1956; McKean in J. Comb. Theory 2:358–382, 1967) giving rise to the Boltzmann equation. We solve the conjecture raised by Kac (Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III, pp. 171–197, 1956), motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates.

Motivated by the inspirative paper by Grünbaum (Arch. Ration. Mech. Anal. 42:323–345, 1971), we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators (consistency estimates), along with differentiability estimates on the flow of the nonlinear limit equation (stability estimates). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac).

Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of entropic chaoticity, as defined by Carlen et al. (Kinet. Relat. Models 3:85–122, 2010). (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and true Maxwell molecules interactions. The proof of the stability estimates for these models requires significant analytic efforts and new estimates.


Kac’s program Kinetic theory Master equation Mean-field limit Quantitative Uniform in time Jump process Collision process Boltzmann equation Maxwell molecules Non-cutoff Hard spheres 

Mathematics Subject Classification

82C40 76P05 54C70 60J75 



We thank the Mathematics Department of Chalmers University for the invitation in November 2008, where the abstract method was devised and the related joint work [58] with Bernt Wennberg was initiated. We thank Ismaël Bailleul, Thierry Bodineau, François Bolley, Anne Boutet de Monvel, José Alfredo Cañizo, Eric Carlen, Nicolas Fournier, François Golse, Arnaud Guillin, Maxime Hauray, Joel Lebowitz, Pierre-Louis Lions, Richard Nickl, James Norris, Mario Pulvirenti, Judith Rousseau, Laure Saint-Raymond and Cédric Villani for fruitful comments and discussions, and Amit Einav for his careful proofreading of parts of the manuscript. We would also like to mention the inspiring courses by Pierre-Louis Lions at Collège de France on “Mean-Field Games” in 2007–2008 and 2008–2009, which triggered our interest in the functional analysis aspects of this topic. Finally we thank the anonymous referees for helpful suggestions on the presentation.


