Elastic collision measure and weak collision operator

Each velocity is in ${\mathbb{R}}^3$ measured in metres per second, $dv={\lambda }_3$ is three-dimensional Lebesgue measure, and $v_{\ast }=1\mathrm{m}{\mathrm{s}}^{-1}$ is the fixed reference speed. The elastic collision set is $Q_{\mathrm{e}\mathrm{l}}:=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in {\mathbb{R}}^{12}:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\},$ with its relative Borel sigma-algebra. A collision measure is a nonnegative countably additive measure $\nu$ on this set, invariant under particle exchange in either pair and under interchange of incoming and outgoing pairs. Symmetry is a model hypothesis; it does not follow merely from the conservation equations.

The collision model declares an input domain ${\mathcal{D}}_{\nu }$ contained in the finite-valued, positive, Lebesgue-measurable functions $f:{\mathbb{R}}^3\to (0,\mathrm{\infty })$ for which each of the four coordinate evaluations $f(v),f(w),f(v^{\mathrm{\prime }}),f(w^{\mathrm{\prime }})$ on $Q_{\mathrm{e}\mathrm{l}}$ is $\nu$-measurable. For each $f\in {\mathcal{D}}_{\nu }$, it supplies a chosen measurable locally integrable representative $Q_{\nu }(f)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^3,dv)$. For finite-valued complex Lebesgue-measurable $\phi$, set $A:=f(v)f(w),B:=f(v^{\mathrm{\prime }})f(w^{\mathrm{\prime }}),\delta \phi :=\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }})\mathrm{.}$ Let ${\mathcal{T}}_f$ be the class of those $\phi$ for which $\delta \phi$ is $\nu$-measurable and both integrals ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }\phi (v)Q_{\nu }(f)(v)\mathrm{\mid }dv,{\int }_{Q_{\mathrm{e}\mathrm{l}}}\mathrm{\mid }(B-A)\delta \phi \mathrm{\mid }d\nu$ are finite. For each $f\in {\mathcal{D}}_{\nu }$, the model declares a complex linear subspace ${\mathcal{S}}_f\subseteq {\mathcal{T}}_f$ containing every compactly supported smooth complex-valued velocity test $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{C})$. Require the weak collision identity ${\int }_{{\mathbb{R}}^3}\phi (v)Q_{\nu }(f)(v)dv=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(B-A)\delta \phi d\nu$ for every $\phi \in {\mathcal{S}}_f$. For the moment-balance results below, additionally require $1$, $v_1,v_2,v_3$, and $\mathrm{\mid }v{\mathrm{\mid }}^2$ to lie in ${\mathcal{S}}_f$; for the entropy result additionally require $\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})\in {\mathcal{S}}_f$, where $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ is a constant reference density with the same units as $f$. Each required membership includes the stated measurability and absolute-convergence conditions. The collision measure and weak identity are adopted mathematical model data; they do not assert a microscopic derivation or that every density and measure admit such an operator. In a physical application, normalize the units of $\nu$ so $Q_{\nu }(f)$ has units of $f$ per second; no specific cross-section or measured collision rate is fixed here.

Current published elastic collision-kernel definition

Tong, §§2.2–2.3, full lecture notes

Golse, §3.1, full chapter