Steady State Thermodynamic Functions for Radiation Field

Several attempts to construct the thermodynamical formulation for non-equilibrium steady states have been made along the context by Oono and Paniconi, called the steady state thermodynamics (SST). In this paper we study the SST for radiation field. Consider a cavity in a large black body, and put an another black body in it. Let the temperature of the outer black body be different from that of the inner one, and keep their temperatures unchanged. Then the radiation field between the black bodies are retained in a non-equilibrium steady state with a stationary energy flow. (The inner black body may be considered as a representative of a black hole with the Hawking radiation, and the cavity be the universe including the black hole.) According to the total momentum carried by photons par a unit time in passing through a virtual unit area, the pressure for the SST is defined. Then, taking this pressure as the basis, we propose the definitions of steady state thermodynamic functions and state variables satisfying the plausible properties: the convexity/concavity of thermodynamic functions, the intensivity/extensivity of state variables, the Gibbs-Duhem relation and the Legendre transformation among thermodynamic functions.