Properties of Stationary Nonequilibrium States in the Thermostatted Lorentz Gas I: the One Particle System

We study numerically and analytically the properties of the stationary state of a particle moving under the influence of an electric field $\text{bE}$ in a two dimensional periodic Lorentz gas with the energy kept constant by a Gaussian thermostat. Numerically the current appears to be a continuous function of $\text{bE}$ whose derivative varies very irregularly, possibly in a discontinuous manner. We argue for the non differentibility of the current as a function of $\text{bE}$ utilizing a symbolic description of the dynamics based on the discontinuities of the collision map. The decay of correlations and the behavior of the diffusion constant are also investigated.