Interior exchange conditions from entropy maximization

Given: Two named subsystems in independent separated states, fixed total coordinates, an explicitly declared feasible set $\mathcal{F}$, an attained global entropy-maximizing equilibrium point interior to every exchange direction used, $C^2$ entropy on the local charts, $S_U>0$, and SI coordinates. No empirical premise is used.

Stable equilibrium requires an attained global entropy maximum: For a stable equilibrium of the declared isolated compound, the model adopts an attained global maximum of the summed entropy on the exact feasible endpoint set. It assumes attainment; it proves no dynamics or uniqueness.

Isolated compound and feasible equilibrium set: $\mathcal{F}$ is the declared subset of the product of constituent state domains satisfying the specified totals and exactly the allowed exchanges; retained barriers fix the associated coordinates.

Concave macroscopic fundamental entropy and Additive separated systems and tagged bulk scaling: On the selected smooth single-phase charts the entropy has finite $C^2$ derivatives and $S_U>0$, and independent separated constituents have additive entropy and the declared additive extensive coordinates.

Thermal contact family and constraints: A diathermal contact permits thermal energy transfer between constituents while forbidding matter transfer and mechanical or composition work across that contact. Here it supports the energy-exchange variation only; [F1] and the stated bilateral-feasibility hypothesis govern the other named exchange channels.

Entropic temperature, pressure, and molar chemical potentials: Each $D_{c_i}$ uses dimensionless coordinates, and its physical chart is the image under ${\mathrm{\Phi }}_{c_i}$; the entropy on physical tuples is the composition with ${\mathrm{\Phi }}_{c_i}^{-1}$. Write $(U_i^{\ast },V_i^{\ast },n_i^{\ast })={\mathrm{\Phi }}_{c_i}(x_i^{\ast })$ for the physical SI tuple of subsystem $i$. The derivative identities are $S_{i,U}=1\mathrm{/}{T}_i$, $S_{i,V}=p_i\mathrm{/}{T}_i$, and $S_{i,n_j}=-{\mu }_{i,j}\mathrm{/}{T}_i$, with other physical coordinates fixed.

SI quantities and unit conventions: The coordinates $U,V,n_j$ use joules, cubic metres, and moles; $T$ is in kelvins, $p$ in pascals, and ${\mu }_j$ in joules per mole.

Mechanical pressure-volume work coupling: Along a $C^1$ path of equilibrium states for the specified simple-compressible chart, a constant-area piston satisfies $V=V_{\ast }+A\mathrm{\ell }$ with $A>0$. Such a path is quasistatic in the postulate's stated equilibrium-path sense, mechanically equilibrated when its net normal force vanishes, and frictionless when its tangential contact force vanishes. The postulate adopts $p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=p_c$ and $\delta W_{\mathrm{o}\mathrm{n}}=-p_{c}dV$ on such a path; this is a physical assumption, not a mathematical consequence.

The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$: For differentiable maps between open Euclidean sets, the derivative of a composition is the composition of the derivatives. This is the mathematical chain rule used to differentiate entropy along each feasible one-parameter exchange.