Advertisement

Communications in Mathematical Physics

, Volume 154, Issue 3, pp 569–601 | Cite as

Steady-state electrical conduction in the periodic Lorentz gas

  • N. I. Chernov
  • G. L. Eyink
  • J. L. Lebowitz
  • Ya. G. Sinai
Article

Abstract

We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a “thermostat” constructed according to Gauss’ principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohm's transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimensiond=2, the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions.

Keywords

Lyapunov Exponent Entropy Production Ergodic Measure Linear Response Theory Markov Partition 
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.
    Abramov, L.M.: On the Entropy of a Flow. Dokl. Akad. Nauk SSSR128, 873–875 (1959)Google Scholar
  2. 2.
    Bunimovich, L.A., Sinai, Ya.G.: Markov Partitions for Dispersed Billiards. Commun. Math. Phys.73, 247–280 (1980)CrossRefGoogle Scholar
  3. 3.
    Bunimovich, L.A., Sinai, Ya.G.: Statistical Properties of Lorentz Gas with Periodic Configuration of Scatterers. Commun. Math. Phys.78, 479–497 (1981)Google Scholar
  4. 4.
    Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Markov Partitions for Two-Dimensional Hyperbolic Billiards. Russ. Math. Surv.45, 105–152 (1990)Google Scholar
  5. 5.
    Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Statistical Properties of Two-Dimensional Hyperbolic Billiards. Russ. Math. Surv.46, 47–106 (1991)Google Scholar
  6. 6.
    Bunimovich, L.A.: A Theorem on Ergodicity of Two-Dimensional Hyperbolic Billiards. Commun. Math. Phys.130, 599–621 (1990)CrossRefGoogle Scholar
  7. 7.
    Chernov, N.I.: The Ergodicity of a Hamiltonian System of Two Particles in an External Field. Physica D53, 233–239 (1991)CrossRefGoogle Scholar
  8. 8.
    Chernov, N.I.: Ergodic and Statistical Properties of Piecewise Linear Hyperbolic Automorphisms of the 2-Tours. J. Stat. Phys.69, 111–134 (1992)CrossRefGoogle Scholar
  9. 9.
    Chernov, N.I.: Statistical Properties of the Periodic Lorentz Gas: Multidimensional Case. In preparationGoogle Scholar
  10. 10.
    Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Berlin, Heidelberg, New York: Springer 1982Google Scholar
  11. 11.
    Donnay, V., Liverani, C.: Potentials on the Two-Torus for Which the Hamiltonian Flow is Ergodic. Commun. Math. Phys.135, 267–302 (1991)CrossRefGoogle Scholar
  12. 12.
    Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Liquids. San Diego, CA: Academic Press 1990Google Scholar
  13. 13.
    Eyink, G.L., Lebowit, J.L., Spohn, H.: Microscopic Origin of Hydrodynamic Behavior: Entropy Production and the Steady State. Chaos/Xaoc Soviet-American Perspectives on Nonlinear Science. New York: American Institute of Physics, 1990, pp. 367–391Google Scholar
  14. 14.
    Eyink, G.L., Lebowitz, J.L., Spohn, H.: Hydrodynamics of Stationary Non-Equilibrium States for Some Stochastic Lattice Gas Models. Commun. Math. Phys.132, 253–283 (1990)Google Scholar
  15. 15.
    Gallavotti, G., Ornstein, D.: Billiards and Bernoulli schemes. Commun. Math. Phys.38, 83–101 (1974)CrossRefGoogle Scholar
  16. 16.
    Gauss, K.F.: Uber ein neues allgemeines Grundgesetz der Mechanik. J. Reine Angew. Math.IV, 232–235 (1829)Google Scholar
  17. 17.
    Goldstein, S., Kipnis, C., Ianiro, N.: Stationary States for a Mechanical System with Stochastic Boundary Conditions. J. Stat. Phys.41, 915–939 (1985)CrossRefGoogle Scholar
  18. 18.
    Goldstein, S., Lebowitz, J.L., Presutti, E.: Mechanical Systems with Stochastic Boundaries. Colloquia Mathematicae Societatis Janos Bolyai27, Random Fields. Amsterdam: North-Holland 1981Google Scholar
  19. 19.
    de Groot, S., Masur, P.: Nonequilibrium Thermodynamics. Amsterdam: North-Holland 1962Google Scholar
  20. 20.
    Hoover, W.G.: Computational Statistical Mechanics. Amsterdam; Elsevier 1991Google Scholar
  21. 21.
    Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Gröningen: Wolters-Noordhoff 1971Google Scholar
  22. 22.
