Single-species elastic Boltzmann model

Let $\mathrm{\Omega }={\prod }_{j=1}^3(0,L_j)$, with $L_j>0$ measured in metres and opposite faces identified, and let $I\subseteq \mathbb{R}$ be an open time interval measured in seconds. The primitive state variable $f$ is a one-particle number-density field $f:I\times \mathrm{\Omega }\times {\mathbb{R}}^3\to [0,\mathrm{\infty })$, measurable for the product Lebesgue sigma-algebra and periodic in $x$, where $x$ is in metres and velocity $v$ is in metres per second; $f{d}^{3}x{d}^{3}v$ counts particles, so $f$ has units ${\mathrm{s}}^3{\mathrm{m}}^{-6}$. For solutions, require $f$ and its weak time and spatial derivatives ${\mathrm{\partial }}_{t}f$ and ${\mathrm{\nabla }}_{x}f$ to be represented locally by integrable functions, so $f$, ${\mathrm{\partial }}_{t}f$, and $v\cdot {\mathrm{\nabla }}_{x}f$ are integrable on every compact subset of the product domain. Only translational kinetic energy $m\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}2$ is modeled; no internal molecular modes are included. The single-particle mass $m>0$ in kilograms and Boltzmann constant $k_B>0$ in joules per kelvin are primitive fixed parameters of this formulation; neither is derived from the collision equation.

An elastic collision quadruple is $(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in ({\mathbb{R}}^3{)}^4$ satisfying $v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }}$ and $\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2$; write $Q_{\mathrm{e}\mathrm{l}}$ for this closed set and give it the Borel sets inherited from $({\mathbb{R}}^3{)}^4$. A collision-rate measure is a function $\nu$ on those sets with $\nu (\mathrm{\varnothing })=0$, values in $[0,\mathrm{\infty }]$, and $\nu ({\bigcup }_n{E}_n)={\sum }_n\nu (E_n)$ for pairwise disjoint measurable $E_n$. This nonnegative countably additive $\nu$ is a primitive collision-rate input, invariant under each particle swap and under interchanging incoming and outgoing pairs, with rate normalization chosen so the collision operator has units of $f$ per second. For the collision equation, restrict each velocity section $f(x,\cdot ,t)$ to the collision domain ${\mathcal{D}}_{\nu }$ of positive finite-valued measurable densities; a nonnegative ambient state outside ${\mathcal{D}}_{\nu }$ is not assigned a collision operator by this postulate.

Adopt one-sided incoming molecular chaos: for each directed elastic collision, at each collision point and time, the incoming two-particle velocity density used by the collision term is $f(x,v,t)f(x,w,t)$. In the gain term, the product of the densities at $(v^{\mathrm{\prime }},w^{\mathrm{\prime }})$ applies the same assumption to the reversed collision, for which $(v^{\mathrm{\prime }},w^{\mathrm{\prime }})$ is the incoming pair; this uses the adopted invariance of $\nu$ under interchange of incoming and outgoing pairs. No separate factorization of the outgoing pair of a fixed forward collision is imposed, and no derivation from reversible many-particle mechanics or propagation of chaos is asserted.

For $A=f(x,v,t)f(x,w,t)$ and $B=f(x,v^{\mathrm{\prime }},t)f(x,w^{\mathrm{\prime }},t)$, adopt a collision operator $Q_{\nu }(f)$, with a jointly measurable locally integrable representative, whose weak action for every compact smooth velocity test and every moment or logarithmic test for which both sides converge absolutely is

${\int }_{{\mathbb{R}}^3}\psi (v)Q_{\nu }(f)(x,v,t)d^{3}v=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(B-A)(\psi (v)+\psi (w)-\psi (v^{\mathrm{\prime }})-\psi (w^{\mathrm{\prime }}))d\nu \mathrm{.}$

The adopted kinetic equation is

${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f)$

as an equality of locally integrable functions almost everywhere. These equations and the incoming factorization are physical model postulates, not deductions. This item asserts no empirical measurement, universal dilute-gas threshold, or solution existence.

Current draft mathematical definition of the elastic collision-rate measure and weak operator: elastic collision measure and weak operator.

MIT OpenCourseWare 8.333, Lecture 8, §III.D (dilute-gas BBGKY scaling and short-range collision correlations; contextual material)

David Tong, Lectures on Kinetic Theory, §§2.2.1–2.2.3 (incoming-only factorization in equations (2.21)–(2.24), printed pp. 22–23; compare the later before-and-after factorization, printed pp. 31–32)