Specified equilibrium particle exchange

For a selected fixed-composition material branch, adopt the local state function and energy convention of Restricted local equilibrium parcel with adopted material state function and Rest fluid state, frame and geometric current definitions. In that state map, $T>0$ is the derivative of the supplied energy per marker with respect to entropy per marker, and the chemical potential is $\mu =H-T{\sigma }_p$ in joules per marker. Rest energy stays in the energy convention used for $\mu$ and for any particle generator; a rest-energy subtraction made in a different convention must be tracked explicitly.

For particle exchange, select and declare one of these physical model protocols:

• Finite-cell exchange. Supply $r\ge 1$ finitely many independent cells with open convex state domains $U_i\subseteq {\mathbb{R}}^2$, coordinates $(E_i,n_i)$, and concave entropy functions $S_i\in C^1(U_i,\mathbb{R})$. Here $E_i$ is in joules, $n_i$ is a continuous dimensionless particle-number coordinate, and $S_i$ is in joules per kelvin. Fix dimensionless lapse values $N_i>0$, a common charge $q$ in coulombs, and potential values ${\mathrm{\Phi }}_i$ in volts in a stated common reference. Put $U={\prod }_i{U}_i$ and $S_{\mathrm{t}\mathrm{o}\mathrm{t}}={\sum }_i{S}_i$. The adopted exchange conserves $\mathcal{E}={\sum }_i(N_i{E}_i+q{\mathrm{\Phi }}_i{n}_i)$ and $\mathcal{N}={\sum }_i{n}_i$. At a supplied feasible interior point $x^{\ast }\in U$, with these totals fixed and ${\mathrm{\partial }}_{E_i}{S}_i(x_i^{\ast })>0$ for every $i$, adopt the following equilibrium criterion: $D{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}(x^{\ast })[h]=0$ for every $h=(\delta E_1,\delta n_1,\dots ,\delta E_r,\delta n_r)$ satisfying ${\sum }_i(N_i\delta E_i+q{\mathrm{\Phi }}_i\delta n_i)=0$ and ${\sum }_i\delta n_i=0$. All such independent infinitesimal variations are admitted. Define $1\mathrm{/}{T}_i={\mathrm{\partial }}_{E_i}{S}_i(x_i^{\ast })$ and ${\mu }_i\mathrm{/}{T}_i=-{\mathrm{\partial }}_{n_i}{S}_i(x_i^{\ast })$, so $T_i>0$ is in kelvins and ${\mu }_i$ is in joules per particle. This criterion is an adopted condition on any selected equilibrium point, not a consequence of conservation, an existence assertion, or a relaxation law. Independent finite entropy-exchange equilibrium supplies its conditional consequences. Identification of these variables with another material state map is an additional model identification, not inferred here.

• Charged kinetic exchange. On the selected stationary domain, supply a species of rest mass $m>0$, charge $q$, smooth local fields $T>0$ and $\mu$, a global smooth real potential $A$ with $F=dA$, and a smooth compensator ${\mathrm{\Lambda }}_{\xi }$ satisfying ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$. Use the future massive shell, charged transport, and dimensionless exponential ansatz exactly as defined in Charged particle and Liouville data on a stationary Lorentzian domain. Adopt whole-shell stationarity ${\mathcal{L}}_{F,q}f=0$ as the exchange criterion. This is a model assumption on every future mass-shell momentum, not an assertion that collisions drive a gas to this ansatz.

The two protocols are separate model choices. This postulate does not derive either protocol from gravity, assert a global electrostatic potential when no single-valued symmetry compensator exists, establish a Gibbs state or material realization, or identify finite entropy variables with kinetic fields without an explicit model assumption.