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Entropy Production and Convergence to Equilibrium

  • Chapter
Entropy Methods for the Boltzmann Equation

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1916))

Abstract

This set of notes was used to complement my short course on the convergence to equilibrium for the Boltzmann equation, given at Institut Henri Poincaré in November\2-December 2001, as part of the Hydrodynamic limits program organized by Stefano Olla and François Golse. The informal style is in accordance with the fact that this is neither a reference book nor a research paper. The reader can use my review paper, A review of mathematical topics in collisional kinetic theory, as a reference source to dissipate any ambiguity with respect to notation for instance. Apart from minor corrections here and there, the main changes with respect to the original version of the notes were the addition of a final section to present some more recent developments and open directions, and the change of the sign convention for the entropy, to agree with physical tradition. Irene Mazzella is warmly thanked for kindly typesetting a preliminary version of this manuscript.

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References

  1. Alexandre, R. and Villani, C. (2004). On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21(1):61-95.

    MATH  MathSciNet  Google Scholar 

  2. Ané, C., Blachère, S., Chafaı, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., and Scheffer, G. (2000). Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris. With a preface by Dominique Bakry and Michel Ledoux.

    Google Scholar 

  3. Arkeryd, L. (1983). L∞ estimates for the space-homogeneous Boltzmann equation. J. Statist. Phys., 31(2):347-361.

    Article  MATH  MathSciNet  Google Scholar 

  4. Arkeryd, L. (1988). Stability in L1 for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 103(2):151-167.

    Article  MATH  MathSciNet  Google Scholar 

  5. Ball, K., Barthe, F., and Naor, A. (2003). Entropy jumps in the presence of a spectral gap. Duke Math. J., 119(1):41-63.

    MATH  MathSciNet  Google Scholar 

  6. Baranger, C. and Mouhot, C. (2005). Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Mat. Iberoamericana, 21 (3):819-841.

    MATH  MathSciNet  Google Scholar 

  7. Blachman, N. M. (1965). The convolution inequality for entropy powers. IEEE Trans. Inform. Theory, 2:267-271.

    Article  MathSciNet  Google Scholar 

  8. Bobylev, A. V. (1988). The theory of the nonlinear, spatially uniform Boltzmann equation for Maxwellian molecules. Sov. Sci. Rev. C. Math. Phys., 7:111-233.

    MathSciNet  Google Scholar 

  9. Bobylev, A. V. and Cercignani, C. (1999). On the rate of entropy production for the Boltzmann equation. J. Statist. Phys., 94(3-4):603-618.

    Article  MATH  MathSciNet  Google Scholar 

  10. Bobylev, A. V. and Cercignani, C. (2002a). Exact eternal solutions of the Boltzmann equation. J. Statist. Phys., 106(5-6):1019-1038.

    Article  MATH  MathSciNet  Google Scholar 

  11. Bobylev, A. V. and Cercignani, C. (2002b). The inverse Laplace transform of some an-alytic functions with an application to the eternal solutions of the Boltzmann equation. Appl. Math. Lett., 15(7):807-813.

    Article  MATH  MathSciNet  Google Scholar 

  12. Bobylev, A. V. and Cercignani, C. (2002c). Self-similar solutions of the Boltzmann equation for non-Maxwell molecules. J. Statist. Phys., 108(3-4):713-717.

    Article  MATH  MathSciNet  Google Scholar 

  13. Bonami, A. (1970)Étude des coefficients de Fourier des fonctions de Lp (G). Ann. Inst. Fourier (Grenoble), 20(fasc. 2):335-402 (1971).

    MATH  MathSciNet  Google Scholar 

  14. Caflisch, R. (1980). The Boltzmann equation with a soft potential. Comm. Math. Phys., 74:71-109.

    Article  MATH  MathSciNet  Google Scholar 

  15. Caputo, P. (2003). Uniform Poincaré inequalities for unbounded conservative systems: the non-interacting case. Stochastic Process. Appl. 106, no.2, pp. 223-244.

