Violation of Onsager's theorem in approximate master equation approaches
  The consistency with Onsager's theorem is examined for commonly used perturbative approaches, such as the Redfield and second-order von Neumann master equations, for thermoelectric transport through nanostructures. We study a double quantum dot, which requires coherences between states for a correct description, and we find that these perturbative approaches violate Onsager's theorem. We show that the deviations from the theorem scale with the lead-coupling strength in an order beyond the one considered systematically in the respective approach. 
Kevin Marc Seja, Gediminas Kir\v{s}anskas, Carsten Timm, and Andreas   Wacker
cond-mat.mes-hall