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


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.


Lyapunov Exponent Entropy Production Ergodic Measure Linear Response Theory Markov Partition 
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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

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