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 stat e 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 Gaussia n thermostat. Numerically the current appears to be a continuous function o f E whose derivative varies very irregularly, possibly in a discontinuous m anner. we argue for the non differentiability of the current as a function of E utilizing a symbolic description of the dynamics based on the disconti nuities of the collision map. The decay of correlations and the behavior of the diffusion constant are also investigated.