Non-equilibrium stationary states from the equation of motion of open systems

Abstract

Non-equilibrium stationary states are solved from the equation of motion of open systems via a Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY)-like hierarchy. As an example, we demonstrate this approach for the Redfield equation, which is derived from the whole dynamic equation of motion of a central system coupled to two baths through the projector technique. Generalization to other equations of motion is straightforward. For non-interacting central systems, the first equation out of the hierarchy is closed, whereas for interacting systems it is coupled to higher-order equations in the hierarchy. In the case of interacting systems, two systematic approximations, in the form of perturbation theories, are proposed to truncate and solve the hierarchy. A non-equilibrium Wick's theorem is proved to provide a basis for the perturbation theories. As a test of reliability of the proposed methods, we apply them to small systems, where it is also possible to apply other exact direct methods. Consistent results were found.