    van Kampen, N.: The Case Against Linear Response Theory. Physica Norvegica5, 279–284 (1971)Google Scholar
  23. 23.
    Katok, A., Strelcyn, J.-M.: Invariant Manifolds, Entropy, and Billiards; Smooth Maps with Singularities. Lecture Notes in Mathematics, vol.1222, New York: Springer 1986Google Scholar
  24. 24.
    Katz, S., Lebowitz, J.L., Spohn, H.: Nonequilibrium Steady States of Stochastic Lattice Gas Models of Fast Ionic Conductors. J. Stat. Phys.34, 497–537 (1984)CrossRefGoogle Scholar
  25. 25.
    Krámli, A., Simányi, N., Száss, D.: A “Transversal” Fundamental Theorem for Semi-Dispersing Billiards. Commun. Math. Phys.129, 535–560 (1990)CrossRefGoogle Scholar
  26. 26.
    Kubo, R.: Statistical Mechanical Theory of Irreversible Processes. I. J. Phys. Soc. Jap.12, 570–586 (1957)Google Scholar
  27. 27.
    Lebowitz, J.L.: Stationary Nonequilibrium Gibbsian Ensembles. Phys. Rev.114, 1192–1202 (1959)CrossRefGoogle Scholar
  28. 28.
    Lebowitz, J.L., Bergmann, P.G.: Irreversible Gibbsian Ensembles. Ann. Phys.1, 1–23 (1957)CrossRefGoogle Scholar
  29. 29.
    McLennan, J.A. Jr.: Statistical Mechanics of the Steady State. Phys. Rev.115, 1405–1409 (1959)CrossRefGoogle Scholar
  30. 30.
    Moran, B., Hoover, W.: Diffusion in a Periodic Lorentz Gas. J. Stat. Phys.48, 709–726 (1987)CrossRefGoogle Scholar
  31. 31.
    Morris, G.P., Evans, D.J., Cohen, E.G.D., van Beijeren, H.: Phys. Rev. Lett.62, 1579 (1989)CrossRefGoogle Scholar
  32. 32.
    Ornstein, D.S., Weiss, B.: Statistical Properties of Chaotic Systems. Bull. Am. Math. Soc.24, 11–116 (1991)Google Scholar
  33. 33.
    Ruelle, D.: Thermodynamic Formalism. New York: Addison-Wesley 1978Google Scholar
  34. 34.
    Sinai, Ya.G.: Dynamical Systems with Elastic Reflections. Ergodic Properties of Dispersing Billiards. Russ. Math. Surv.25, 137–189 (1970)Google Scholar
  35. 35.
    Sinai, Ya.G., Chernov, N.I.: Ergodic Properties of some Systems of 2-Dimensional Discs and 3-Dimensional Spheres. Russ. Math. Surv.42, 181–207 (1987)Google Scholar
  36. 36.
    Sinai, Ya.G.: Hyperbolic Billiards. Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990Google Scholar
  37. 37.
    Toda, M., Kubo, R., Hashitume, N.: Statistical Physics II. Non-equilibrium Statistical Mechanics. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  38. 38.
    Vaienti, S.: Ergodic Properties of the Discontinuous Sawtooth Map. J. Statist. Phys.67 (1992) (to appear)Google Scholar
  39. 39.
    Vul, E.B., Sinai, Ya.G., Khanin, K.M.: Feigenbaum Universality and Thermodynamic Formalism. Russ. Math. Surv.39, 1–40 (1984)Google Scholar
  40. 40.
    Wojtkowski, M.: Principles for the Design of Billiards with Nonvanishing Lyapunov Exponents. Commun. Math. Phys.105, 391–414 (1986)CrossRefGoogle Scholar
  41. 41.
    Yamada, T., Kawasaki, K.: Nonlinear Effects in the Shear Viscosity of a Critical Mixture. Prog. Theor. Phys.38, 1031–1051 (1967)Google Scholar
  42. 42.
    Young L.-S.: Bowen-Ruelle Measures for Certain Piecewise Hyperbolic Maps. Trans. Am. Math. Soc.281, 41–48 (1985)Google Scholar
  43. 43.
    Young, L.-S.: Dimension, Entropy and Lyapunov Exponents. Erg. Th. and Dyn. Syst.2, 109–124 (1982)Google Scholar
  44. 44.
    Zubarev, D.N.: The Statistical Operator for Nonequilibrium Systems. Sov. Phys. Dokl.6, 776–778 (1962)Google Scholar
  45. 45.
    Zubarev, D.N.: Nonequilibrium Statistical Thermodynamics. New York: Consultants 1974.Google Scholar
  46. 46.
    Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G., Derivation of Ohm's Law in a Deterministic Mechanical Model. Submitted to Phys. Ref. Let.Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • N. I. Chernov
    • 1
    • 2
  • G. L. Eyink
    • 3
  • J. L. Lebowitz
    • 3
  • Ya. G. Sinai
    • 4
  1. 1.Center for Dynamical Systems and Nonlinear StudiesGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Joint Institute for Nuclear Research, DubnaMoscowRussia
  3. 3.Departments of Mathematics and PhysicsRutgers UniversityNew BrunswickUSA
  4. 4.L.D. Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscow V-334Russia

Personalised recommendations