  We study the Lyapunov exponents for a moving, charged particle in a
two-dimensional Lorentz gas with randomly placed, non-overlapping hard disk
scatterers placed in a thermostatted electric field, $\vec{E}$. The low density
values of the Lyapunov exponents have been calculated with the use of an
extended Lorentz-Boltzmann equation. In this paper we develop a method to
extend these results to higher density, using the BBGKY hierarchy equations and
extending them to include the additional variables needed for calculation of
Lyapunov exponents. We then consider the effects of correlated collision
sequences, due to the so-called ring events, on the Lyapunov exponents. For
small values of the applied electric field, the ring terms lead to
non-analytic, field dependent, contributions to both the positive and negative
Lyapunov exponents which are of the form ${\tilde{\epsilon}}^{2}
\ln\tilde{\epsilon}$, where $\tilde{\epsilon}$ is a dimensionless parameter
proportional to the strength of the applied field. We show that these
non-analytic terms can be understood as resulting from the change in the
collision frequency from its equilibrium value, due to the presence of the
thermostatted field, and that the collision frequency also contains such
non-analytic terms.
