Thermodynamics based on the Hahn-Banach Theorem: The Clausius inequality

ed, permit analysis of those theories within a common framework. In the spirit of work by COLEMAN and Ow~N, we seek to develop a " theory of theories" which, for example, makes possible isolation of precisely those attributes a particular theory must have so that within its framework one or another of the two statements takes on validity. We shall be especially interested in separating those aspects of the Clausius inequality that depend only on the force of the Second Law from those that depend upon additional structure that might or might not be present within a given theory. Insofar as all theories presumably have the Second Law built into them, this will enable us to identify results which transcend the details of special theories. In Section 2 we provide some mathematical preliminaries. In particular, we state the version of the Hahn-Banach Theorem that plays a role in proof of virtually every theorem in the main body of this article. When not used explicitly it manifests itself in the guise of Lemma 6.1 or 6.2. In Section 3 we abstract those features of theories of cyclic processes that will concern us. It seems to us that if, within a particular theory, statements (A) and (B) can be made precise and tested for validity two things are essential. First, the notion of state must be rendered concrete within the theory. I f temperature is to be a "function of state", then the domain of "states" on which that function is to take values should be described clearly at the outset. Second, the cyclic processes admitted for consideration within the theory should be delineated to the extent that the integral in (1.1) can, at least in principle, be calculated for each. With these ideas in mind we take a theory to be described by specification of two sets, Z and cg, which carry the required information. A meaningful interpretation of these must await Section 3. Here we can only attempt a vaguely suggestive discussion. The set 27, called the set of state descriptions (or, less formally, the set of states), serves to specify the manner in which states of material points are described within a particular theory. Roughly speaking, elements of 27 are the "values" states might conceivably take. Thus, in a theory of a particular gas, elements of 27 might be pairs (p, v), where p is the pressure at a material point and v is the specific volume. In a theory of an elastic solid, elements of 27 might be taken to be pairs (e, F), where e is the internal energy density and F is the deformation gradient. In a theory that takes as primitive the existence of a hotness manifold, elements of Z might be taken to be the "hotnesses" material points can experience. In any ease, we presume that 27 is endowed with a Hausdorff topology. Moreover, we presume throughout the main body of this article that 27 may be taken to be compact. In effect we restrict our attention to processes in which no material point experiences a state outside some fixed compact set, perhaps very large. That we impose this restriction at the outset results from a decision to sacrifice a degree of generality in exchange for a presentation substantially less encumbered by technical considerations. To compensate for this we relax the compactness assumption in Appendix E. There we show that, in the absence of compactness, important theorems in the main body of this article require modification, and we indicate what modifications need be made. The set cg, called the set of cyclic heating measures, carries information about heat exchange in those cyclic processes a particular theory admits. With each 208 M. FEINBERG & R. LAVINE such process there is associated a signed Borel measure on Z' that provides an account of net heat receipt by the body suffering the process according to the states experienced by its material points as they exchange heat with the exterior of the body. More precisely, if q~ is the measure associated with a process and A ( 27 is a (Borel) set of states, then ~,(A) is the net amount of heat received during the entire process (from the exterior of the body) by material points experiencing states in A. The set of measures derived in this way from all cyclic processes admitted within a particular theory is what we call ~. This rather terse interpretation of c~ is elaborated upon considerably in Section 3, where we also attribute to c~ a natural convexity property one would expect in a reasonable theory. The pair (Z, c~) taken to characterize a particular theory we call a cyclic heating system (Definition 3.1). A Kelvin-Planck system is a cyclic heating system that, in a sense made precise in Definition 3.2, respects the Kelvin-Planck statement of the Second Law. In terms that are not quite precise a Kelvin-Planck systems is a cyclic heating system for which no nonzero cyclic heating measure takes non-negative values in every Borel set. Thus, if heat is absorbed by a body suffering a cyclic process that body must emit heat as well. In Section 4 we prove (Theorem 4.1) that every Kelvin-Planck system (Z, c~) admits a continuous function T: 27 ~ P, P denoting the positive real numbers, such that f ~ 0 , V,E~. (1.2)