Exact expression for Drude conductivity in one-dimension with an   arbitrary potential
  An exact expression for the Drude conductivity in one dimension is derived under the presence of an arbitrary potential. In getting the conductivity the influence of the electric field on the crystal potential is taken into account. This coupling leads to a systematic deformation of the potential and consequently to a significant modification of the charge transport. The corrections to the conventional Drude conductivity are determined by the configurational part of the partition function. The activation energy for the conductivity process is expressed by a combination of the free energy of the underlying equilibrium system. The electric current is calculated in the linear response regime by solving the Smoluchowski equation. The steady state solution differs significantly from the equilibrium distribution. In case of a tight binding potential the conductivity offers corrections depending on the amplitude of the potential. As a further application we discuss nanocontacts with piecewise constant potentials. The electric conductivity is corrected by the potential height. 
Michael Schulz, Steffen Trimper