    MATH  MathSciNet  Google Scholar 

  16. Carleman, T. (1932). Sur la théorie de l’equation intégrodifférentielle de Boltzmann. Acta Math., 60:369-424.

    Google Scholar 

  17. Carlen, E. A. (1991a). Some integral identities and inequalities for entire functions and their application to the coherent state transform. J. Funct. Anal., 97(1):231-249.

    Article  MATH  MathSciNet  Google Scholar 

  18. Carlen, E. A. (1991b). Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. Funct. Anal., 101(1):194-211.

    Article  MATH  MathSciNet  Google Scholar 

  19. Carlen, E. A. and Carvalho, M. C. (1992). Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Statist. Phys., 67(3-4):575-608.

    Article  MATH  MathSciNet  Google Scholar 

  20. Carlen, E. A. and Carvalho, M. C. (1994). Entropy production estimates for Boltzmann equations with physically realistic collision kernels. 74(3-4):743-782.

    MATH  MathSciNet  Google Scholar 

  21. Carlen, E. A., Carvalho, M. C., and Gabetta, E. (2000). Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. Comm. Pure Appl. Math., 53(3):370-397.

    Article  MATH  MathSciNet  Google Scholar 

  22. Carlen, E. A., Carvalho, M. C., and Loss, M. (2001). Many-body aspects of approach to equilibrium. In Séminaire: E´quations aux Dérivées Partielles, 2000-2001, Sémin. É qu. Dériv. Partielles, pages Exp. No. XIX, 12. École Polytech., Palaiseau.

    Google Scholar 

  23. Carlen, E. A., Carvalho, M. C., and Loss, M. (2003). Determination of the spectral gap for Kac’s master equation and related stochastic evolution. Acta Math., 191(1):1-54.

    Article  MATH  MathSciNet  Google Scholar 

  24. Carlen, E. A., Carvalho, M. C., and Wennberg, B. (1997). Entropic convergence for solutions of the Boltzmann equation with general physical initial data. Transport Theory Statist. Phys., 26(3):373-378.

    MATH  MathSciNet  Google Scholar 

  25. Carlen, E. A., Esposito, R., Lebowitz, J. L., Marra, R., and Rokhlenko, A. (1998). Kinetics of a model weakly ionized plasma in the presence of multiple equilibria. Arch. Rational Mech. Anal., 142(3):193-218.

    Article  MATH  MathSciNet  Google Scholar 

  26. Carlen, E. A., Ester, G., and Regazzini, E. (2007). On the rate of explosion for infinite energy solutions of the spatially homogeneous Boltzmann equation. Preprint.

    Google Scholar 

  27. Carlen, E. A., Gabetta, E., and Toscani, G. (1999). Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Comm. Math. Phys., 199(3):521-546.

    Article  MATH  MathSciNet  Google Scholar 

  28. Carlen, E. A. and Lu, X. (2003). Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums. J. Statist. Phys., 112(1-2):59-134.

    Article  MATH  MathSciNet  Google Scholar 

  29. Carlen, E. A. and Soffer, A. (1991). Entropy production by block variable summation and central limit theorems. Comm. Math. Phys., 140:339-371.

    Article  MATH  MathSciNet  Google Scholar 

  30. Carrillo, J. A., McCann, R. J., and Villani, C. (2003). Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana, 19(3):971-1018.

    MATH  MathSciNet  Google Scholar 

  31. Cercignani, C. (1982). H -theorem and trend to equilibrium in the kinetic theory of gases. Arch. Mech., 34:231-241.

    MATH  MathSciNet  Google Scholar 

  32. Cover, T. M. and Thomas, J. A. (1991). Elements of information theory. Wiley Series in Telecommunications. John Wiley & Sons Inc., New York. A Wiley-Interscience Publication.

    Google Scholar 

  33. Dembo, A. (1989). Simple proof of the concavity of the entropy power with respect to added Gaussian noise. IEEE Trans. Inform. Theory, 35(4):887-888.

    Article  MathSciNet  Google Scholar 

  34. Dembo, A., Cover, T., and Thomas, J. (1991). Information theoretic inequalities. IEEE Trans. Inform. Theory, 37(6):1501-1518.

