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.
1. Muschik, W. and Berezovski, A. Thermodynamic interaction between two discrete systems in non-equilibrium. J. Non-Equilib. Thermodyn., 2004, 29, 237–255.
https://doi.org/10.1515/JNETDY.2004.053
2. Schottky, W. Thermodynamik. Springer, Berlin, 1929.
https://doi.org/10.1007/978-3-642-88482-5
3. Kestin, J. A Course in Thermodynamics, Vol. I. Hemisphere, Washington, 1979.
4. Muschik, W. Aspects of Non-Equilibrium Thermodynamics. World Scientific, Singapore, 1990.
https://doi.org/10.1142/0991
5. Hoffmann, K.-H., Burzler, J. M. and Schubert, S. Endoreversible thermodynamics. J. Non-Equilib. Thermodyn., 1997, 22, 311–355.
6. de Groot, S. R. and Mazur, P. Non-Equilibrium Thermodynamics. North Holland, Amsterdam, 1963.
https://doi.org/10.1063/1.3050930
7. Muschik, W. and Gümbel, S. Does Clausius’ inequality exist for open discrete systems? J. Non-Equilib. Thermodyn., 1999, 24, 97–106.
https://doi.org/10.1515/JNETDY.1999.005
8. Breuer, H.-P. and Petruccione, F. The Theory of Open Quantum Systems. Oxford University Press, 2002.
9. Muschik, W. Fundamentals of non-equilibrium thermodynamics. In Non-Equilibrium Thermodynamics with Application to Solids (Muschik, W., ed.). Springer, Wien, 1993, 1–63.
https://doi.org/10.1007/978-3-7091-4321-6_1
10. Muschik, W. Empirical foundation and axiomatic treatment of non-equilibrium temperature. Arch. Rat. Mech. Anal., 1977, 66, 379–401.
https://doi.org/10.1007/BF00248902
11. Muschik, W. and Brunk, G. A concept of non-equilibrium temperature. Int. J. Eng. Sci., 1977, 15, 377–389.
https://doi.org/10.1016/0020-7225(77)90047-7
12. Muschik, W. and Riemann, H. Intensification of Clausius inequality. J. Non-Equilib. Thermodyn., 1979, 4, 17–30.
https://doi.org/10.1515/jnet.1979.4.1.17
13. Muschik, W. Extended formulation of the second law for open discrete systems. J. Non-Equilib. Thermodyn., 1983, 8, 219–228.
https://doi.org/10.1515/jnet.1983.8.3.219
14. Keller, J. U. Ein Beitrag zur Thermodynamik fluider Systeme. Physica, 1971, 53, 602–620.
https://doi.org/10.1016/0031-8914(71)90117-0
15. Muschik, W. Existence of non-negative entropy production. In Proceedings of the 5th International Symposium on Continuum Models of Discrete Systems, Nottingham, 14–20 July 1985 (Spencer, A. J. M., ed.). Balkema, Rotterdam, 1987, 39–45.
16. Kestin, J. Local-equilibrium formalism applied to mechanics of solids. Int. J. Solids Struct., 1992, 29, 1827–1836.
https://doi.org/10.1016/0020-7683(92)90174-R
