Skip to main content
SpringerLink
Log in

Kac’s program in kinetic theory

  • Published:
Inventiones mathematicae Aims and scope

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Kac in fact called this notion “Boltzmann’s property” in [42] as a clear tribute to the fundamental intuition of Boltzmann.

  2. We refer to [20] for a bibliographic discussion, see also [10] where [54] is quoted as the first paper proving this result.

  3. I.e. invariant according to permutations of the particles.

  4. By extensivity we mean here that the functional measuring the distance between two distributions should behave additively with respect to the tensor product.

References

  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. 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)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  5. Barthe, F., Cordero-Erausquin, D., Maurey, B.: Entropy of spherical marginals and related inequalities. J. Math. Pures Appl. 86(2), 89–99 (2006)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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. Carlen, E., Carvalho, M.C., Loss, M.: Spectral gap for the Kac model with hard collisions. Work in progress (personal communication)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carrapatoso, K.: Quantitative and qualitative Kac’s chaos on the Boltzmann sphere. hal-00694767

  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)

    MathSciNet  Google Scholar 

  16. Cercignani, C.: On the Boltzmann equation for rigid spheres. Transp. Theory Stat. Phys. 2(3), 211–225 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cercignani, C.: The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol. 67. Springer, New York (1988)

    Book  MATH  Google Scholar 

  18. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York (1994)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dobrić, V., Yukich, J.E.: Asymptotics for transportation cost in high dimensions. J. Theor. Probab. 8(1), 97–118 (1995)

    Article  MATH  Google Scholar 

  23. Einav, A.: On Villani’s conjecture concerning entropy production for the Kac master equation. Kinet. Relat. Models 4(2), 479–497 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Einav, A.: A counter example to Cercignani’s conjecture for the d dimensional Kac model. arXiv:1204.6031v1 (2012)

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  28. Escobedo, M., Mischler, S.: Scalings for a ballistic aggregation equation. J. Stat. Phys. 141(3), 422–458 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fournier, N., Méléard, S.: Monte Carlo approximations and fluctuations for 2d Boltzmann equations without cutoff. Markov Process. Relat. Fields 7, 159–191 (2001)

    MATH  Google Scholar 

  30. Fournier, N., Méléard, S.: A stochastic particle numerical method for 3d Boltzmann equation without cutoff. Math. Comput. 71, 583–604 (2002)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  34. Graham, C., Méléard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25, 115–132 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. Grünbaum, F.A.: Propagation of chaos for the Boltzmann equation. Arch. Ration. Mech. Anal. 42, 323–345 (1971)

    Article  MATH  Google Scholar 

  36. Hauray, M., Mischler, S.: On Kac’s chaos and related problems. hal-00682782

  37. Hewitt, E., Savage, L.J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Janvresse, E.: Spectral gap for Kac’s model of Boltzmann equation. Ann. Probab. 29(1), 288–304 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Kolokoltsov, V.N.: Nonlinear Markov Processes and Kinetic Equations. Cambridge Tracts in Mathematics, vol. 182. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  46. Lions, P.-L.: Théorie des jeux de champ moyen et applications (mean field games). In: Cours du Collège de France. http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp, 2007–2009

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  50. Maslen, D.K.: The eigenvalues of Kac’s master equation. Math. Z. 243(2), 291–331 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  51. Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 157, 49–88 (1867)

    Article  Google Scholar 

  52. McKean, H.P.: Fluctuations in the kinetic theory of gases. Commun. Pure Appl. Math. 28(4), 435–455 (1975)

    Article  MathSciNet  Google Scholar 

  53. McKean, H.P. Jr.: An exponential formula for solving Boltzmann’s equation for a Maxwellian gas. J. Comb. Theory 2, 358–382 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  56. Mischler, S.: Sur le Programme de Kac (concernant les Limites de Champ Moyen). Séminaire EDP-X, Décembre 2010, Preprint arXiv. http://hal.archives-ouvertes.fr/hal-00726384

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16(4), 467–501 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  61. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  63. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    Book  MATH  Google Scholar 

  64. Peyre, R.: Some ideas about quantitative convergence of collision models to their mean field limit. J. Stat. Phys. 136(6), 1105–1130 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  65. Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, vol. II. Probability and Its Applications. Springer, New York (1998)

    MATH  Google Scholar 

  66. Spohn, H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3(4), 445–455 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  67. Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monograph in Physics. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  68. Sznitman, A.-S.: Équations de type de Boltzmann, spatialement homogènes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 66(4), 559–592 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  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. Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrscheinlichkeitstheor. Verw. Geb. 46(1), 67–105 (1978/79)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  73. Villani, C.: Limite de champ moyen. Cours de DEA, 2001–2002, ÉNS, Lyon

  74. Villani, C.: Fisher information estimates for Boltzmann’s collision operator. J. Math. Pures Appl. 77(8), 821–837 (1998)

    MathSciNet  MATH  Google Scholar 

  75. Villani, C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234(3), 455–490 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  76. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics Series, vol. 58. Am. Math. Soc., Providence (2003)

    MATH  Google Scholar 

  77. Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  78. Wild, E.: On Boltzmann’s equation in the kinetic theory of gases. Proc. Camb. Philos. Soc. 47, 602–609 (1951)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Stéphane Mischler

  2. DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Clément Mouhot

Authors
  1. Stéphane Mischler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Clément Mouhot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clément Mouhot.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mischler, S., Mouhot, C. Kac’s program in kinetic theory. Invent. math. 193, 1–147 (2013). https://doi.org/10.1007/s00222-012-0422-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-012-0422-3

Keywords

Mathematics Subject Classification

Search

Navigation