On measures, convex cones, and foundations of thermodynamics I. Systems with vector-valued actions

Abstract

In this two-part paper, a theory of non-equilibrium thermodynamic systems (“with memory”) is developed. The emphasis is laid upon the possibility of presenting the non-equilibrium thermodynamics deductively starting from the basic laws in a form which is capable of a direct experimental verification. This first part introduces the concept of a system with a vector-valued action which underlies the concept of a thermodynamic system (introduced in the second part). A special case of a vector-valued action is a real-valued action; the theory of real-valued actions with the Clausius property due to Coleman and Owen is briefly sketched. A subintegrating functional for a vector-valued action is a non-zero linear functional whose composition with the action gives a real-valued action with the Clausius property. For special, actions which have relevance to thermodynamics the existence of a subintegrating functional with some extra properties is equivalent to the existence of the absolute temperature scale or to the existence of the mechanical equivalent of a unit of heat. The main purpose of the first part of the paper is to give conditions for the existence of a subintegrating functional which is positive on a given set of vectors. These conditions will be shown in the second part to be the abstract analogues of the verbal statements of the laws of thermodynamics.