Rectangular local-equilibrium field data

Let $I\subseteq \mathbb{R}$ be an open interval measured in seconds, let $L_1,L_2,L_3>0$ be lengths in metres, and set $\mathrm{\Omega }={\prod }_{i=1}^3(0,L_i)\subseteq {\mathbb{R}}^3$ with closure $\bar{\mathrm{\Omega }}$. A rectangular local-equilibrium field record consists of an open temperature interval $\mathcal{T}\subset (0,\mathrm{\infty })$, a temperature field $T:\bar{\mathrm{\Omega }}\times I\to \mathcal{T}$ whose time derivative through order one and spatial derivatives through order two extend continuously to $\bar{\mathrm{\Omega }}\times I$, a function $u\in C^1(\mathcal{T},\mathbb{R})$ of internal-energy density in $\mathrm{J}{\mathrm{m}}^{-3}$, and its volumetric heat capacity $c(T):=u^{\mathrm{\prime }}(T)>0$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. The coordinates $x_i$ are lengths in metres, and $T$ is measured in kelvins, using SI quantities and unit conventions.

The record may also include a thermal-energy flux-density field $j_Q:\bar{\mathrm{\Omega }}\times I\to {\mathbb{R}}^3$ with components in $\mathrm{W}{\mathrm{m}}^{-2}$ and a conductivity field $K:\bar{\mathrm{\Omega }}\times \mathcal{T}\to {\mathbb{R}}^{3\times 3}$ with entries in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. For any unit vector $n$, $j_Q\cdot n$ is the signed thermal-energy transfer rate per unit area in direction $n$, positive when energy flows along $n$. The flux and tensor are field data here; no relation between them is imposed by this Definition. In Euclidean coordinates, $\mathrm{\nabla }T=({\mathrm{\partial }}_{x_1}T,{\mathrm{\partial }}_{x_2}T,{\mathrm{\partial }}_{x_3}T)$. The classical divergence $\mathrm{div}{j}_Q(x,t):={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}(j_Q{)}_i(x,t)$ is defined at interior points where these three partial derivatives exist. No spatial differentiability of an optional flux is assumed here; any balance or result using its divergence throughout a domain must impose the needed regularity. On a flat face, $n$ denotes its outward unit normal.

This is a specified regular field class on a fixed rigid body. It does not assert that every physical medium has local-equilibrium fields, nor does it adopt a heat equation, a boundary condition, Fourier's law, or an entropy-production law.

The label “local-equilibrium field record” names this model's field class; it is not a claim about microscopic equilibration.