A concave entropy supremum not attained on an open energy-partition set

Take two independent separated one-species systems $A$ and $B$ with material constitutions $c_A,c_B$ inside an isolated compound. Write $D_A:=D_{c_A}$ and $D_B:=D_{c_B}$ for their equilibrium domains; each is $(0,\mathrm{\infty }{)}^3$ in the dimensionless coordinates $({\widehat{U}}_i,{\widehat{V}}_i,{\widehat{n}}_i)$ of Equilibrium state domains, and each full domain is designated as an open single-phase chart. Fix positive volumes and amounts ${\widehat{V}}_A,{\widehat{V}}_B,{\widehat{n}}_A,{\widehat{n}}_B$, total internal energy $U_A+U_B=1\mathrm{J}$, and a thermal-contact protocol that permits internal energy transfer while keeping volumes and amounts fixed, as in Thermal contact family and constraints. Adopt the retained product endpoint set and the separately specified total maps below:

$\varphi (x)=((x,{\widehat{V}}_A,{\widehat{n}}_A),(1-x,{\widehat{V}}_B,{\widehat{n}}_B)),\mathcal{F}=\{\varphi (x):0<x<1\}\subseteq D_A\times D_B\mathrm{.}$

Thus $x={\widehat{U}}_A=U_A\mathrm{/}(1\mathrm{J})$ and $1-x={\widehat{U}}_B=U_B\mathrm{/}(1\mathrm{J})$. With $s_{\ast }=1\mathrm{J}{\mathrm{K}}^{-1}$, adopt the toy constitutive entropies $S_A=2s_{\ast }{\widehat{U}}_A$ and $S_B=s_{\ast }{\widehat{U}}_B$. They are finite, continuous, affine (hence concave), $C^2$, and have positive $S_U$ on the stated open domains. For each $\lambda >0$, use the same formula in the own coordinates of the distinct tags $c_A^{(\lambda )}$ and $c_B^{(\lambda )}$. Then the coordinate scaling ${\delta }_{\lambda }$ of Additive separated systems and tagged bulk scaling gives $S_{c_A^{(\lambda )}}({\delta }_{\lambda }{x}_A)=\lambda S_A(x_A)$ and $S_{c_B^{(\lambda )}}({\delta }_{\lambda }{x}_B)=\lambda S_B(x_B)$. Adopt $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x_A,x_B):=S_A(x_A)+S_B(x_B)$ as additional entropy data on the retained product endpoints, independently of simple aggregate composition. Its value on $\mathcal{F}$ is $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(\varphi (x))=s_{\ast }(1+x)$. The model-declared feasible set is an open energy-partition interval, so its entropy has a finite supremum but no attained maximum. This is a conditional model example, not a claim about a real material.

For complete contact data, set $E:=(0,1)$ and take $Z_{A\bigsqcup B}=Z_{A\bigsqcup B}^{\mathrm{e}\mathrm{q}}=\mathcal{F}$, with constituent maps ${\xi }_A,{\xi }_B$ the two coordinate projections and contact family $\mathcal{T}:=\{(c_A,x_A):x_A\in D_A\}\cup \{(c_B,x_B):x_B\in D_B\}$. The two constitutions share one species label. Give the aggregate constitution the domain $D_{A\bigsqcup B}=(0,\mathrm{\infty }{)}^3$, and set ${\xi }_{A\bigsqcup B}(\varphi (x))=(1,{\widehat{V}}_A+{\widehat{V}}_B,{\widehat{n}}_A+{\widehat{n}}_B)$. This is a supplied aggregate coordinate map for process records, not an assertion of an aggregate entropy identity. For each $x,y\in E$ with $y\ge x$, admit the record $P_{x,y}$ of Thermodynamic process records and transfer conventions on $I=[0,1]$ seconds: its aggregate endpoints are both this constant coordinate tuple, $z_{x,y}(t)=\varphi (x)$ for $0\le t<1$ and $z_{x,y}(1)=\varphi (y)$, its control record is $(x,y)$, and its signed transfer record is the real number $x-y$. Declare the control-record set to be $\{(x,y)\in E^2:y\ge x\}$ and the transfer-record set to be $\mathbb{R}$. Each transfer record supplies $H_{P_{x,y}}^{\mathrm{i}\mathrm{n}},H_{P_{x,y}}^{\mathrm{o}\mathrm{u}\mathrm{t}}:[0,1]\to [0,\mathrm{\infty })$, both identically zero, with numerical values in joules; these maps are nondecreasing and start at zero. No path regularity is required by the supplier. The outer boundary admits no matter or energy transfer; fixed constituent volumes and amounts and zero contact work are retained constraints. Supply fixed positive constituent and compound masses and zero endpoint center-of-mass velocities in one Galilean chart, and no separately tracked external-field energy. Thus these are core closed records, with $Q[P_{x,y}]=W_{\mathrm{o}\mathrm{n}}[P_{x,y}]=0$. For the contact-record set ${\mathcal{P}}_T=\{P_{x,y}:x,y\in E,y\ge x\}$ supply $q_{A\to B}[P_{x,y}]=x-y$ in joule magnitudes as in Thermal contact family and constraints. The constituent energy changes are $(y-x)\mathrm{J}$ and $(x-y)\mathrm{J}$. Each feasible endpoint has its self-record; nonzero transfers occur when $y>x$. No contact-equilibrium designation or relaxation law is assumed.

To check consistency with the separately scoped aggregate postulates, take ${\mathcal{C}}_{AB}:=\{(x_A,x_B)\in D_A\times D_B:{\widehat{U}}_A={\widehat{U}}_B\}$ and let ${\kappa }_{AB}$ sum their three coordinates. Adopt $S_{A\bigsqcup B}(\widehat{U},\widehat{V},\widehat{n})=\frac{3}{2}{s}_{\ast }\widehat{U}$. On this nonempty admission set, $S_A+S_B=3s_{\ast }{\widehat{U}}_A=S_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))$, and all dimensioned coordinates add. Use the same aggregate formula on its scaled tags and all positive-coordinate scaling domains. Only the retained partition $x=1\mathrm{/}2$ is in this aggregate admission set; there the product entropy and aggregate entropy both equal $3s_{\ast }\mathrm{/}2$. All other retained partitions have separately adopted product entropy, so no aggregate entropy identity is asserted for them. Totals and entropy on these complete partitioned endpoints are additional declared maps allowed by Isolated compound and feasible equilibrium set. The maximum criterion of Stable equilibrium requires an attained global entropy maximum is used on their stated entropy sum; no aggregate-composition or accessibility theorem is claimed.