Lyapunov Exponents from Kinetic Theory for a Dilute, Field-driven Lorentz Gas

Abstract

Positive and negative Lyapunov exponents for a dilute, random, two-dimensional Lorentz gas in an applied field, $\vec{E}$, in a steady state at constant energy are computed to order $E^2$. The results are: ${\lambda }_{\pm }={\lambda }_{\pm }^0-a_{\pm }(qE/mv{)}^2{t}_0$ where ${\lambda }_{\pm }^0$ are the exponents for the field-free Lorentz gas, $a_{+}=11/48,a_{-}=7/48$, $t_0$ is the mean free time between collisions, $q$ is the charge, $m$ the mass and $v$ is the speed of the particle. The calculation is based on an extended Boltzmann equation in which a radius of curvature, characterizing the separation of two nearby trajectories, is one of the variables in the distribution function. The analytical results are in excellent agreement with computer simulations. These simulations provide additional evidence for logarithmic terms in the density expansion of the diffusion coefficient.

7 pages, revtex, 3 postscript figures

References in corpus (4)

• Steady-State Electrical Conduction in the Periodic Lorentz Gas
• Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients
• Lyapunov Exponents and KS Entropy for the Lorentz Gas at Low Densities
• Boltzmann and Statistical Mechanics

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• Transport properties in chaotic and non-chaotic many particles systems
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• B. B. G. K. Y. Hierarchy Methods for Sums of Lyapunov Exponents for Dilute Gases
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• Thermodynamic formalism for field driven Lorentz gases