Generalized entropy and extensions of the second law of thermodynamics   and the Clausius relation for quantum systems in steady heat conduction   states
  This paper has been withdrawn by the author. For the reason, see the bottom paragraph of this abstract.   By generalizing Tasaki's work on the second law of thermodynamics for an adiabatic process between two equilibrium states of a macroscopic quantum compound system, we obtain an extension of the second law to a transient adiabatic process that takes a macroscopic quantum compound system consisting of a system of interest and two heat reservoirs from an initial equilibrium state to a final non-equilibrium state, where the system of interest is in a steady heat conduction state. For the system of interest, we define its generalized entropy so that it becomes an experimentally measurable quantity and exactly satisfies an extension of the Clausius relation.   This paper has been withdrawn by the author because the generalized entropy presented in this article violates the second law when the system in a steady heat conduction state is detached from the two reservoirs and then attached to a reservoir at the temperature T^bar defined in Sec.1.2 so that it will reach an equilibrium state at T^bar. In this irreversible process, the total entropy for the system and the reservoir remains constant and thus violates the second law. 
Hiroshi Matsuoka
cond-mat.stat-mech