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

Positive and negative Lyapunov exponents for a dilute, random, two-dimensional Lorentz gas in an applied field, →E, in a steady state at constant energy are computed to order E². The results are ${{λ}}_{±{}}{{0ex}{0ex}}={{0ex}{0ex}}{{λ}}_{±{}}⁰{{-}a}_{±{}}(qE/mv{)}²{t}₀$ where ${{λ}}_{±{}}⁰$ are the exponents for the field-free Lorentz gas, ${a}₊{{0ex}{0ex}}={{0ex}{0ex}}11/48$, ${a}_{{-}}{{0ex}{0ex}}={{0ex}{0ex}}7/48$, ${t}₀$ is the mean free time between collisions, $q$ is the charge, $m$ is 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.