Anomalous current in periodic Lorentz gases with infinite horizon

Electric current is studied in a two-dimensional periodic Lorentz gas in the presence of a weak homogeneous electric field. When the horizon is finite, that is, free flights between collisions are bounded, the resulting current $\mathbf{J}$ is proportional to the voltage difference $\mathbf{E}$, that is, $\mathbf{J}=\frac12\mathbf{D}^*\mathbf{E}+o(\|\mathbf{E}\|)$, where $\mathbf{D}^*$ is the diffusion matrix of a Lorentz particle moving freely without an electric field (see a mathematical proof in [1]). This formula agrees with Ohm's classical law and the Einstein relation. Here the more difficult model with an infinite horizon is investigated. It is found that infinite corridors between scatterers allow the particles (electrons) to move faster, resulting in an abnormal current (causing ‘superconductivity’). More precisely, the current is now given by $\mathbf{J}=\frac12\mathbf{D}\mathbf{E}\bigl|\log\|\mathbf{E}\|\bigr|+\mathscr{O}(\|\mathbf{E}\|)$, where $\mathbf{D}$ is the ‘superdiffusion’ matrix of a Lorentz particle moving freely without an electric field. This means that Ohm's law fails in this regime, but the Einstein relation (suitably interpreted) still holds. New results are also obtained for the infinite-horizon Lorentz gas without external fields, complementing recent studies by Szász and Varjú [2].
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