Kinetic number, mass and thermal observables

Work in three spatial and velocity dimensions, with $x$ in metres, $v$ in metres per second, and a nonnegative measurable one-particle number density $f(x,v,t)$ as defined by the kinetic-density model. Adopt Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ as in that definition, and put $v_{\ast }=1\mathrm{m}{\mathrm{s}}^{-1}$ and ${\omega }_3(v)={\sum }_{j=0}^3(\mathrm{\mid }v\mathrm{\mid }\mathrm{/}{v}_{\ast }{)}^j$. Assume ${\int }_{{\mathbb{R}}^3}{\omega }_3(v)f(x,v,t)d^{3}v<\mathrm{\infty }$ at the state under consideration. Use the primitive fixed particle mass $m>0$ in kilograms and Boltzmann constant $k_B>0$ in joules per kelvin from the adopted Boltzmann model. The number density, mass density, mean velocity, peculiar velocity, translational temperature, stress, heat flux, and energy below are derived moment quantities. Where $n(x,t)>0$, define the number density, mass density, mean velocity, peculiar velocity, and translational temperature by

$n(x,t)={\int }_{{\mathbb{R}}^3}f{d}^{3}v,\rho (x,t)=mn(x,t),u(x,t)=\frac{1}{n(x,t)}{\int }_{{\mathbb{R}}^3}vf{d}^{3}v,c(x,v,t)=v-u(x,t),$

$T(x,t)=\frac{m}{3k_{B}n(x,t)}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }v-u(x,t){\mathrm{\mid }}^{2}f(x,v,t)d^{3}v\mathrm{.}$

Use $u$ only where $n>0$, and use $T$ in the positive-temperature branch only where the centered second moment is strictly positive. Define the kinetic stress tensor, heat-flux vector, and total kinetic-energy density by

$P(x,t)=m{\int }_{{\mathbb{R}}^3}c\otimes cf{d}^{3}v,q(x,t)=\frac{m}{2}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }c{\mathrm{\mid }}^{2}cf{d}^{3}v,E(x,t)=\frac{m}{2}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }v{\mathrm{\mid }}^{2}f{d}^{3}v\mathrm{.}$

Here $n$ is in ${\mathrm{m}}^{-3}$, $\rho$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $u$ in $\mathrm{m}{\mathrm{s}}^{-1}$, $T$ in kelvins, $P$ in pascals, $q$ in $\mathrm{W}{\mathrm{m}}^{-2}$, and $E$ in $\mathrm{J}{\mathrm{m}}^{-3}$. The definitions impose no isotropy on $P$; $p$ will denote a separate scalar pressure only in a specified local-equilibrium conclusion.

Let $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ be a constant reference density with the same units as $f$. At $f=0$, define both $f\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$ and $f\mathrm{\mid }\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})\mathrm{\mid }$ to be zero, their continuous extensions from $f>0$; their products with $v$ or $\mathrm{\mid }v\mathrm{\mid }$ are also zero there. Where both $\int f\mathrm{\mid }\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})\mathrm{\mid }{d}^{3}v$ and $\int \mathrm{\mid }v\mathrm{\mid }f\mathrm{\mid }\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})\mathrm{\mid }{d}^{3}v$ are finite, define the dimensionless-log kinetic entropy integral, its flux, the physical kinetic entropy density, and its flux by

$h={\int }_{{\mathbb{R}}^3}f\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})d^{3}v,J_H={\int }_{{\mathbb{R}}^3}vf\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})d^{3}v,s_{\mathrm{k}\mathrm{i}\mathrm{n}}=-k_{B}h,j_{\mathrm{k}\mathrm{i}\mathrm{n}}=-k_B{J}_H\mathrm{.}$

Thus $h$ and $J_H$ have units ${\mathrm{m}}^{-3}$ and ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, respectively; $s_{\mathrm{k}\mathrm{i}\mathrm{n}}$ has units $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$ and $j_{\mathrm{k}\mathrm{i}\mathrm{n}}$ has units $\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$. These are kinetic-model observables; $s_{\mathrm{k}\mathrm{i}\mathrm{n}}$ is not the fine-grained Gibbs entropy of a reversible many-particle state.

Current draft K01 definition of a nonnegative measurable kinetic density with finite absolute velocity moments through order 3: kinetic density and moment domain.

Current draft single-species Boltzmann postulate supplying the fixed mass and Boltzmann-constant parameters: single-species elastic Boltzmann model.

MIT OpenCourseWare 8.333, Lecture 10, §III.G (collision moments, pressure tensor, and heat flux)

David Tong, Lectures on Kinetic Theory, §§2.4.1–2.4.2 (kinetic moments and ideal-fluid variables)