The thermal vacuum for non-equilibrium steady state

Our purpose is to construct a theoretical description of non-equilibrium steady state (NESS), employing thermo field dynamics (TFD). TFD is the operator-based formalism of thermal quautum field theory, where every degree of freedom is doubled and thermal averages are given by expectation values of the thermal vacuum\footnote{H. Umezawa, \textit{Thermo Field Dynamics and Condensed States} (Elsevier Science Ltd, 1982).}. To specify the thermal vacuum for NESS is a non-trivial issue, and we attempt it on the analogy between the superoperator formalism and TFD\footnote{M. Schmutz, Z. Phys. B \textbf{30}, 97 (1978); Y. Nakamura and Y. Yamanaka, Ann. Phys. (N.Y.) \textbf{331}, 51 (2013).}. Using the thermal vacuum thus obtained, we analyze the NESS which is realized in the two-reservoir model. It will be shown that the NESS vacuum of the model coincides with the fixed point solutions of the quantum transport equation derived by the self-consistent renormalization of the self-energy in non-equilibrium TFD \footnote{H. Umezawa, \textit{Advanced Field Theory: Micro, Macro, and Thermal Physics} (AIP, 1993); Y. Nakamura and Y. Yamanaka, Ann. Phys. (N.Y.) \textbf{331}, 51 (2013).} .