Properties of Stationary Nonequilibrium States in the Thermostatted Periodic 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 E 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 E whose derivative varies very irregularly, possibly in a discontinuous manner. We argue for the non differentiability of the current as a function of E 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.