    Article  MATH  MathSciNet  Google Scholar 

  35. Desvillettes, L. (1989). Entropy dissipation rate and convergence in kinetic equations. Comm. Math. Phys., 123(4):687-702.

    Article  MATH  MathSciNet  Google Scholar 

  36. Desvillettes, L. (1990). Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Rational Mech. Anal., 110(1):73-91.

    Article  MATH  MathSciNet  Google Scholar 

  37. Desvillettes, L. (1993). Some applications of the method of moments for the homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 123(4):387-395.

    Article  MATH  MathSciNet  Google Scholar 

  38. Desvillettes, L. (2000). Convergence to the thermodynamical equilibrium. In G. Iooss,  O. G. and Nouri, A., editors, Trends in Applications of Mathematics to Mechanics, Monographs and Surveys in Pure and Applied Mathematics, pages 115-126. Chapman and Hall, Boca Raton.

    Google Scholar 

  39. Desvillettes, L. and Villani, C. (2000). On the spatially homogeneous Landau equation for hard potentials. II. H -theorem and applications. Comm. Partial Differential Equations, 25(1-2):261-298.

    Article  MATH  MathSciNet  Google Scholar 

  40. Desvillettes, L. and Villani, C. (2001). On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math., 54(1):1-42.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. Eckmann, J.-P. and Hairer, M. (2003). Spectral properties of hypoelliptic operators. Comm. Math. Phys., 235(2):233-253.

    Article  MATH  MathSciNet  Google Scholar 

  43. Fisher, R. A. (1925). Theory of statistical estimation. Math. Proc. Cambridge Philos. Soc., 22:700-725.

    Article  MATH  Google Scholar 

  44. Gabetta, E., Toscani, G., and Wennberg, B. (1995). Metrics for probability distribu-tions and the trend to equilibrium for solutions of the Boltzmann equation. J. Statist. Phys., 81:901-934.

    Article  MATH  MathSciNet  Google Scholar 

  45. Gallay, T. and Wayne, C. E. (2002). Invariant manifolds and the long-time asymptot- ics of the Navier-Stokes and vorticity equations on R2 . Arch. Ration. Mech. Anal., 163(3):209-258.

    Article  MATH  MathSciNet  Google Scholar 

  46. Gustafsson, T. (1986). Lp -estimates for the nonlinear spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 92(1):23-57.

    MATH  MathSciNet  Google Scholar 

  47. Gustafsson, T. (1988). Global Lp -properties for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 103:1-38.

    Article  MATH  MathSciNet  Google Scholar 

  48. Helffer, B. and Nier, F. (2005). Hypoelliptic estimates and spectral theory for FokkerPlanck operators and Witten Laplacians, volume 1862 of Lecture Notes in Mathematics. Springer-Verlag, Berlin.

    Google Scholar 

  49. Hérau, F. and Nier, F. (2004). Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal., 171(2):151-218.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  51. Kac, M. (1956). Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III, pages 171-197, Berkeley and Los Angeles. University of California Press.

    Google Scholar 

  52. Ledoux, M. (1995). L’algèbre de Lie des gradients itérés d’un générateur markovien développements de moyennes et d’entropies. Ann. Scient. Ec. norm. sup., 28:435-460.

    MATH  MathSciNet  Google Scholar 

  53. Lieb, E. (1982). Comment on : “Approach to Equilibrium of a Boltzmann-Equation Solution”. Phys. Rev. Lett., 48(15):1057.

    Article  MathSciNet  Google Scholar 

  54. Lieb, E. H. (1978). Proof of an entropy conjecture of Wehrl. Comm. Math. Phys., 62(1):35-41.

    Article  MATH  MathSciNet  Google Scholar 

  55. Linnik, Y. V. (1959). An information-theoretic proof of the central limit theorem with the Lindenberg condition. Theory Probab. Appl., 4:288-299.

