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, $\stackrel{\ensuremath{\rightarrow}}{E}$, in a steady state at constant energy are computed to order ${E}^{2}$. The results are ${\ensuremath{\lambda}}_{\ifmmode\pm\else\textpm\fi{}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\lambda}}_{\ifmmode\pm\else\textpm\fi{}}^{0}{\ensuremath{-}a}_{\ifmmode\pm\else\textpm\fi{}}(\mathrm{qE}/\mathrm{mv}{)}^{2}{t}_{0}$ where ${\ensuremath{\lambda}}_{\ifmmode\pm\else\textpm\fi{}}^{0}$ are the exponents for the field-free Lorentz gas, ${a}_{+}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}11/48$, ${a}_{\ensuremath{-}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}7/48$, ${t}_{0}$ 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.