Equilibrium state domains

Fix a system $s$ as in Thermodynamic systems and permitted exchanges. A material constitution $c$ specifies a finite ordered list of named species $({\sigma }_1,\dots ,{\sigma }_r)$, with $r\in \mathbb{N}$ (The natural numbers $\mathbb{N}$ (von Neumann)) and $r\ge 1$, and the model parameters that identify the material. An equilibrium-preparation label in the set $E_c$ selects a single macrostate that the model designates as an equilibrium state for constitution $c$. The equilibrium designation is primitive model input here; it is not a dynamical equilibration theorem or a global-stability claim. For each admitted preparation, its mechanical model supplies a positive total mass and a defined center-of-mass velocity as in Galilean inertial charts and the center-of-mass rest frame, so the allowed boost can select a center-of-mass rest chart in which its internal energy is recorded. A preparation for which these mechanical data are unavailable lies outside this framework. A statistical preparation that instead supplies a normalized probability measure on a declared measurable space is as in Probability measures and probability spaces and belongs to a separate statistical branch; it is an element of $E_c$ only if that model also assigns it a single equilibrium macrostate.

Each $e\in E_c$ is assigned a unique coordinate tuple $(U,V,n_1,\dots ,n_r)$, where $U$ is the adopted internal-energy coordinate in joules as in Galilean inertial charts and the center-of-mass rest frame, $V>0$ is volume in cubic metres, and each $n_j>0$ is an amount in moles; the SI unit assignments are those of SI quantities and unit conventions. These are real-valued mole amounts in the bulk continuous chart, not integer particle counts. Define the dimensionless numerical coordinates $\widehat{U}=U\mathrm{/}(1\mathrm{J})$, $\widehat{V}=V\mathrm{/}(1{\mathrm{m}}^3)$, and ${\widehat{n}}_j=n_j\mathrm{/}(1\mathrm{m}\mathrm{o}\mathrm{l})$. The representation map ${\pi }_c:E_c\to {\mathbb{R}}^{r+2}$ is a function as in A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain, sends $e$ to $(\widehat{U},\widehat{V},{\widehat{n}}_1,\dots ,{\widehat{n}}_r)$, and the equilibrium state domain is its image $D_c:={\pi }_c(E_c)\subseteq {\mathbb{R}}^{r+2}$ using inclusion as in Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$. For $x=(\widehat{U},\widehat{V},{\widehat{n}}_1,\dots ,{\widehat{n}}_r)\in D_c$, write $U_c(x):=\widehat{U}\mathrm{J}$, $V_c(x):=\widehat{V}{\mathrm{m}}^3$, and $n_{c,j}(x):={\widehat{n}}_j\mathrm{m}\mathrm{o}\mathrm{l}$ for its dimensioned coordinates. Preparation labels with the same coordinate tuple represent the same thermodynamic state in this framework. Statistical preparations that specify a probability law belong to a separate branch and are not used in this state-domain definition.

For fixed $n=(n_1,\dots ,n_r)$ in a chart used by the model, write ${\mathrm{\Gamma }}_{c,n}=\{(\widehat{U},\widehat{V}):(\widehat{U},\widehat{V},\widehat{n})\in D_c\}\subseteq {\mathbb{R}}^2$, with inclusion notation as in Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$. This designated slice is a specified nonempty open convex subset. With the dimensionless Euclidean norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ (a norm by Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation), openness means that every $z\in {\mathrm{\Gamma }}_{c,n}$ has some $ϵ>0$ for which $\{w\in {\mathbb{R}}^2:\mathrm{\parallel }w-z{\mathrm{\parallel }}_2<ϵ\}\subseteq {\mathrm{\Gamma }}_{c,n}$. Convexity means that $(1-t)z_0+t{z}_1\in {\mathrm{\Gamma }}_{c,n}$ whenever $z_0,z_1\in {\mathrm{\Gamma }}_{c,n}$ and $t\in [0,1]$, as in A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points. Both properties are assumptions about the particular slice, not conclusions about every $D_c$; no coordinate tuple is presumed physically realizable merely because it lies in an ambient Euclidean space. A zero amount for an absent species belongs only to a separately specified boundary domain and is not in this positive-mole chart.

In a simple-compressible chart, volume $V$ is the only mechanical coordinate in the equilibrium representation; composition changes, fields, polarization, surface terms, and other mechanical coordinates require a separately declared state domain and exchange rules. No pressure or reversible-work formula follows from this definition.