Regular single-species elastic collision data

Fix a particle mass $m_p>0$ in kilograms and velocity space $V={\mathbb{R}}^3$ with components in $\mathrm{m}{\mathrm{s}}^{-1}$. The elastic collision set is $Q_{\mathrm{e}\mathrm{l}}=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in V^4:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\}\mathrm{.}$ Thus each allowed event conserves momentum $m_p(v+w)$ and kinetic energy $m_p(\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2)\mathrm{/}2$. Equip $Q_{\mathrm{e}\mathrm{l}}$ with the Borel sets generated by its relative topology.

Collision-rate data consist of a nonnegative, countably additive, $\sigma$-finite Borel measure $\nu$ on these Borel sets. It is invariant under exchanging $v$ with $w$, exchanging $v^{\mathrm{\prime }}$ with $w^{\mathrm{\prime }}$, and exchanging the incoming pair $(v,w)$ with the outgoing pair $(v^{\mathrm{\prime }},w^{\mathrm{\prime }})$: for each of these maps $S$ and Borel set $E$, $\nu (S^{-1}E)=\nu (E)$. This is the three-dimensional physical restriction of the collision set and exchange conventions in Elastic collision model and weak collision identity, with $\sigma$-finiteness stated explicitly for the product integrals used by the entropy theorem. For physical applications of that weak identity, at a fixed position and time let $f:V\to (0,\mathrm{\infty })$ be the one-particle number density per spatial volume and velocity volume: $f(v)d^{3}v$ is number per spatial volume in the velocity element. Particle number is dimensionless, $d^{3}v$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-3}$, and $f$ has units ${\mathrm{s}}^3{\mathrm{m}}^{-6}$. A collision term $Q_{\nu }(f)$, when a representative satisfying the absolutely convergent weak identity exists, has units ${\mathrm{s}}^2{\mathrm{m}}^{-6}$, namely $f$ per second. Test functions $\phi :V\to \mathbb{R}$ are measurable and have one common unit $U_{\phi }$ for their values (dimensionless for the tests $1$ and $\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$, where $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ has the units of $f$). Assign the measure $\nu$ itself units ${\mathrm{m}}^9{\mathrm{s}}^{-7}$: its numerical values are taken in these SI units, not as a density relative to an unstated measure on $Q_{\mathrm{e}\mathrm{l}}$. Thus ${\int }_V\phi Q_{\nu }(f)d^{3}v$ and the collision integral of $f^2$ times the test bracket both have units $U_{\phi }{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. The magnitude and distribution of $\nu$ remain model inputs; dimensional consistency fixes neither a numerical collision rate nor existence of $Q_{\nu }(f)$. The measure is a declared model input; no differential cross section or microscopic limit is inferred from this Definition.