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Recent Results on the Periodic Lorentz Gas

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Nonlinear Partial Differential Equations

Part of the book series: Advanced Courses in Mathematics - CRM Barcelona ((ACMBIRK))

Abstract

The kinetic theory of gases was proposed by J. Clerk Maxwell [34, 35] and L. Boltzmann [5] in the second half of the XIXth century. Because the existence of atoms, on which kinetic theory rested, remained controversial for some time, it was not until many years later, in the XXth century, that the tools of kinetic theory became of common use in various branches of physics such as neutron transport, radiative transfer, plasma and semiconductor physics, etc.

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Bibliography

  1. S. Blank, N. Krikorian, Thom’s problem on irrational flows. Internat. J. Math. 4 (1993), 721–726.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Bleher, Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J. Statist. Phys. 66 (1992), 315–373.

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Boca, A. Zaharescu, The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit. Commun. Math. Phys. 269 (2007), 425–471.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Boldrighini, L. A. Bunimovich, Ya. G. Sinai, On the Boltzmann equation for the Lorentz gas. J. Statist. Phys. 32 (1983), 477–501.

    Google Scholar 

  5. L. Boltzmann, Weitere Studien ¨uber das W¨armegleichgewicht unter Gasmolek ¨ulen. Wiener Berichte 66 (1872), 275–370.

    Google Scholar 

  6. J. Bourgain, F. Golse, B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas. Commun. Math. Phys. 190 (1998), 491–508.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Bunimovich, Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78 (1980/81), 479–497.

    Google Scholar 

  8. L. Bunimovich, N. Chernov, Ya. G. Sinai, Statistical properties of twodimensional hyperbolic billiards. Russian Math. Surveys 46 (1991), 47–106.

    Google Scholar 

  9. E. Caglioti, F. Golse, On the distribution of free path lengths for the periodic Lorentz gas III. Commun. Math. Phys. 236 (2003), 199–221.

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Caglioti, F. Golse, The Boltzmann–Grad limit of the periodic Lorentz gas in two space dimensions. C. R. Math. Acad. Sci. Paris 346 (2008), 477–482.

    MathSciNet  MATH  Google Scholar 

  11. E. Caglioti, F. Golse, On the Boltzmann-Grad Limit for the Two Dimensional Periodic Lorentz Gas. J. Stat. Phys. 141 (2010), 264–317.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Cercignani, On the Boltzmann equation for rigid spheres. Transport Theory Statist. Phys. 2 (1972), 211–225.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Cercignani, R. Illner, M. Pulvirenti, The mathematical theory of dilute gases. Applied Mathematical Sciences, 106. Springer-Verlag, New York, 1994.

    Google Scholar 

  14. P. Dahlqvist, The Lyapunov exponent in the Sinai billiard in the small scatterer limit. Nonlinearity 10 (1997), 159–173.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Desvillettes, V. Ricci, Nonmarkovianity of the Boltzmann–Grad limit of a system of random obstacles in a given force field. Bull. Sci. Math. 128 (2004), 39–46.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Drude, Zur Elektronentheorie der Metalle. Annalen der Physik 306 (3) (1900), 566–613.

    Article  Google Scholar 

  17. H. S. Dumas, L. Dumas, F. Golse, Remarks on the notion of mean free path for a periodic array of spherical obstacles. J. Statist. Phys. 87 (1997), 943–950.

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Gallavotti, Divergences and approach to equilibrium in the Lorentz and the wind–tree–models. Phys. Rev. (2) 185 (1969), 308–322.

    Google Scholar 

  19. G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota interna no. 358, Istituto di Fisica, Univ. di Roma (1972). Available as preprint mp-arc-93-304.

    Google Scholar 

  20. G. Gallavotti, Statistical Mechanics: a Short Treatise, Springer, Berlin- Heidelberg (1999).

    MATH  Google Scholar 

  21. F. Golse, On the statistics of free-path lengths for the periodic Lorentz gas. Proceedings of the XIVth International Congress on Mathematical Physics (Lisbon 2003), 439–446, World Scientific, Hackensack NJ, 2005.

    Google Scholar 

  22. F. Golse, The periodic Lorentz gas in the Boltzmann–Grad limit. Proceedings of the International Congress of Mathematicians, Madrid 2006, vol. 3, 183– 201, European Math. Soc., Z¨urich, 2006.

    Google Scholar 

  23. F. Golse, The periodic Lorentz gas in the Boltzmann–Grad limit (joint work with J. Bourgain, E. Caglioti and B. Wennberg). Oberwolfach Report

    Google Scholar 

  24. 54/2006, vol. 3 (2006), no. 4, 3214, European Math. Soc., Z¨urich, 2006.