  1. 1.
    Alonso, R., Cañizo, J.A., Gamba, I., Mouhot, C.: A new approach to the creation and propagation of exponential moments in the Boltzmann equation. Commun. Part. Differ. Equ. (2012). doi: 10.1080/03605302.2012.715707 Google Scholar
  2. 2.
    Ambrosio, L., Gigliand, N., Savare, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zurich, vol. 2005. Birkhäuser, Basel (2005) zbMATHGoogle Scholar
  3. 3.
    Arkeryd, L., Caprino, S., Ianiro, N.: The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation. J. Stat. Phys. 63(1–2), 345–361 (1991) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bardos, C., Golse, F., Mauser, N.J.: Weak coupling limit of the N-particle Schrödinger equation. Methods Appl. Anal. 7(2), 275–293 (2000). Cathleen Morawetz: a great mathematician MathSciNetzbMATHGoogle Scholar
  5. 5.
    Barthe, F., Cordero-Erausquin, D., Maurey, B.: Entropy of spherical marginals and related inequalities. J. Math. Pures Appl. 86(2), 89–99 (2006) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bobylëv, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev., C, Math. Phys. Rev. 7, 111–233 (1988) zbMATHGoogle Scholar
  7. 7.
    Boltzmann, L.: Weitere studien über das wärmegleichgewicht unter gasmolekülen. Sitzungsber. Sächs. Akad. Wiss. Leipz., Math.-Nat. Wiss. Kl. 66, 275–370 (1872). Translation: Further studies on the thermal equilibrium of gas molecules, in Kinetic Theory, vol. 2, pp. 88–174, ed. S.G. Brush, Pergamon, Oxford (1966) zbMATHGoogle Scholar
  8. 8.
    Boltzmann, L.: Lectures on Gas Theory. University of California Press, Berkeley (1964). Translated by S.G. Brush. Reprint of the 1896–1898 Edition. Reprinted by Dover Publications, 1995 Google Scholar
  9. 9.
    Carlen, E., Carvalho, M.C., Loss, M.: Spectral gap for the Kac model with hard collisions. Work in progress (personal communication) Google Scholar
  10. 10.
    Carlen, E.A., Carvalho, M.C., Le Roux, J., Loss, M., Villani, C.: Entropy and chaos in the Kac model. Kinet. Relat. Models 3(1), 85–122 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Carlen, E.A., Carvalho, M.C., Loss, M.: Determination of the spectral gap for Kac’s master equation and related stochastic evolution. Acta Math. 191(1), 1–54 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Carlen, E.A., Gabetta, E., Toscani, G.: Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 199(3), 521–546 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Carlen, E.A., Geronimo, J.S., Loss, M.: Determination of the spectral gap in the Kac model for physical momentum and energy-conserving collisions. SIAM J. Math. Anal. 40(1), 327–364 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Carrapatoso, K.: Quantitative and qualitative Kac’s chaos on the Boltzmann sphere. hal-00694767 Google Scholar
  15. 15.
    Carrillo, J.A., Toscani, G.: Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma Ser. 7 6, 75–198 (2007) MathSciNetGoogle Scholar
  16. 16.
    Cercignani, C.: On the Boltzmann equation for rigid spheres. Transp. Theory Stat. Phys. 2(3), 211–225 (1972) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Cercignani, C.: The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol. 67. Springer, New York (1988) zbMATHCrossRefGoogle Scholar
  18. 18.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York (1994) zbMATHGoogle Scholar
  19. 19.
    Di Blasio, G.: Differentiability of spatially homogeneous solutions of the Boltzmann equation in the non Maxwellian case. Commun. Math. Phys. 38, 331–340 (1974) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Diaconis, P., Freedman, D.: A dozen de Finetti-style results in search of a theory. Ann. Inst. Henri Poincaré B, Probab. Stat. 23(2), 397–423 (1987) MathSciNetzbMATHGoogle Scholar
  21. 21.
    DiPerna, R.J., Lions, P.-L.: Global solutions of Boltzmann’s equation and the entropy inequality. Arch. Ration. Mech. Anal. 114(1), 47–55 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Dobrić, V., Yukich, J.E.: Asymptotics for transportation cost in high dimensions. J. Theor. Probab. 8(1), 97–118 (1995) zbMATHCrossRefGoogle Scholar
  23. 23.
    Einav, A.: On Villani’s conjecture concerning entropy production for the Kac master equation. Kinet. Relat. Models 4(2), 479–497 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Einav, A.: A counter example to Cercignani’s conjecture for the d dimensional Kac model. arXiv:1204.6031v1 (2012)
  25. 25.
    Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. Commun. Pure Appl. Math. 59(12), 1659–1741 (2006) CrossRefGoogle Scholar
  26. 26.
    Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167(3), 515–614 (2007) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross–Pitaevskii equation for the dynamics of Bose-Einstein condensate. Ann. Math. 172(1), 291–370 (2010) CrossRefGoogle Scholar
  28. 28.
    Escobedo, M., Mischler, S.: Scalings for a ballistic aggregation equation. J. Stat. Phys. 141(3), 422–458 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Fournier, N., Méléard, S.: Monte Carlo approximations and fluctuations for 2d Boltzmann equations without cutoff. Markov Process. Relat. Fields 7, 159–191 (2001) zbMATHGoogle Scholar
  30. 30.
    Fournier, N., Méléard, S.: A stochastic particle numerical method for 3d Boltzmann equation without cutoff. Math. Comput. 71, 583–604 (2002) zbMATHGoogle Scholar
  31. 31.
    Fournier, N., Mouhot, C.: On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Commun. Math. Phys. 283(3), 803–824 (2009) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Gabetta, G., Toscani, G., Wennberg, B.: Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81(5–6), 901–934 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Graham, C., Méléard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25, 115–132 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Grünbaum, F.A.: Propagation of chaos for the Boltzmann equation. Arch. Ration. Mech. Anal. 42, 323–345 (1971) zbMATHCrossRefGoogle Scholar
  36. 36.
    Hauray, M., Mischler, S.: On Kac’s chaos and related problems. hal-00682782 Google Scholar
  37. 37.
