Single-species BGK relaxation operator

Adopt Countable Choice, denoted ${\mathrm{A}\mathrm{C}}_{\omega }$, as in the kinetic-density definition. Fix the particle mass $m>0$ in kilograms, Boltzmann constant $k_B>0$ in joules per kelvin, and velocity Lebesgue measure $dv$ from the kinetic-density definition. Put $v_{\ast }=1\mathrm{m}{\mathrm{s}}^{-1}$ and ${\omega }_3(v)={\sum }_{j=0}^3(\mathrm{\mid }v\mathrm{\mid }\mathrm{/}{v}_{\ast }{)}^j$. Here $v\in {\mathbb{R}}^3$ is measured in metres per second, and $f$ is a number density in velocity space with units ${\mathrm{s}}^3{\mathrm{m}}^{-6}$. For a finite-valued nonnegative measurable velocity density $f:{\mathbb{R}}^3\to [0,\mathrm{\infty })$ with finite absolute third moment $\int {\omega }_3(v)f(v)dv<\mathrm{\infty }$, set $n:=\int fdv$. Where $n>0$, set $u:=n^{-1}\int vfdv$ and $T:=m(3k_{B}n{)}^{-1}\int \mathrm{\mid }v-u{\mathrm{\mid }}^{2}fdv$; then $n$ has units ${\mathrm{m}}^{-3}$, $u$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, and $T$ has units kelvins. Define the admissible BGK input domain ${\mathcal{D}}_{\mathrm{B}\mathrm{G}\mathrm{K}}:=\{f:\int {\omega }_{3}fdv<\mathrm{\infty },f\ge 0\text{ finite-valued measurable},n>0,\int \mathrm{\mid }v-u{\mathrm{\mid }}^{2}fdv>0\}\mathrm{.}$ Fix a relaxation time $\tau >0$ measured in seconds. For $f\in {\mathcal{D}}_{\mathrm{B}\mathrm{G}\mathrm{K}}$, the temperature $T$ is strictly positive, $\theta :=k_{B}T\mathrm{/}m$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$, and the moment-matched Maxwellian $M[f]$ and relaxation operator are defined by the following formulas; $M[f]$ has the same units as $f$: $M[f](v):=n(2\pi \theta {)}^{-3\mathrm{/}2}\exp (-\mathrm{\mid }v-u{\mathrm{\mid }}^2\mathrm{/}(2\theta )),Q_{\mathrm{B}\mathrm{G}\mathrm{K}}(f):=\frac{M[f]-f}{\tau }\mathrm{.}$ The displayed formula defines the single-species Bhatnagar–Gross–Krook (BGK) relaxation operator pointwise on ${\mathcal{D}}_{\mathrm{B}\mathrm{G}\mathrm{K}}$, with $Q_{\mathrm{B}\mathrm{G}\mathrm{K}}(f)$ in units of $f$ per second. The corresponding accelerated homogeneous model with dimensionless parameter $ϵ>0$ is ${\mathrm{\partial }}_{t}f=Q_{\mathrm{B}\mathrm{G}\mathrm{K}}(f)\mathrm{/}ϵ$. Outside ${\mathcal{D}}_{\mathrm{B}\mathrm{G}\mathrm{K}}$ this item assigns no BGK operator. This is a prescribed relaxation model. It is distinct from the binary collision operator and does not by itself assert a derivation from microscopic mechanics or an inhomogeneous hydrodynamic limit.

Tong, §2.5.1, full lecture notes

Zamponi, §3.2, full lecture notes