Local-equilibrium caloric entropy density

For the local-equilibrium medium of Rectangular local-equilibrium field data, adopt a local entropy-density map $s:\mathcal{T}\to \mathbb{R}$ measured in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$ with $\frac{ds}{dT}(T)=\frac{c(T)}{T},$ where $c(T)=u^{\mathrm{\prime }}(T)>0$ is the volumetric heat capacity and $T>0$ is measured in kelvins. Equivalently, for a fixed reference $T_{\mathrm{r}\mathrm{e}\mathrm{f}}\in \mathcal{T}$ and reference density $s_{\mathrm{r}\mathrm{e}\mathrm{f}}\in \mathbb{R}$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$, set $s(T)=s_{\mathrm{r}\mathrm{e}\mathrm{f}}+{\int }_{T_{\mathrm{r}\mathrm{e}\mathrm{f}}}^T\frac{c(\theta )}{\theta }d\theta \mathrm{.}$ The volume integral of this field is the entropy assigned by this continuum model. This adopts a local thermodynamic relation extending the macroscopic entropy framework of Concave macroscopic fundamental entropy; it does not prove that every evolving medium has such a local entropy density or identify it with an arbitrary finite-system entropy function.

For constant $c(T)=c_0>0$, the adopted relation gives $s(T)=s_{\mathrm{r}\mathrm{e}\mathrm{f}}+c_0\log (T\mathrm{/}{T}_{\mathrm{r}\mathrm{e}\mathrm{f}})$. This is a restricted constitutive specialization, not a universal equation of state.