Two-temperature steady-state thermodynamics for a radiation field

A candidate for a consistent steady-state thermodynamics is constructed for a radiation field in vacuum sandwiched by two black bodies of different temperatures. Because of the collisionless nature of photons, a steady state of a radiation field is completely determined by the temperatures of the two black bodies. Then the zeroth, first, second and third laws can be extended to steady states, where the idea of local steady states plays an important role in the system whose geometrical shape is anisotropic and inhomogeneous. The thermodynamic formalism presented in this paper does not include an energy flux as a state variable. This is consistent with the notable conclusion by [C. Essex, Adv. Thermodyn. 3 (1990) 435; Planet. Space. Sci. 32 (1984) 1035] that, contrary to the success in the irreversible thermodynamics for dissipative systems, a nonequilibrium radiation field does not obey the bilinear formalism of the entropy production rate using an energy flux and its conjugate force. Although the formalism given in this paper may be unique to a radiation field, a nonequilibrium order parameter of steady states of a radiation field is explicitly defined. This order parameter denotes that the geometrical shape of the system determines how a steady state is far from an equilibrium. The higher the geometrical symmetry, the more distant the steady state.