    Google Scholar 

  25. F. Golse, On the periodic Lorentz gas in the Boltzmann–Grad scaling. Ann. Facult´e des Sci. Toulouse 17 (2008), 735–749.

    Google Scholar 

  26. F. Golse, B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas II. M2AN Mod´el. Math. et Anal. Num´er. 34 (2000), 1151– 1163.

    Google Scholar 

  27. H. Grad, Principles of the kinetic theory of gases, in: Handbuch der Physik, S. Fl¨ugge ed. Band XII, 205–294, Springer-Verlag, Berlin 1958.

    Google Scholar 

  28. R. Illner, M. Pulvirenti, Global validity of the Boltzmann equation for twoand three-dimensional rare gas in vacuum. Erratum and improved result: “Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum” [Commun. Math. Phys. 105 (1986), 189–203] and “Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum” [ibid. 113 (1987), 79–85] by M. Pulvirenti. Commun. Math. Phys. 121 (1989), 143– 146.

    Google Scholar 

  29. A. Ya. Khinchin, Continued Fractions. The University of Chicago Press, Chicago, Ill.-London, 1964.

    Google Scholar 

  30. O. E. Lanford III, Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974). Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, 1–111.

    Google Scholar 

  31. H. Lorentz, Le mouvement des ´electrons dans les m´etaux. Arch. N´eerl. 10 (1905), 336–371.

    Google Scholar 

  32. J. Marklof, A. Str¨ombergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. 172 (2010), 1949–2033.

    Google Scholar 

  33. J. Marklof, A. Str¨ombergsson, The Boltzmann–Grad limit of the periodic Lorentz gas. Ann. of Math. 174 (2011), 225–298.

    Google Scholar 

  34. J. Marklof, A. Str¨ombergsson, Kinetic transport in the two-dimensional periodic Lorentz gas. Nonlinearity 21 (2008), 1413–1422.

    Google Scholar 

  35. J. Clerk Maxwell, Illustration of the Dynamical Theory of Gases I & II. Philos. Magazine and J. of Science 19 (1860), 19–32 & 20 (1860) 21–37.

    Google Scholar 

  36. J. Clerk Maxwell, On the Dynamical Theory of Gases. Philos. Trans. R. Soc. London, 157 (1867), 49–88.

    Article  Google Scholar 

  37. H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conference Series in Mathematics, 84. American Mathematical Society, Providence, RI, 1994.

    Google Scholar 

  38. V. Ricci, B. Wennberg, On the derivation of a linear Boltzmann equation from a periodic lattice gas. Stochastic Process. Appl. 111 (2004), 281–315.

    Article  MathSciNet  MATH  Google Scholar 

  39. L. A. Santal´o, Sobre la distribuci´on probable de corp´usculos en un cuerpo, deducida de la distribuci´on en sus secciones y problemas an´alogos. Revista Uni´on Mat. Argentina 9 (1943), 145–164.

    Google Scholar 

  40. C. L. Siegel, ¨Uber Gitterpunkte in convexen K¨orpern und ein damit zusammenh ¨angendes Extremalproblem. Acta Math. 65 (1935), 307–323.

    Article  MathSciNet  Google Scholar 

  41. Ya. G. Sinai, Topics in Ergodic Theory. Princeton Mathematical Series, 44. Princeton University Press, Princeton, NJ, 1994.

    Google Scholar 

  42. V. T. S´os, On the distribution mod. 1 of the sequence . Ann. Univ. Sci. Univ. Budapest. E¨otv¨os Math. 1 (1958), 127–134.

    Google Scholar 

  43. J. Suranyi, ¨Uber die Anordnung der Vielfahren einer reelen Zahl mod. 1. Ann. Univ. Sci. Univ. Budapest. E¨otv¨os Math. 1 (1958), 107–111.

    Google Scholar 

  44. H. Spohn, The Lorentz process converges to a random flight process. Commun. Math. Phys. 60 (1978), 277–290.

    Article  MathSciNet  MATH  Google Scholar 

  45. D. Szasz, Hard Ball Systems and the Lorentz Gas. Encyclopaedia of Mathematical Sciences, 101. Springer, 2000.

    Google Scholar 

  46. S. Ukai, N. Point, H. Ghidouche, Sur la solution globale du probl`eme mixte de l’´equation de Boltzmann non lin´eaire. J. Math. Pures Appl. (9) 57 (1978), 203–229.

    Google Scholar 

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Golse, F. (2012). Recent Results on the Periodic Lorentz Gas. In: Nonlinear Partial Differential Equations. Advanced Courses in Mathematics - CRM Barcelona. Springer, Basel. https://doi.org/10.1007/978-3-0348-0191-1_2

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