Defined classical Jüttner kinetic preparation

Let $(M,g,N_0)$ be the smooth time-oriented Lorentzian geometry of Thermodynamics-GR fields, currents, reference normalization and units, with signature $(-,+,+,+)$ and physical length coordinates $x^0=ct$. On an open region $O\subseteq M$, take a smooth observer congruence $u$ whose value at each event is a future unit timelike observer in the sense of Observers, rest spaces and the spatial projection at an event and Observer congruences, their observer curves and adapted charts. The congruence is a field of observer directions, not an adapted coordinate chart or a tetrad. Let $n:O\to (0,\mathrm{\infty })$ be a smooth proper number density in ${\mathrm{m}}^{-3}$ and $T:O\to (0,\mathrm{\infty })$ a smooth proper-temperature scalar in kelvins. These fields are prescribed model data.

Choose one species with rest mass $m>0$ in kilograms, put $M_{\ast }:=m{c}^2>0$ in joules, choose $k_B>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$, and choose a reference action $h_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ in $\mathrm{J}\mathrm{s}$. At an event $x\in O$, the future massive shell ${\mathcal{H}}_{M_{\ast },x}^{+}$ and its tetrad-independent measure $d{\mu }_{M_{\ast }}$ are those of Canonical future massive mass-shell measure. For each $P\in {\mathcal{H}}_{M_{\ast },x}^{+}$, define the positive observer-measured energy

$E_u(P):=-g_x(P,u_x)>0,\alpha (x):=\frac{1}{k_{B}T(x)}\mathrm{.}$

In any future orthonormal tetrad whose time axis is $u_x$, write the spatial energy-momentum coordinates as $\mathbf{p}\in {\mathbb{R}}^3$, so $E_u(P)=\sqrt{M_{\ast }^2+\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2}$ and $d^3\mathbf{p}=E_u(P)d{\mu }_{M_{\ast }}(P)$. The spatial mechanical momentum is $\mathbf{p}\mathrm{/}c$; $\mathbf{p}$ itself is the spatial part of the energy four-vector $P$.

Define

$Z_{M_{\ast }}(\alpha ):=\frac{4\pi M_{\ast }^2}{\alpha }{K}_2(\alpha M_{\ast }),K_j(z):={\int }_0^{\mathrm{\infty }}{e}^{-z\cosh t}\cosh (jt)dt(j=1,2)\mathrm{.}$

and the observer-rest momentum probability law

$d{\nu }_{u,x}(\mathbf{p}):=Z_{M_{\ast }}(\alpha (x){)}^{-1}{e}^{-\alpha (x)E_u(P)}{d}^3\mathbf{p}\mathrm{.}$

Set

$\beta :=\frac{u}{k_{B}T},\zeta :=\log \text{\hspace{0.17em}⁣}\left(\frac{n{h}_{\mathrm{r}\mathrm{e}\mathrm{f}}^3{c}^3}{Z_{M_{\ast }}(\alpha )}\right),f(x,P):=e^{\zeta (x)+g_x(\beta (x),P)}\mathrm{.}$

Finally define the model number current and stress-energy tensor by the convergent shell moments

$J^a(x):=(h_{\mathrm{r}\mathrm{e}\mathrm{f}}^3{c}^3{)}^{-1}{\int }_{{\mathcal{H}}_{M_{\ast },x}^{+}}{P}^{a}f(x,P)d{\mu }_{M_{\ast }}(P),{\mathsf{T}}^{ab}(x):=(h_{\mathrm{r}\mathrm{e}\mathrm{f}}^3{c}^3{)}^{-1}{\int }_{{\mathcal{H}}_{M_{\ast },x}^{+}}{P}^a{P}^{b}f(x,P)d{\mu }_{M_{\ast }}(P)\mathrm{.}$

The mathematical suppliers Jüttner partition, probability, and perfect-fluid moments and Probability measures and probability spaces give $0<Z_{M_{\ast }}<\mathrm{\infty }$, show that ${\nu }_{u,x}$ has total mass one, and evaluate the current and stress moments:

$J^a=n{u}^a,{\mathsf{T}}^{ab}=(\epsilon +p)u^a{u}^b+p{g}^{ab},\epsilon =n(M_{\ast }\frac{K_1(\alpha M_{\ast })}{K_2(\alpha M_{\ast })}+\frac{3}{\alpha }),p=\frac{n}{\alpha }=n{k}_{B}T\mathrm{.}$

Here $\epsilon$ and $p$ are the exact moments of this one-species kinetic model. They are not automatically identified with an independently adopted material state function or stress tensor. The suppliers use the stated countable-choice strength ${\mathrm{A}\mathrm{C}}_{\omega }$ only; this definition adds no arbitrary-index choice principle. The scale $h_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is a declared phase-space unit convention, not Planck's constant or a measured parameter.

This is an exact classical model definition. It does not claim that an actual material realizes this distribution, that collisions drive it to equilibrium, that the full field $f$ solves a transport equation, or that any quantum-statistical or empirical law follows.