On the violation of Ohm's law for bounded interactions: A one dimensional system

We consider an infinite Hamiltonian system in one space dimension, given by a charged particle subjected to a constant electric field and interacting with an infinitely extended system of particles. We discuss conditions on the particle/medium interaction which are necessary for the charged particle to reach a finite limiting velocity. We assume that the background system is initially in an equilibrium Gibbs state and we prove that for bounded interactions the average velocity of the charged particle increases linearly in time. This statement holds for any positive intensity of the electric field, thus contradicting Ohm's law.