On measures, convex cones, and foundations of thermodynamics II. Thermodynamic systems

Abstract

In this two-part paper, a theory of non-equilibrium thermodynamic systems (“with memory”) is developed. Emphasis is laid upon the possibility of presenting non-equilibrium thermodynamics deductively starting from the basic laws in a form which is capable of a direct experimental verification. In the present second part mathematical conditions corresponding to statements of the second law due to Carnot, Clausius, Kelvin, and Planck are formulated, and some new mathematical forms of the first law are exhibited. In this way, similarities and differences between the two laws are revealed. As consequences of the appropriate versions of the first and the second laws there is proved the existence of Joule's mechanical equivalent of a unit of heat, the energy, the absolute temperature scale, and entropy. The absolute temperature and entropy are defined in and away from equilibrium and satisfy the entropy inequality for processes which are not restricted by the requirement to be quasi-static. The central concept of the theory is the heat distribution measure which contains all the necessary information about the way heat is gained by the system during the process. This approach suggests an interpretation of the absolute temperature different from the one given by Carathéodory (“absolute temperature=integrating denominator”): here the reciprocal function of the absolute temperature scale gives a functional which separates the set of all heat distribution measures corresponding to cyclic processes from the convex cone of positive measures.