Separate entropy-only radiation state maps

Fix a real parameter $a_R>0$ and let the temperature coordinate range over the open interval $T\in (0,\mathrm{\infty })$. Define the entropy-only radiation state maps

${\epsilon }_T(T):=a_R{T}^4,p_T(T):=\frac{1}{3}{a}_R{T}^4,s_T(T):=\frac{4}{3}{a}_R{T}^3\mathrm{.}$

Here ${\epsilon }_T$ and $p_T$ are scalar-valued maps on $(0,\mathrm{\infty })$, and $s_T$ is the entropy coordinate as a scalar-valued map on the same interval. Put $h_T(T):={\epsilon }_T(T)+p_T(T)$. No particle-number coordinate, particle-number current, or chemical-potential coordinate is part of this state space.

Also define maps on the positive entropy-coordinate interval by

$\tau (s):={\left(\frac{3s}{4a_R}\right)}^{1\mathrm{/}3},{\epsilon }_s(s):=a_R\tau (s{)}^4,p_s(s):=\frac{1}{3}{\epsilon }_s(s),s\in (0,\mathrm{\infty })\mathrm{.}$

The displayed formulas define functions on the stated domains. In the conventional SI physical reading, $T$ is measured in K, $a_R$ in J m${}^{-3}$ K${}^{-4}$, ${\epsilon }_T,p_T$ in J m${}^{-3}$, and $s_T$ in J m${}^{-3}$ K${}^{-1}$. These units do not assign an empirical value to $a_R$. In particular, this definition does not derive a quantum blackbody coefficient or assert that any physical system realizes these maps.