Positive and negative Lyapunov exponents for a dilute, random,two-dimensional Lorentz gas in an applied field, $\vec{E}$, in a steady stateat 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. Thecalculation is based on an extended Boltzmann equation in which a radius ofcurvature, characterizing the separation of two nearby trajectories, is one ofthe variables in the distribution function. The analytical results are inexcellent agreement with computer simulations. These simulations provideadditional evidence for logarithmic terms in the density expansion of thediffusion coefficient.
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