Isolated compound and feasible equilibrium set

Fix a finite list of named subsystems $s_1,\dots ,s_m$, where $m\in \mathbb{N}$ and $m\ge 1$ (The natural numbers $\mathbb{N}$ (von Neumann)), each having its own equilibrium state domain $D_{c_i}$ as defined in Equilibrium state domains. Their separated endpoint state space is the Cartesian product (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$)

$\mathcal{D}:=D_{c_1}\times \cdots \times D_{c_m}\mathrm{.}$

If the boundary also includes named auxiliaries or work devices $d_1,\dots ,d_k$ not already among the $s_i$, where $k\in \mathbb{N}$, supply endpoint state sets $Y_j$ and energy maps $E_j:Y_j\to \mathbb{R}$ with numerical values interpreted in joules. Supply also every coordinate map needed for their matter, volume, retained constraints and separately tracked mechanical or external-field energy. The complete endpoint space is

${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}:=\mathcal{D}\times Y_1\times \cdots \times Y_k\mathrm{.}$

For $k=0$ use ${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}=\mathcal{D}$. Write an endpoint as $(\mathbf{x},\mathbf{y})$ and obtain constituent states by coordinate projection. In the separated additive energy model, declare $E_{\mathrm{t}\mathrm{o}\mathrm{t}}(\mathbf{x},\mathbf{y})={\sum }_{i=1}^m{U}_{c_i}(x_i)+{\sum }_{j=1}^k{E}_j(y_j)\mathrm{J}$, with finite sums as in Finite sums and finite products, by recursion and the empty auxiliary sum zero. Every tracked endpoint energy contribution must be included in these maps or in an additional explicitly supplied contribution; an interacting model must supply its total-energy map on ${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}$. For each entropy-bearing auxiliary supply $S_j^{\mathrm{a}\mathrm{u}\mathrm{x}}:Y_j\to \mathbb{R}$ with values in $\mathrm{J}{\mathrm{K}}^{-1}$. If the complete separated split has an adopted additive entropy model, its total includes those maps as well as the $S_{c_i}$; omitting an auxiliary requires an explicitly constant entropy contribution or a supplied reduced total-entropy function.

An isolated compound consists of all these constituents inside a declared outer boundary, with no matter or energy transfer across that boundary during the admitted process, in the sense of Thermodynamic systems and permitted exchanges. Internal transfers may change auxiliary energy and entropy. On pairs admitted to the simple aggregate composition map of Additive separated systems and tagged bulk scaling, totals and entropy satisfy that postulate and Concave macroscopic fundamental entropy. A retained partition endpoint in ${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}$ is not automatically such an admitted aggregate state: summing its coordinates can discard partition information. For endpoints outside that composition's admitted set, total-energy and total-entropy maps must be supplied separately; no aggregate entropy identity is inferred.

For specified totals and retained barriers, an equilibrium feasible set is the model-declared subset $\mathcal{F}\subseteq {\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}$ of complete endpoint tuples satisfying those totals and exactly the exchange permissions in the declared protocol. Its thermodynamic projection is $\mathrm{\Pi }(\mathcal{F}):=\{\mathbf{x}\in \mathcal{D}:\mathrm{\exists }\mathbf{y}\text{ with }(\mathbf{x},\mathbf{y})\in \mathcal{F}\}$; that projection alone does not record variable auxiliary states, energy or entropy. When $k=0$ the original convention $\mathcal{F}\subseteq \mathcal{D}$ applies. Totals use all supplied endpoint maps and the dimensioned coordinate maps $U_c,V_c,n_{c,j}$ of Equilibrium state domains, rather than treating numerical coordinates as quantities with units. For two constituents without auxiliaries, thermal contact through Thermal contact family and constraints at fixed volumes and mole amounts gives the model-declared allowed energy partitions with prescribed total energy and any further retained constraints. Volume or species exchange changes $\mathcal{F}$ only when explicitly permitted; retained walls fix the corresponding coordinates.

The set $\mathcal{F}$ may be empty, noncompact, or otherwise fail to contain an entropy-maximizing state. This definition imposes no existence, compactness, convexity, uniqueness, or attainment assumption.