On the integrability of differential forms related to nonequilibrium entropy and irreversible thermodynamics

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Abstract
The Boltzmann equation has been used in the literature to show that the entropy differential consists of a Pfaffian form in the space of conserved and nonconserved variables (moments) and a term related to the energy dissipation due to the irreversible processes in the system. The said Pfaffian form is called the compensation differential. In this paper, the integrability of the compensation differential is examined by means of the theory of differential forms. The integrability conditions turn out to be generalized forms of the Maxwell relations in equilibrium thermodynamics. It is also shown that the generalized form of the Gibbs–Duhem relation can be seen as an equivalent of the integrability conditions. This conclusion is drawn by using the notion of homotopy operator. The Caratheodory principle is also applied to make the connection with the second law of thermodynamics more intimate than the direct but mathematically more abstract approach using the integrability conditions. The meaning of the integrating factor is clarified by using the notion of contact temperature since the integrating factor, a mathematical function, is endowed with a thermodynamically operational meaning when it is identified with the inverse local absolute temperature through the notion of contact temperature, and the compensation differential is thereby made a thermodynamically meaningful equation governing nonequilibrium processes. Through this study it is shown that in irreversible thermodynamics the compensation differential can play the role parallel to the equilibrium Gibbs relation for the entropy in equilibrium thermodynamics and thus can serve as the foundation on which to formulate a theory of irreversible processes.

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