Scope of transport constitutive laws

Scope.

Use the fields and local-equilibrium regime of Scope of a local-equilibrium continuum branch. For this constitutive relation, assume $T$ has continuous first spatial partial derivatives, defined as in Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, and write

$\mathrm{\nabla }T:=({\mathrm{\partial }}_{x_1}T,{\mathrm{\partial }}_{x_2}T,{\mathrm{\partial }}_{x_3}T)\in {\mathbb{R}}^3\mathrm{.}$

Here the physical coordinate indices $i=1,2,3$ correspond to function indices $i-1$ in The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$.

The components of $\mathrm{\nabla }T$ have units $\mathrm{K}{\mathrm{m}}^{-1}$.

Matrices are elements of $M_{3\times 3}(\mathbb{R})$ as in The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations. In the physical one-based display, $K_{ij}$ denotes the library matrix entry with indices $(i-1,j-1)$, and $v_i$ denotes vector coordinate $i-1$, for $i,j=1,2,3$. The action is $(Kv{)}_i:={\sum }_{j=1}^3{K}_{ij}{v}_j$; $I_3$ is the matrix with entries $1$ on the diagonal and $0$ off it as in Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes. A Fourier heat-transport model may postulate

${\mathbf{j}}_Q(\mathbf{x},t)=-K(\mathbf{x},t)\mathrm{\nabla }T(\mathbf{x},t),$

where ${\mathbf{j}}_Q:\mathcal{O}\times I\to {\mathbb{R}}^3$ is a continuous heat-energy flux field in $\mathrm{W}{\mathrm{m}}^{-2}$ and $K:\mathcal{O}\times I\to M_{3\times 3}(\mathbb{R})$ is a declared conductivity tensor field with entries in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. In an isotropic model $K(\mathbf{x},t)=\kappa (\mathbf{x},t)I_3$, where $\kappa :\mathcal{O}\times I\to [0,\mathrm{\infty })$ is the declared scalar conductivity field. A diffusion model may separately postulate ${\mathbf{j}}_n=-D\mathrm{\nabla }n$, where number density $n:\mathcal{O}\times I\to [0,\mathrm{\infty })$ is in ${\mathrm{m}}^{-3}$, number flux ${\mathbf{j}}_n:\mathcal{O}\times I\to {\mathbb{R}}^3$ is in ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, and $D$ is either a scalar field $D:\mathcal{O}\times I\to \mathbb{R}$ or a matrix field $D:\mathcal{O}\times I\to M_{3\times 3}(\mathbb{R})$ with coefficient units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$; its regularity and positivity conditions are model data.

These are constitutive choices for specified material, geometry, regime, field regularity and boundary conditions. In a kinetic model with microscopic state space $\mathrm{\Xi }$ and allowed collision transitions $\mathcal{C}\subseteq \mathrm{\Xi }\times \mathrm{\Xi }$, a collision invariant is a real-valued function $Q:\mathrm{\Xi }\to \mathbb{R}$ satisfying $Q({\xi }^{\mathrm{\prime }})=Q(\xi )$ for every $(\xi ,{\xi }^{\mathrm{\prime }})\in \mathcal{C}$. If the model has a distribution $f$ on a declared phase space and a collision operator $C[f]$ giving the collision contribution to its evolution, a single-relaxation-time replacement may be $C_{\tau }[f]=-(f-f_0)\mathrm{/}\tau$, with model-declared reference distribution $f_0$ and $\tau >0$. This replacement need not preserve collision invariants unless it is constrained to preserve their integrals against the model's state distribution. Different kinetic, linear-response and quantum transport models use different hypotheses and coefficient definitions; a coefficient derived in one is not thereby universal. No kinetic operator or reciprocity law is defined or asserted here. Microscopic calculations and measurements may determine or test coefficients for a specified model and material, but neither the equilibrium entropy postulates nor dimensional consistency alone determine them.

Any heat equation or transport conclusion additionally needs the relevant conservation balance, sources or work terms, constitutive relation, initial data and boundary conditions. No such evolution equation, Onsager relation, or transport coefficient is proved or adopted by this remark.