    MathSciNet  Google Scholar 

  56. Lions, P.-L. (1994). Compactness in Boltzmann’s equation via Fourier integral oper-ators and applications, I. J. Math. Kyoto Univ., 34(2):391-427.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  58. Mattingly, J. C. and Stuart, A. M. (2002). Geometric ergodicity of some hypo-elliptic diffusions for particle motions. Markov Process. Related Fields, 8(2):199-214. Inhomogeneous random systems (Cergy-Pontoise, 2001).

    MATH  MathSciNet  Google Scholar 

  59. Mattingly, J. C., Stuart, A. M., and Higham, D. J. (2002). Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl., 101(2):185-232.

    MATH  MathSciNet  Google Scholar 

  60. McKean, H. J. (1966). Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Rational Mech. Anal., 21:343-367.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  62. Mouhot, C. (2006). Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Comm. Math. Phys., 261(3):629-672.

    Article  MATH  MathSciNet  Google Scholar 

  63. Mouhot, C. and Strain, R. (2006). Spectral gap and coercivity estimates for the linearized boltzmann collision operator without angular cutoff. Preprint.

    Google Scholar 

  64. Mouhot, C. and Villani, C. (2004). Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Ration. Mech. Anal., 173(2):169-212.

    MATH  MathSciNet  Google Scholar 

  65. Olaussen, K. (1982). Extension of the Boltzmann H theorem. Phys. Rev. A, 25 (6):3393-3395.

    Article  MathSciNet  Google Scholar 

  66. Plastino, A. R. and Plastino, A. (1996). Symmetries of the fokker-planck equation and the fisher-frieden arrow time. Phys. Rev. E, 54:4423.

    Article  Google Scholar 

  67. Pulvirenti, A. and Wennberg, B. (1997). A Maxwellian lower bound for solutions to the Boltzmann equation. Comm. Math. Phys., 183:145-160.

    Article  MATH  MathSciNet  Google Scholar 

  68. Rey-Bellet, L. and Thomas, L. E. (2000). Asymptotic behavior of thermal nonequilib-rium steady states for a driven chain of anharmonic oscillators. Comm. Math. Phys., 215(1):1-24.

    Article  MATH  MathSciNet  Google Scholar 

  69. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Tech. J., 27:379-423, 623-656.

    MATH  MathSciNet  Google Scholar 

  70. Shannon, C. E. and Weaver, W. (1949). The Mathematical Theory of Communication. The University of Illinois Press, Urbana, Ill.

    MATH  Google Scholar 

  71. Stam, A. (1959). Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control, 2:101-112.

    Article  MATH  MathSciNet  Google Scholar 

  72. Talay, D. (2002). Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Related Fields, 8(2):163-198. Inhomogeneous random systems (Cergy-Pontoise, 2001).

    MATH  MathSciNet  Google Scholar 

  73. Tanaka, H. (1973). An inequality for a functional of probability distributions and its application to Kac’s one-dimensional model of a Maxwellian gas. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 27:47-52.

    Article  MATH  MathSciNet  Google Scholar 

  74. Tanaka, H. (1978). Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete, 46(1):67-105.

    Article  MATH  MathSciNet  Google Scholar 

  75. Toscani, G. (1992a). Lyapunov functionals for a Maxwell gas. Arch. Rational Mech. Anal., 119:301-307. NOTA: The argument used to apply the main result in this paper to the Fisher information (a.k.a. Linnik functional) is wrong in dimension 3. See Villani (1998) for a (hopefully) correct proof.

    Google Scholar 

  76. Toscani, G. (1992b). New a priori estimates for the spatially homogeneous Boltzmann equation. Cont. Mech. Thermodyn., 4:81-93.

    Article  MATH  MathSciNet  Google Scholar 

  77. Toscani, G. (1999). Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation. Quart. Appl. Math., 57(3):521-541.

    MATH  MathSciNet  Google Scholar 

  78. Toscani, G. and Villani, C. (1999a). Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Statist. Phys., 94(3-4):619-637.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  80. Villani, C. (1998). Fisher information bounds for Boltzmann’s collision operator. J. Math. Pures Appl., 77:821-837.

