W. Muschik and H. Ehrentraut
An amendment to the second law
J. Non-Equilib. Thermodyn. 21, 1996, 175-192

In continuum thermodynamics the usual form of the Second Law (SL) is local in time and position and runs as follows: "The entropy production density is not negative at each position for all times". This statement is not unique, because after having inserted the constitutive equations into the balance equations we can differently interprete as follows:
i) All solutions of the balance equations have to satisfy the SL, or
ii) There are solutions of the balance equations which satisfy the SL, and others which do not.
The consequences of these two interpretations of the SL are totally different. Statement i) means that the constitutive equations have to be restricted in such a way that all mathematical solutions of the balance equations must satisfy the SL. Consequently i) means restriction of material properties by the SL. Statement ii) means that there are solutions of the balance equations which do not exist in nature because they do not satisfy the SL. Consequently ii) means restriction of processes by the SL.

After having chosen constitutive equations obviously only one of the statements i) and ii) can be true, but which of them ? This question can not be decided by the formulation of the SL given above, because the terms "process" and "constitutive equation" are not included in this formulation. Thus we need some more knowledge for deciding, what is the true statement, i) or ii). This "some more" is an amendment to the SL, because it has to contain more than in SL is stated.

In the following we show that the (nearly self-evident, but never formulated) amendment "There are no reversible process directions in non-equilibrium" allows to prove statement i). As a consequence the entropy production density is a function of state. The connection between the amendment and the stronger Coleman-Mizel formulation of the second law is discussed. Additional constraints for the constitutive equations, called Liu equations and residual dissipation inequality, are consequences of the amendment.