Conditional entropy production in the periodic Boltzmann model

Given: The positive regular periodic solution and collision data described in [A1]–[A3], Boltzmann's constant $k_B>0$, and the exact current mathematical supplier interface [L1].

Periodic Boltzmann model with incoming molecular chaos: on the periodic three-torus, the adopted one-sided incoming-pair closure gives an admissible locally integrable collision term $Q_{\nu }(f)$, the kinetic equation ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f)$ almost everywhere, and the weak collision identity with factor $1\mathrm{/}4$ for absolutely convergent test functions; it is a model postulate, not a many-particle theorem.

The solution satisfies $f>0$, is $C^1$ in $(x,t)$ and $C^2$ in $v$, and is periodic in each spatial coordinate. The reference $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ is constant in $x,v,t$, so $\mathrm{\ell }=\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$ and $\log (A\mathrm{/}B)$ are dimensionless.

For each $t\in I$, $f$, $f\mathrm{\mid }v{\mathrm{\mid }}^2$, and $f\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }$ are integrable over ${\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}\times {\mathbb{R}}^3$. For every compact interval $I_0\subset I$, assume a common $G_{I_0}\in L^1({\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}\times {\mathbb{R}}^3)$ bounds $\mathrm{\mid }(1+\mathrm{\ell }){\mathrm{\partial }}_{t}f\mathrm{\mid }$ almost everywhere for every $t\in I_0$. Also assume $(1+\mathrm{\mid }\mathrm{\ell }\mathrm{\mid })\mathrm{\mid }v\cdot {\mathrm{\nabla }}_{x}f\mathrm{\mid }\in L^1({\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}\times {\mathbb{R}}^3)$ for each $t\in I$. For every $t\in I$, 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 $L^1({\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}})$. In addition to the almost-everywhere weak identity in [A1], assume that the identities for tests $1$ and $\mathrm{\ell }(x,\cdot ,t)$ are absolutely convergent and hold at every $(x,t)$, as required by the exact current interface [L1]. The collision measure is $\sigma$-finite. Also require ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f)$ at every $(x,v,t)$ for this representative, so substitution at each fixed time is licensed.

Regular single-species elastic collision data: in dimension three, $Q_{\mathrm{e}\mathrm{l}}$ consists of collisions conserving pair momentum and squared speed; $\nu$ is a nonnegative $\sigma$-finite Borel measure invariant under the two particle exchanges and incoming/outgoing exchange. The $\sigma$-finiteness here is an explicit physical restriction that makes the spatial/collision product integrals well-defined.

SI quantities and unit conventions: $k_B$ is the positive Boltzmann constant with SI units $\mathrm{J}{\mathrm{K}}^{-1}$.

Conditional Boltzmann entropy inequality and equality form: the current draft supplier assumes a positive $C^1$-in-space-and-time, $C^2$-in-velocity density on a periodic rectangle; finite moments and entropy; a compact-time $L^1$ majorant for $(1+\mathrm{\ell }){\mathrm{\partial }}_{t}f$; transport integrability at each time; an $L^1$ spatial collision-production map; and absolutely convergent weak identities for tests $1$ and $\mathrm{\ell }$ at every point. It states $\frac{d}{dt}H[f](t)=-\frac{1}{4}{\int }_{\mathrm{\Omega }}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(A-B)\log (A\mathrm{/}B)d\nu dr\le 0,$ with equality exactly when $A=B$ for $\nu$-almost every collision at almost every spatial point. The carrier remains draft without independent proof acceptance; this item uses only that inequality and equality condition, not the supplier's additional Maxwellian branch.