    MATH  MathSciNet  Google Scholar 

  81. Villani, C. (1999). Regularity estimates via the entropy dissipation for the spatially ho- mogeneous Boltzmann equation without cut-off. Rev. Mat. Iberoamericana, 15(2):335-352.

    MATH  MathSciNet  Google Scholar 

  82. Villani, C. (2000a). Contribution à l’étude mathématique des collisions en théorie cinétique. Master’s thesis, Univ. Paris-Dauphine, France.

    Google Scholar 

  83. Villani, C. (2000b). Decrease of the Fisher information for solutions of the spatially homogeneous Landau equation with Maxwellian molecules. Math. Models Methods Appl. Sci., 10(2):153-161.

    MATH  MathSciNet  Google Scholar 

  84. Villani, C. (2000c). A short proof of the “concavity of entropy power”. IEEE Trans. Inform. Theory, 46(4):1695-1696.

    Article  MATH  MathSciNet  Google Scholar 

  85. Villani, C. (2002a). Limites hydrodynamiques de l’équation de Boltzmann (d’après C. Bardos, F. Golse, C. D. Levermore, P.-L. Lions, N. Masmoudi, L. Saint-Raymond). Astérisque, (282):Exp. No. 893, ix, 365-405. Séminaire Bourbaki, Vol. 2000/2001.

    Google Scholar 

  86. Villani, C. (2002b). A review of mathematical topics in collisional kinetic theory. In Handbook of mathematical fluid dynamics, Vol. I, pages 71-305. North-Holland, Amsterdam.

    Chapter  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  88. Villani, C. (2006). Hypocoercivity. Preprint, available online via www.umpa.ens-lyon.fr/~cvillani.

  89. Wennberg, B. (1992). On an entropy dissipation inequality for the Boltzmann equation. C.R. Acad. Sci. Paris, Série I,, 315:1441-1446.

    MATH  MathSciNet  Google Scholar 

  90. Wennberg, B. (1993a). Stability and exponential convergence in Lp for the spatially homogeneous Boltzmann equation. Nonlinear Anal., 20(8):935-964.

    Article  MATH  MathSciNet  Google Scholar 

  91. Wennberg, B. (1993b). Stability and exponential convergence in Lp for the spatially homogeneous Boltzmann equation. Nonlinear Anal., 20(8):935-964.

    Article  MATH  MathSciNet  Google Scholar 

  92. Wennberg, B. (1994a). On moments and uniqueness for solutions to the space homogeneous Boltzmann equation. Transport Theory Statist. Phys., 24(4):533-539.

    MathSciNet  Google Scholar 

  93. Wennberg, B. (1994b). Regularity in the Boltzmann equation and the Radon transform. Comm. Partial Differential Equations, 19(11-12):2057-2074.

    Article  MATH  MathSciNet  Google Scholar 

  94. Wennberg, B. (1995). Stability and exponential convergence for the Boltzmann equation. Arch. Rational Mech. Anal., 130(2):103-144.

    Article  MATH  MathSciNet  Google Scholar 

  95. Wennberg, B. (1996). The Povzner inequality and moments in the Boltzmann equation. In Proceedings of the VIII International Conference on Waves and Stability in Continuous Media, Part II (Palermo, 1995), number 45, part II, pages 673-681.

    Google Scholar 

  96. Wennberg, B. (1997). Entropy dissipation and moment production for the Boltzmann equation. J. Statist. Phys., 86(5-6):1053-1066.

    Article  MATH  MathSciNet  Google Scholar 

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  1. ENS, Lyon, France

    C. Villani

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  1. Laboratoire Jacques-Louis Lions (UMR CNRS 7598), Université Pierre et Marie Curie, 175 rue du Chevaleret, 75013, Paris, France

    François Golse

  2. Centre de recherche en mathématiques de la décision (UMR CNRS 7534), Université Paris - Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Stefano Olla

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Villani, C. (2008). Entropy Production and Convergence to Equilibrium. In: Golse, F., Olla, S. (eds) Entropy Methods for the Boltzmann Equation. Lecture Notes in Mathematics, vol 1916. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73705-6_1

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