Elastic collision model and weak collision identity

Fix $d\ge 2$. The elastic collision set is $Q_{\mathrm{e}\mathrm{l}}:=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in {\mathbb{R}}^{4d}:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\},$ with its relative Borel sigma-algebra. A collision measure is a nonnegative countably additive measure $\nu$ on this measurable space. It is symmetric if it is invariant under the exchanges $(v,w)\leftrightarrow (w,v)$, $(v^{\mathrm{\prime }},w^{\mathrm{\prime }})\leftrightarrow (w^{\mathrm{\prime }},v^{\mathrm{\prime }})$, and $(v,w)\leftrightarrow (v^{\mathrm{\prime }},w^{\mathrm{\prime }})$. These invariances are hypotheses on $\nu$, not consequences of energy and momentum conservation.

Fix the velocity measure space $({\mathbb{R}}^d,\mathcal{L}({\mathbb{R}}^d),dv)$, where $dv$ is the Lebesgue set function of Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$ and its countably additive measure-space structure is assumed as part of the data. This definition makes no existence assertion for that structure or for the collision operator. An input density is a finite-valued positive Lebesgue-measurable function $f:{\mathbb{R}}^d\to (0,\mathrm{\infty })$ whose four coordinate evaluations $f(v),f(w),f(v^{\mathrm{\prime }}),f(w^{\mathrm{\prime }})$ on $Q_{\mathrm{e}\mathrm{l}}$ are $\nu$-measurable. Let ${\mathcal{D}}_{\nu }$ be a declared set of such inputs. Put ${\mathcal{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^1:=\{q:{\mathbb{R}}^d\to \mathbb{R}:q\text{ is Lebesgue measurable and }{\int }_K\mathrm{\mid }q\mathrm{\mid }dv<\mathrm{\infty }\text{ for every compact }K\subset {\mathbb{R}}^d\}$. Its elements here are chosen function representatives, rather than equivalence classes. A collision model includes a specified map $Q_{\nu }:{\mathcal{D}}_{\nu }\to {\mathcal{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$; write $Q(f):=Q_{\nu }(f)$.

For $f\in {\mathcal{D}}_{\nu }$, put $A=f(v)f(w)$ and $B=f(v^{\mathrm{\prime }})f(w^{\mathrm{\prime }})$. For a finite-valued complex Lebesgue-measurable function $\phi$ on velocity space, put $\delta \phi :=\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }})$. Define the complete admissible test class ${\mathcal{T}}_f$ to consist of exactly those $\phi$ for which $\delta \phi$ is $\nu$-measurable and ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }\phi (v)Q_{\nu }(f)(v)\mathrm{\mid }dv<\mathrm{\infty },{\int }_{Q_{\mathrm{e}\mathrm{l}}}\mathrm{\mid }(B-A)\delta \phi \mathrm{\mid }d\nu <\mathrm{\infty }\mathrm{.}$ Absolute integrability is in the sense of Integrable real and complex functions, and their integrals. The model also declares, for each $f\in {\mathcal{D}}_{\nu }$, a complex linear testing subspace ${\mathcal{S}}_f\subseteq {\mathcal{T}}_f$ containing at least one nonzero test. Require the following weak collision identity for every $f\in {\mathcal{D}}_{\nu }$ and every $\phi \in {\mathcal{S}}_f$. A formulation adopting all admissible tests specifies ${\mathcal{S}}_f={\mathcal{T}}_f$; a result using only particular tests names those tests and their linear span explicitly: ${\int }_{{\mathbb{R}}^d}\phi (v)Q(f)(v)dv=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(f(v^{\mathrm{\prime }})f(w^{\mathrm{\prime }})-f(v)f(w))(\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }}))d\nu ,$ Both sides are absolutely convergent by the test-class definition. Admissible tests need not be bounded or compactly supported: $1$, velocity coordinates, squared speed, and $\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$ for a positive constant $f_{\mathrm{r}\mathrm{e}\mathrm{f}}$ belong to ${\mathcal{T}}_f$ whenever they meet the displayed measurability and integrability conditions. A consumer invoking one of these tests must establish or assume those conditions and require it to belong to the declared testing domain ${\mathcal{S}}_f$. In particular, the entropy branch uses the span of $1$ and $\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$, with absolute convergence explicitly assumed. The map $Q_{\nu }$, its input domain and its testing domains are adopted model data; the weak identity is their admissibility condition, not an assertion that every positive density and collision measure admit such a map or that these tests alone uniquely determine it. Molecular chaos or a microscopic derivation of the identity is not part of the definition.

The measure's normalization is chosen so $Q(f)$ has units of $f$ per unit time in an application. No specific cross-section or measured collision rate is asserted.