Advertisement

Billiards Systems and the Transport Equation

  • François Golse
Part of the NATO ASI Series book series (NSSB, volume 320)

Abstract

Consider a system of like particles that do not interact between themselves but collide with a periodic array of strictly convex obstacles. This type of dynamical system is referred to as a dispersive billiards systems in this work. The most genuine example of such systems is the so-called Lorentz gas model of hard spheres7: the interaction between the particles and the obstacles is the pure (Descartes) reflection law. In large time, due to multiple collisions and the strict convexity of the obstacles, the distribution of the particles tends to forget its velocity dependence7. Bunimovich and Sinai7 have proved that the density of particles approaches in some sense the solution of a diffusion (heat) equation for large time and on a large space scale. Their proof relies on symbolic dynamics via the construction of a Markov partition8 , 9. The role of the strict convexity of the scatterers is reminiscent of the ergodic properties of the geodesic flow on compact manifolds with negative curvature: see Arnold 1 (although the explanation provided there is not fully rigorous). However, the method used by Bunimovich-Sinai7 does not provide the value of the diffusion coefficient, except through the use of the ubiquitous Kubo series. See also Gaspard-Nicolis11 for an interesting expression of the diffusion coefficient. None of these expressions clearly provides the dependence of the diffusion coefficient in terms of the geometry of the array of scatterers. Numerical experiments have been conducted by Machta-Zwanzig, as well as some asymptotics in the case where the distance between neighboring scatterers goes to zero (see Machtal4).

Keywords

Hard Sphere Diffusion Approximation Strict Convexity Collision Term Finite Horizon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Arnold: Mathematical Methods of Classical Mechanics; Springer Verlag.Google Scholar
  2. 2.
    C. Bardos, L. Dumas, F. Golse: Diffusion de particules par un réffiseau d’obstacles circulaires; to appear in C. R. Acad. Sci.Google Scholar
  3. 3.
    C. Bardos, R. Santos, R. Sentis: Diffusion Approximation and Computation of the Critical Size of a Transport Operator; Trans. of the A.M.S. 284, 617–649, (1984).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    A. Bensoussan, J.-L. Lions, G. Papanicolaou: Asymptotic Study of Periodic Structures; North Holland (1978).Google Scholar
  5. 5.
    A. Bensoussan, J.-L. Lions, G. Papanicolaou: Boundary Layers and Homogenization of Transport Processes; Publ. R.I.M.S. Kyoto 15, 53–115, (1979).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    C. Boldighrini, L. Bunimovich, Ya. Sinai: On the Boltzmann Equation for the Lorentz Gas 32, 477–502, (1983).Google Scholar
  7. 7.
    L. Bunimovich, Ya. Sinai: Statistical Properties of the Lorentz Gas with Periodic Configuration of Scatterers; Commun. Math. Phys. 78, 479–497, (1981).MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    L. Bunimovich, Ya. Sinai: Markov Partitions of Dispersed Billiards; Commun. Math. Phys. 73, 247–280, (1980).MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    L. Bunimovich, Ya. Sinai, N. Chernov: Markov Partitions for Two-Dimensional Hyperbolic Billiards; Russian. Math. Surveys 45, 105–152, (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    C. Cercignani: The Boltzmann Equation, Springer Verlag.Google Scholar
  11. 11.
    P. Gaspard, G. Nicolis: Transport Properties, Lyapunov Exponents, and Entropy per Unit Time; Phys. Rev; Letters 65, 1693–1696, (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    F. Golse: Transport dans les milieux composites fortement contrastés I: le modèle du billard; preprint C. E. A., Limeil (1991).Google Scholar
  13. 13.
    E. Larsen, G. Pomraning, V. Badham: Asymptotic Analysis of Radiative Transfer Problems; J. Quant. Spectrosc. and Radiative Transfer 29, 285–310, (1983).ADSCrossRefGoogle Scholar
  14. 14.
    J. Machta: Power Law Decay of Correlations in a Billiard Problem; J. of Stat. Phys. 32, 555–564, (1983).MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    H. Spohn: The Lorentz Process Converges to a Random Flight Process; Commun. in Math. Phys. 60, 277, (1978).MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • François Golse
    • 1
  1. 1.Département de MathématiquesUniversité Paris VIIParis Cédex 05France

Personalised recommendations