On the Measure-theoretic Foundations of the Second Law of Thermodynamics

In this paper we reconsider the standard formulation of the Second Law of Thermodynamics in the general framework of fields with divergence measure, using Geometric Measure Theory and some recent results on Cauchy interactions. To our knowledge, the first attempt to put the Second Law of Thermodynamics within the framework of Geometric Measure Theory is Gurtin, Williams, Ziemer. Recently, the present authors studied the balance of heat and generalized it to fields having divergence measure and measure-valued sources. Here, with respect to Ref. 5, more general balances are considered, involving inequalities and superadditive entropy production functions, as pointed out in Ref. 4. Two main improvements have been achieved: first, we obtain the existence of temperatures and the Clausius–Duhem inequality with flux vector fields having divergence measure, a degree of generality which seems to be the highest compatible with the representation results of general fluxes (see, for example, the discussion in Degiovanni, Marzocchi, Musesti). To this end, we dropped the absolute continuity of the fluxes with respect to Lebesgue measure: this is not an idle generalization, since it is conceivable to have some form of concentrated heat or entropy fluxes. Second, and perhaps more important, we find that the usual statement of the Second Law of Thermodynamics has to be weakened