    Hewitt, E., Savage, L.J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Ikenberry, E., Truesdell, C.: On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory. I. J. Ration. Mech. Anal. 5, 1–54 (1956) MathSciNetzbMATHGoogle Scholar
  39. 39.
    Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum. Commun. Math. Phys. 105(2), 189–203 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Janvresse, E.: Spectral gap for Kac’s model of Boltzmann equation. Ann. Probab. 29(1), 288–304 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pp. 171–197. University of California Press, Berkeley (1956) Google Scholar
  43. 43.
    Kac, M.: Probability and Related Topics in Physical Sciences, Proceedings of the Summer Seminar, Boulder, CO. Lectures in Applied Mathematics, vol. 1957. Interscience, London (1959). With special lectures by G.E. Uhlenbeck, A.R. Hibbs, and B. van der Pol. Google Scholar
  44. 44.
    Kolokoltsov, V.N.: Nonlinear Markov Processes and Kinetic Equations. Cambridge Tracts in Mathematics, vol. 182. Cambridge University Press, Cambridge (2010) zbMATHGoogle Scholar
  45. 45.
    Lanford, O.E. III: Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications, Recontres, Battelle Res. Inst., Seattle, WA, 1974. Lecture Notes in Phys., vol. 38, pp. 1–111. Springer, Berlin (1975) CrossRefGoogle Scholar
  46. 46.
    Lions, P.-L.: Théorie des jeux de champ moyen et applications (mean field games). In: Cours du Collège de France., 2007–2009
  47. 47.
    Lu, X.: Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation. J. Stat. Phys. 96(3–4), 765–796 (1999) zbMATHCrossRefGoogle Scholar
  48. 48.
    Lu, X., Mouhot, C.: On measure solutions of the Boltzmann equation, part I: moment production and stability estimates. J. Differ. Equ. 252(4), 3305–3363 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Lu, X., Wennberg, B.: Solutions with increasing energy for the spatially homogeneous Boltzmann equation. Nonlinear Anal., Real World Appl. 3(2), 243–258 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Maslen, D.K.: The eigenvalues of Kac’s master equation. Math. Z. 243(2), 291–331 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 157, 49–88 (1867) CrossRefGoogle Scholar
  52. 52.
    McKean, H.P.: Fluctuations in the kinetic theory of gases. Commun. Pure Appl. Math. 28(4), 435–455 (1975) MathSciNetCrossRefGoogle Scholar
  53. 53.
    McKean, H.P. Jr.: An exponential formula for solving Boltzmann’s equation for a Maxwellian gas. J. Comb. Theory 2, 358–382 (1967) MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Mehler, F.G.: Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnungn. Crelle’s J. 1866, 161–176 (1966) CrossRefGoogle Scholar
  55. 55.
    Méléard, S.: Asymptotic behaviour of some interacting particle systems; McKean–Vlasov and Boltzmann models. In: Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995. Lecture Notes in Math., vol. 1627, pp. 42–95. Springer, Berlin (1996) CrossRefGoogle Scholar
  56. 56.
    Mischler, S.: Sur le Programme de Kac (concernant les Limites de Champ Moyen). Séminaire EDP-X, Décembre 2010, Preprint arXiv.
  57. 57.
    Mischler, S., Mouhot, C., Rodriguez Ricard, M.: Cooling process for inelastic Boltzmann equations for hard spheres. I. The Cauchy problem. J. Stat. Phys. 124(2–4), 655–702 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Mischler, S., Mouhot, C., Wennberg, M.: A new approach to quantitative propagation of chaos for drift, diffusion and jump processes. arXiv:1101.4727 (2011)
  59. 59.
    Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16(4), 467–501 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Mouhot, C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 261(3), 629–672 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983) zbMATHCrossRefGoogle Scholar
  64. 64.
    Peyre, R.: Some ideas about quantitative convergence of collision models to their mean field limit. J. Stat. Phys. 136(6), 1105–1130 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, vol. II. Probability and Its Applications. Springer, New York (1998) zbMATHGoogle Scholar
  66. 66.
    Spohn, H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3(4), 445–455 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monograph in Physics. Springer, Berlin (1991) zbMATHCrossRefGoogle Scholar
  68. 68.
    Sznitman, A.-S.: Équations de type de Boltzmann, spatialement homogènes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 66(4), 559–592 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Sznitman, A.-S.: Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math., vol. 1464, pp. 165–251. Springer, Berlin (1991) Google Scholar
  70. 70.
    Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrscheinlichkeitstheor. Verw. Geb. 46(1), 67–105 (1978/79) CrossRefGoogle Scholar
  71. 71.
    Tanaka, H.: Some probabilistic problems in the spatially homogeneous Boltzmann equation. In: Theory and Application of Random Fields, Bangalore, 1982. Lecture Notes in Control and Inform. Sci., vol. 49, pp. 258–267. Springer, Berlin (1983) CrossRefGoogle Scholar
  72. 72.
    Toscani, G., Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94(3–4), 619–637 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Villani, C.: Limite de champ moyen. Cours de DEA, 2001–2002, ÉNS, Lyon Google Scholar
  74. 74.
    Villani, C.: Fisher information estimates for Boltzmann’s collision operator. J. Math. Pures Appl. 77(8), 821–837 (1998) MathSciNetzbMATHGoogle Scholar
  75. 75.
    Villani, C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234(3), 455–490 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics Series, vol. 58. Am. Math. Soc., Providence (2003) zbMATHGoogle Scholar
  77. 77.
    Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009) zbMATHCrossRefGoogle Scholar
  78. 78.
    Wild, E.: On Boltzmann’s equation in the kinetic theory of gases. Proc. Camb. Philos. Soc. 47, 602–609 (1951) MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.CEREMADE, UMR CNRS 7534Université Paris-DauphineParis Cedex 16France
  2. 2.DPMMS, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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