Kinetic density, velocity, covariance stress and heat flux

Use the velocity space ${\mathbb{R}}^3$ and Lebesgue measure $dv$ fixed by Cutoff elastic-reflection collision measure and gain. Let $\mathrm{\Omega }={\prod }_{j=1}^3(\mathbb{R}\mathrm{/}{L}_j\mathbb{Z})$ be a flat periodic spatial domain with periods $L_j>0$ (in metres in the physical interpretation), and let $I\subseteq \mathbb{R}$ be an open time interval. Let $F:\mathrm{\Omega }\times {\mathbb{R}}^3\times I\to [0,\mathrm{\infty })$ be a measurable number density in position and velocity. At each $(r,t)$ where the integrals below are finite, assume $0<n(r,t):={\int }_{{\mathbb{R}}^3}F(r,v,t)dv<\mathrm{\infty }$ and $\int \mathrm{\mid }v{\mathrm{\mid }}^{3}F(r,v,t)dv<\mathrm{\infty }$. Fix a particle mass $m>0$ in kilograms, with number count dimensionless. In SI, $F$ has units ${\mathrm{m}}^{-3}(\mathrm{m}{\mathrm{s}}^{-1}{)}^{-3}$.

Define $(x\otimes y{)}_{ij}=x_i{y}_j$, ${\mathrm{\nabla }}_r\cdot T$ by $({\mathrm{\nabla }}_r\cdot T{)}_i={\sum }_j{\mathrm{\partial }}_{r_j}{T}_{ij}$, and $A:B={\sum }_{i,j}{A}_{ij}{B}_{ij}$. Define the number density $n$, mass density $\rho =mn$, mean velocity $u=n^{-1}\int vFdv$, kinetic covariance momentum flux

$\mathrm{\Pi }=m{\int }_{{\mathbb{R}}^3}(v-u)\otimes (v-u)Fdv,$

internal kinetic-energy density $e=\frac{1}{2}\mathrm{tr}\mathrm{\Pi }$, and heat-flux vector

$q=\frac{m}{2}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }v-u{\mathrm{\mid }}^2(v-u)Fdv\mathrm{.}$

Their SI units are respectively ${\mathrm{m}}^{-3}$, $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $\mathrm{m}{\mathrm{s}}^{-1}$, pascals, $\mathrm{J}{\mathrm{m}}^{-3}$, and $\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. The tensor $\mathrm{\Pi }$ is symmetric because each integrand $(v-u)\otimes (v-u)$ is symmetric, and it is positive semidefinite because for every $\xi \in {\mathbb{R}}^3$, ${\xi }^{\mathsf{T}}\mathrm{\Pi }\xi =m\int [\xi \cdot (v-u){]}^{2}Fdv\ge 0$. It is the kinetic momentum-flux covariance; in the balance law below it enters with a plus sign, while a continuum tensile-stress variable defined from it would be $-\mathrm{\Pi }$.