A nonconstant local-equilibrium temperature field

Use the one-coordinate convention of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ explicitly: $\iota :\mathbb{R}\to {\mathbb{R}}^1$ sends $a$ to the function on $\{0\}$ with value $\iota (a)(0)=a$, and the inverse takes the coordinate at index zero. In every scalar display below, a one-coordinate parameter or field is written through this bijection. Thus $\mathcal{Z}$ means $\iota ((0,\mathrm{\infty }))$, the scalar formula for $z$ denotes its $\iota$ image, $C(e,y)=e$ means $C(e,y)=\iota (e)$, $c(a)=a$ is the identity on $\mathcal{Z}$, and $q(\iota (a))=(a,0)$; temperature and inequalities use the scalar coordinate $a$. This is a notation convention, not a set-theoretic identity between a real and a function.

Take $\mathcal{O}=(0,L{)}^3$ in metre coordinates and $I=(0,\tau )$ in second coordinates, with $L>0\mathrm{m}$ and $\tau >0\mathrm{s}$. Choose ${\mathrm{\Theta }}_0>0\mathrm{K}$ and dimensionless $0<\alpha <1$. Instantiate $m=1$ and the local-state parameter set $\mathcal{Z}=(0,\mathrm{\infty })\subseteq {\mathbb{R}}^1$, and define its one-coordinate field $z:\mathcal{O}\times I\to \mathcal{Z}$ by

$z(\mathbf{x},t)=\frac{{\mathrm{\Theta }}_0}{1\mathrm{K}}(1+\alpha \frac{x_1}{L}),T(\mathbf{x},t)=(1\mathrm{K})z(\mathbf{x},t)={\mathrm{\Theta }}_0(1+\alpha \frac{x_1}{L})\mathrm{.}$

For the homogeneous comparison cell, supply $\mathcal{X}=(0,\mathrm{\infty })\times \mathbb{R}$ with states $(e,y)$, physical energy $(1\mathrm{J})e$, and numerical entropy $s(e,y)=\ln e-y^2$, where $\ln :(0,\mathrm{\infty })\to \mathbb{R}$ is the natural logarithm of The natural logarithm as the inverse of the exponential function and physical entropy is $(1\mathrm{J}{\mathrm{K}}^{-1})s(e,y)$. The coordinate $y$ is dimensionless. Set $r=1$, $C(e,y)=e$, $c(a)=a$, and $B_a=\mathcal{X}$ for every $a\in \mathcal{Z}$, so the fixed-energy admissible set is $A_a=\{(a,y):y\in \mathbb{R}\}$. Its entropy-maximizing set is $E_a=\{(a,0)\}$. Define ${\mathcal{E}}_{\mathrm{e}\mathrm{q}}={\bigcup }_{a>0}{E}_a$ and supply $q(a)=(a,0)$; assign this comparison state temperature $(1\mathrm{K})a$. Stipulate the local-equilibrium approximation with this map, so the local constitutive state at $(\mathbf{x},t)$ is $q(z(\mathbf{x},t))$. The field is static and nonconstant; this stipulation is a model assumption, not an assertion that a physical sample realizes the profile.