Conditional gradient-model entropy production

Let $O$ be a nonempty open region of a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, with its metric-compatible torsion-free connection. Use the $C^2$ state map $(v,{\sigma }_p):O\to X$ and the associated $n=v^{-1}$, $s=n{\sigma }_p$, $T=e_{{\sigma }_p}$, $p=-e_v$, and $\mu =e+pv-T{\sigma }_p$ of Rest fluid state, frame and geometric current definitions, with $n,T>0$ and the fixed-composition branch adopted by Restricted gradient constitutive dissipative model. The frame is the smooth future unit timelike observer congruence $u$ of that definition, not a chart or tetrad. Define ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $\theta ={\mathrm{\nabla }}_a{u}^a$, and $a_b=u^c{\mathrm{\nabla }}_c{u}_b$. Let ${\sigma }_{ab}$ be the spatial symmetric trace-free part of ${\mathrm{\nabla }}_{(a}{u}_{b)}$, namely ${\mathrm{\Delta }}_a{}^c{\mathrm{\Delta }}_b{}^d{\mathrm{\nabla }}_{(c}{u}_{d)}-(\theta \mathrm{/}3){\mathrm{\Delta }}_{ab}$, where parentheses denote averaged symmetrization. Let the fields in Restricted gradient constitutive dissipative model satisfy its source-free stress and particle balances, closures, and declared applicability regime, including the separately required future-timelike condition on $J$. The observer congruence is smooth and the state function and map are $C^2$ as in Rest fluid state, frame and geometric current definitions. To match the exact hypotheses of Frame-qualified dissipative entropy cancellation, assume $\mathrm{\Pi }$, $Q^a$, $V^a$, and ${\pi }^{ab}$ are smooth on $O$; the transport coefficients remain $C^1$ as in the postulate. The balances and entropy-current divergence are then classical pointwise expressions. Put $Y_a={\mathrm{\nabla }}_{a}T+T{a}_a,Z_a={\mathrm{\nabla }}_a\text{\hspace{0.17em}⁣}\left(\frac{\mu }{T}\right),a_b=u^c{\mathrm{\nabla }}_c{u}_b\mathrm{.}$ Then the entropy current $S^a=s{u}^a+Q^a\mathrm{/}T-\mu V^a\mathrm{/}T$ has the exact pointwise divergence ${\mathrm{\nabla }}_a{S}^a=\frac{\zeta c{\theta }^2+2\eta c{\sigma }^{ab}{\sigma }_{ab}}{T}+\frac{\kappa }{c{T}^2}{\mathrm{\Delta }}^{ab}{Y}_a{Y}_b+L{\mathrm{\Delta }}^{ab}{Z}_a{Z}_b\ge 0,$ where $\theta ={\mathrm{\nabla }}_a{u}^a$, ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, and ${\sigma }_{ab}$ is the spatial symmetric trace-free shear defined in the postulate. Thus this selected branch satisfies the adopted local entropy-current admissibility condition of Local entropy-current admissibility in a separately defined model. With length coordinates $x^0=ct$, the divergence has units J K${}^{-1}$ m${}^{-4}$ and $c{\mathrm{\nabla }}_a{S}^a$ is the physical production rate per volume in J K${}^{-1}$ m${}^{-3}$ s${}^{-1}$.

This is a conditional model deduction for a sufficiently regular solution already supplied in the declared regime. It does not assert that the model has a solution, that the constitutive law describes nature, or that the branch is causal, linearly stable, or globally entropy-increasing. It uses no empirical premise.