Mathematical formulation of the heat-engine theory of thermodynamics including negative absolute temperatures

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Abstract
In order to treat negative absolute temperatures in the heat-engine theory of thermodynamics with logical consistency, a mathematical scheme is proposed which consists of three basic concepts, cycles, reservoirs and heats, and three axioms (1) the existence of at least one irreversible cycle, (2) the existence of a reversible cycle operating between any two reservoirs, (3) the scaling of the size of a cycle and the combination of two cycles. The axiom (1) is the weakest form of the second law of thermodynamics. A basic theorem of the heat-engine theory, a stronger version of Carnot's theorem is derived, and based on it the meaning of temperature is clarified and various forms of the second law are investigated to examine the possibility of negative absolute temperatures. The set of cycles is represented as a half-space in a vector space, and the absolute temperatures are related to a normal vector of the hyperplane which supports the half-space.

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