Anomalous Current in Periodic Lorentz Gases with Infinite Horizon

We study electrical current in two-dimensional periodic Lorentz gas in the presence of a weak homogeneous electric field. When the horizon is finite, i.e. the free flights between collisions are bounded, the resulting current J is proportional to the voltage difference E, i.e. J = 1 2 D∗E + o(‖E‖), where D∗ is the diffusion matrix of the Lorentz particle moving freely without electrical field (see a mathematical proof in [11]). This formula agrees with classical Ohm’s law and the Einstein relation. Here we investigate the more difficult model with infinite horizon. We find that infinite corridors between scatterers allow the particles (electrons) move faster resulting in an abnormal current (causing ‘superconductivity’). Precisely, the current is now given by J = 1 2 DE ∣ log ‖E‖ ∣ ∣+O(‖E‖), where D is the ‘superdiffusion’ matrix of the Lorentz particle moving freely without electrical field. This means that Ohm’s law fails in this regime, but the Einstein relation (suitably interpreted) still holds. We also obtain new results for the infinite horizon Lorentz gas without external fields, complementing recent studies by Szasz and Varju [28].