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The Role of Second Law of Thermodynamics in Continuum Physics: A Muschik and Ehrentraut Theorem Revisited

Second law of thermodynamics imposes that in any thermodynamic process the entropy production must be nonnegative. In continuum physics such a requirement is fulfilled by postulating the constitutive equations which represent the material properties of the bodies in such a way that second law of thermodynamics is satisfied in arbitrary processes. Such an approach, first assumed in some pioneering papers by Coleman and Noll \cite{ColNol} and Coleman and Mizel \cite{ColMiz}, in practice regards second law of thermodynamics as a restriction on the constitutive equations, which must guarantee that any solution of the balance laws satisfies also the entropy inequality. As observed by Muschik and Ehrentraut \cite{MusEhr}, this is a useful operative assumption, but not a consequence of general physical laws. Indeed, a different point of view, which regards second law of thermodynamics as a restriction on the thermodynamic processes, i.e., on the solutions of the system of balance laws, is possible. This is tantamount to assume that there are solutions of the balance laws which satisfy the entropy inequality, and solutions which do not satisfy it. In order to decide what is the correct approach, Muschik and Erhentraut postulated an amendment to the second law, which makes explicit the evident but rather hidden assumption that in any point of the body the entropy production is zero if, and only if, this point is thermodynamic equilibrium. Then they proved that, given the amendment, second law of thermodynamics is necessarily a restriction on the constitutive equations and not on the thermodynamic processes. In the present paper we revisit their proof, lighting up some geometric aspects which were hidden in Ref. \cite{MusEhr}. Moreover, we propose an alternative formulation of second law of thermodynamics which incorporates the amendment.