A non-equilibrium contact between two discrete systems by an inert partition is considered. One of these two systems, the equilibrium environment reservoir, is controlling the other non-equilibrium system which is described by two levels of different accuracy: firstly as an undecomposed system and secondly as an endoreversible composite system of non-interacting subsystems. The intensive variables of the system in its undecomposed description are non-equilibrium contact quantities which are defined by inequalities induced by the second law. The intensive variables of the system in its description as a composite system are given by the equilibrium variables of the reversible subsystems. The different accuracy of the two descriptions leads to the introduction of the concept of compound deficiency. In particular, the sub-additivity of the entropy rates belonging to the different descriptions is caused by compound deficiency. Finally, the relations between different forms of the Clausius inequality of closed systems are derived by using the concept of compound deficiency.
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