A condition equivalent to the existence of the absolute temperature scale satisfying the Clausius inequality is exhibited. The condition is close to the classical statements of the second law of thermodynamics due to Clausius and others. It is formulated as an assertion about cyclic processes of bodies whose totality is called a universe of bodies. The universe can contain general (local or not) bodies with memory but the validity of the result (i.e., the mentioned equivalence) requires that it must contain enough elastic bodies. With each universe of bodies one can associate a collection of Borel measures on the real line, and the restrictions imposed on the universe of bodies induce certain properties of this collection. Also the condition equivalent to the existence of the absolute temperature scale as well as the Clausius inequality can be stated completely in terms of the collection of measures. The proof of the main result is given in this more general setting without recourse to bodies.

background or methods

A condition equivalent to the existence of entropy in classical thermodynamics

On the Clausius inequality

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 Caratheodory (“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.

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

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 1. Systems with Internal Variables . . . . . . . . . . . . . . . . . . . . . 302 2. Adiabatic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 3. Thermodynamic Axiom . . . . . . . . . . . . . . . . . . . . . . . . . 308 4. Entropy and Temperature in Equilibrium . . . . . . . . . . . . . . . . . 311 5. Entropy away from Equilibrium . . . . . . . . . . . . . . . . . . . . . 315 6. Clausius-Planck Inequality . . . . . . . . . . . . . . . . . . . . . . . . 321 7. Non-Uniqueness of Thermostatic Functions . . . . . . . . . . . . . . . . 322 8. Non-Uniqueness of Thermodynamic Functions . . . . . . . . . . . . . . . 325 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

A condition equivalent to the existence of entropy in classical thermodynamics

Citations

On the Clausius inequality

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

A condition equivalent to the existence of non-equilibrium entropy and temperature for materials with internal variables

Efficiency and the existence of entropy in classical thermodynamics

A note on the existence of entropy in classical thermodynamics

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