Entropy production in the adopted collision model

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$: every $\mathbb{N}$-indexed family of nonempty sets has a choice function. Let $\mathrm{\Omega }={\prod }_{j=1}^3(\mathbb{R}\mathrm{/}{L}_j\mathbb{Z})$ be a periodic cell with $L_j>0$ in metres, and let $I$ be an open time interval in seconds. Let $f:I\times \mathrm{\Omega }\times {\mathbb{R}}^3\to (0,\mathrm{\infty })$ be periodic, continuously differentiable in $t,x$, and twice continuously differentiable in $v$, with $v$ in metres per second and $f$ in ${\mathrm{s}}^3{\mathrm{m}}^{-6}$. Fix $k_B>0$ in joules per kelvin, $v_{\ast }=1\mathrm{m}{\mathrm{s}}^{-1}$, ${\omega }_3(v)={\sum }_{j=0}^3(\mathrm{\mid }v\mathrm{\mid }\mathrm{/}{v}_{\ast }{)}^j$, and a constant $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ with the same units as $f$; put $\mathrm{\ell }=\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$. On each compact time interval, assume integrable velocity majorants uniform in $x,t$ for each of $f{\omega }_3$, $f\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }$, $\mathrm{\mid }v\mathrm{\mid }f\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }$, $(1+\mathrm{\mid }\mathrm{\ell }\mathrm{\mid })\mathrm{\mid }{\mathrm{\partial }}_{t}f\mathrm{\mid }$, and $(1+\mathrm{\mid }\mathrm{\ell }\mathrm{\mid })\mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }{\mathrm{\nabla }}_{x}f\mathrm{\mid }$. Define the elastic collision set

$Q_{\mathrm{e}\mathrm{l}}=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in ({\mathbb{R}}^3{)}^4:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\}$

with its relative Borel sigma-algebra, and let $\nu$ be the nonnegative symmetric collision-rate measure on $Q_{\mathrm{e}\mathrm{l}}$. At every $(x,t)$ assume $f(x,\cdot ,t)$ belongs to the collision domain and the weak identity

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

holds absolutely for $\psi =1$ and $\psi =\mathrm{\ell }(x,\cdot ,t)$, where $A=f(x,v,t)f(x,w,t)$ and $B=f(x,v^{\mathrm{\prime }},t)f(x,w^{\mathrm{\prime }},t)$. For each $t\in I$, assume that the map $x⟼{\int }_{Q_{\mathrm{e}\mathrm{l}}}\mathrm{\mid }(A-B)\log (A\mathrm{/}B)\mathrm{\mid }d\nu$ is measurable and belongs to $L^1(\mathrm{\Omega },dx)$. Assume that $f$ solves ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f)$ for Lebesgue-almost every velocity at every fixed $(x,t)$. This sectionwise equality is an additional solution hypothesis beyond the adopted product-almost-everywhere equation.

$A(x,v,w,t)=f(x,v,t)f(x,w,t),B(x,v^{\mathrm{\prime }},w^{\mathrm{\prime }},t)=f(x,v^{\mathrm{\prime }},t)f(x,w^{\mathrm{\prime }},t)\mathrm{.}$

Define

$D(x,t)=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(A-B)\log (A\mathrm{/}B)d\nu \mathrm{.}$ Under the adopted collision-rate normalization, $D$ has units ${\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$.

Then $D(x,t)\ge 0$, and the physical kinetic entropy density and flux $s_{\mathrm{k}\mathrm{i}\mathrm{n}}=-k_B\int f\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})d^{3}v$ and $j_{\mathrm{k}\mathrm{i}\mathrm{n}}=-k_B\int vf\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})d^{3}v$ satisfy

${\mathrm{\partial }}_t{s}_{\mathrm{k}\mathrm{i}\mathrm{n}}+{\mathrm{\nabla }}_x\cdot j_{\mathrm{k}\mathrm{i}\mathrm{n}}=k_{B}D\ge 0.$

Here $s_{\mathrm{k}\mathrm{i}\mathrm{n}}$ and $j_{\mathrm{k}\mathrm{i}\mathrm{n}}$ have units $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$ and $\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$, respectively.

Consequently, on the periodic cell, ${\mathcal{S}}_{\mathrm{k}\mathrm{i}\mathrm{n}}(t)={\int }_{\mathrm{\Omega }}{s}_{\mathrm{k}\mathrm{i}\mathrm{n}}(x,t)d^{3}x$ has units $\mathrm{J}{\mathrm{K}}^{-1}$ and is nondecreasing. Its rate has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}$. This is a conditional deduction of the adopted collision model; it does not assert increase of fine-grained Hamiltonian Gibbs entropy or that the model describes nature.

Countable Choice inherited from the kinetic measure and Gaussian-moment conventions

MIT OpenCourseWare 8.333, Lecture 9, §III.E (complete H-theorem proof)

David Tong, Lectures on Kinetic Theory, §2.3 (H theorem and local entropy balance)