Nonnegative kinetic density and moment domain

Adopt Countable Choice, denoted ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $I\subseteq \mathbb{R}$ be an open time interval measured in seconds, let $L_1,L_2,L_3>0$ be periods measured in metres, and let $\mathrm{\Omega }:={\prod }_{j=1}^3(\mathbb{R}\mathrm{/}{L}_j\mathbb{Z})\mathrm{.}$ Write $C:={\prod }_{j=1}^3[0,L_j)$ and let $\pi :{\mathbb{R}}^3\to \mathrm{\Omega }$ be the quotient map. Define $dx$ on Borel subsets $E\subseteq \mathrm{\Omega }$ by $dx(E):={\lambda }_3(C\cap {\pi }^{-1}(E)),$ and use its completion as the spatial measure. Thus $dx$ is the pushforward of three-dimensional Lebesgue measure restricted to the half-open fundamental cell $C$, is not divided by the cell volume, and satisfies $dx(\mathrm{\Omega })=L_1{L}_2{L}_3$. Velocity $v\in {\mathbb{R}}^3$ is measured in metres per second and $dv$ is three-dimensional Lebesgue measure. Time integration uses Lebesgue measure $dt$ on $I$; product integrals use the corresponding product measures. In this definition $fdxdv$ is a number count, so a density $f$ has units ${\mathrm{m}}^{-3}(\mathrm{m}{\mathrm{s}}^{-1}{)}^{-3}$.

Set $v_{\ast }:=1\mathrm{m}{\mathrm{s}}^{-1}$. For $k=2,3$, define the dimensionless polynomial weight ${\omega }_k(v):={\sum }_{j=0}^k(\mathrm{\mid }v\mathrm{\mid }\mathrm{/}{v}_{\ast }{)}^j$. A kinetic density is a finite-valued measurable function $f:I\times \mathrm{\Omega }\times {\mathbb{R}}^3\to [0,\mathrm{\infty })$ for which, at every $(t,x)$ under discussion, ${\int }_{{\mathbb{R}}^3}{\omega }_3(v)f(t,x,v)dv<\mathrm{\infty }\mathrm{.}$ Define number density $n:=\int fdv$ and mass density $\rho :=mn$ for a fixed particle mass $m>0$ in kilograms. Where $n>0$, define $u:=\frac{1}{n}{\int }_{{\mathbb{R}}^3}vfdv,c:=v-u,T:=\frac{m}{3k_{B}n}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }c{\mathrm{\mid }}^{2}fdv,$ for a fixed Boltzmann constant $k_B>0$ in joules per kelvin. The velocity is $u$ in metres per second and the temperature is $T$ in kelvins. Where also $T>0$, define the candidate monatomic Maxwellian $M_{n,u,T}(v):=n{\left(\frac{m}{2\pi k_{B}T}\right)}^{3\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}(-\frac{m\mathrm{\mid }v-u{\mathrm{\mid }}^2}{2k_{B}T})\mathrm{.}$ Its normalization and moments are established by a separate proposition; this formula is a definition here. At $n=0$, $u$ and $T$ are undefined, and statements using them assume $n>0$. A zero centered second moment gives $T=0$ and does not define the positive-temperature Maxwellian.

Define the pressure tensor, heat-flux vector, and kinetic-energy density by $P:=m{\int }_{{\mathbb{R}}^3}c\otimes cfdv,q:=\frac{m}{2}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }c{\mathrm{\mid }}^{2}cfdv,E:=\frac{m}{2}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }v{\mathrm{\mid }}^{2}fdv\mathrm{.}$ Here $(a\otimes b{)}_{ij}=a_i{b}_j$. The finite absolute third moment makes these integrals finite. Their SI units are respectively pascals, watts per square metre, and joules per cubic metre. This is the monatomic translational model: no internal rotational or vibrational degrees of freedom and no isotropy assumption are included.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

product-measures-and-the-fubini-tonelli-theorems

Tong, §§2.2.2–2.4.2, full lecture notes

Golse, §§3.1–3.2, full chapter