Logarithmic contact model with coherent joins

Fix a finite list of scaled components with amounts $t_j>0$ and labels $i_j\in I$. Write ${\alpha }_j=t_j{c}_{i_j}$, ${\beta }_j=t_j{r}_{i_j}$, $d_j=t_j{u}_{i_j}^{\ast }$, and $e_j=t_j{v}_{i_j}^{\ast }$, with $C={\sum }_j{\alpha }_j>0$. For a split of total energy $U>0$ into positive $U_j$ with ${\sum }_j{U}_j=U$, put $p_j={\alpha }_j\mathrm{/}C$ and $r_j=U_j\mathrm{/}{\alpha }_j$. Then ${\sum }_j{p}_j{r}_j=U\mathrm{/}C$. By [F5] and monotonicity of $\log$ from [F3], ${\sum }_j{\alpha }_j\log (U_j\mathrm{/}{d}_j)={\sum }_j{\alpha }_j\log ({\alpha }_j\mathrm{/}{d}_j)+C{\sum }_j{p}_j\log r_j\le {\sum }_j{\alpha }_j\log ({\alpha }_j\mathrm{/}{d}_j)+C\log (U\mathrm{/}C)={\sum }_j{\alpha }_j\log ({\alpha }_{j}U\mathrm{/}(C{d}_j))\mathrm{.}$ Equality in the finite weighted Jensen step occurs only when all $r_j$ are equal: [F6] gives strict concavity for any unequal pair with positive weights, and grouping terms applies this two-point strict inequality inductively. Since the weighted average is $U\mathrm{/}C$, equality requires $r_j=U\mathrm{/}C$, or $U_j={\alpha }_{j}U\mathrm{/}C$. This allocation is positive and feasible, so the maximum is attained uniquely. The work-coordinate terms do not depend on the energy split, giving exactly the displayed join entropy. Grouping nested joins is the same finite maximization over all $U_j$, so the formula is associative under grouping and permutation.

On each fixed composition, define accessibility by nondecrease of the displayed entropy. Reflexivity and transitivity follow from the real order. A common spectator adds the same entropy to both sides; positive scaling multiplies each entropy comparison by a positive factor. For a simple scaled type, replacing $(U,V,t)$ by $(sU,sV,st)$ leaves both log ratios unchanged and multiplies their coefficients and constant term by $s$. In a join, scaling every amount, energy, and work coordinate by $s$ likewise leaves ${\alpha }_{j}U\mathrm{/}(C{d}_j)$ and $V_j\mathrm{/}{e}_j$ unchanged and scales all coefficients and additive constants by $s$. Extensivity gives ${\sigma }_{(s+t)i}((s+t)x)={\sigma }_{si}(sx)+{\sigma }_{ti}(tx)$, proving reversible splitting. For the stability condition, if $\sigma (A)+{ϵ}_n\sigma (Z_0)\le \sigma (B)+{ϵ}_n\sigma (Z_1)$ for all $n$ and ${ϵ}_n\downarrow 0$, the four entropies are finite. Moreover, the premise includes equality of aggregate composition: $\nu (A)-\nu (B)={ϵ}_n(\nu (Z_1)-\nu (Z_0))$ for every $n$. Choose two different values of ${ϵ}_n$, which exist because the positive sequence tends to zero; subtracting these equalities gives $\nu (Z_1)=\nu (Z_0)$, and then $\nu (A)=\nu (B)$. Taking limits in the entropy inequality gives $\sigma (A)\le \sigma (B)$, now with the same-composition condition required by the accessibility definition. This verifies the stated accessibility and stability hypotheses on the declared comparison domains.

Every simple scaled or finite joined type has entropy of the form $f(U,V)=C\log U+{\sum }_j{\beta }_j\log V_j+K$, where $C,{\beta }_j>0$ and $K$ is constant on that type. Its chart is a nonempty open convex positive orthant. Strict concavity of $\log$ from [F6] gives coordinatewise concavity of $f$, hence $tf(x)+(1-t)f(y)\le f(tx+(1-t)y)$ for $0<t<1$. The two encoded states have the same composition, so the entropy-defined order proves the convex-combination access axiom. Increasing $U$ at fixed work tuple strictly increases $f$, providing a strict successor and S1 on every such type.

For the type in step 1.3, the forward sector through $(U_0,V_0)$ is the superlevel set of $f$, with boundary graph $U=U_0{\prod }_j(V_j\mathrm{/}{V}_{0,j}{)}^{-{\beta }_j\mathrm{/}C}$ over the positive work orthant. Differentiation by [F4] gives graph slopes $-{\beta }_{j}U\mathrm{/}(C{V}_j)$, so put $P_j(U,V)={\beta }_{j}U\mathrm{/}(C{V}_j)$. Concavity gives $f(y)\le f(x)+Df(x)\cdot (y-x)$; hence $f(y)\ge f(x)$ implies $Df(x)\cdot (y-x)\ge 0$, proving that the tangent plane supports the sector. The energy derivative $C\mathrm{/}U$ is nonzero. Any supporting normal annihilates every boundary tangent vector, by testing a graph curve with that tangent for both parameter signs. The tangent space has codimension one, so this supporting plane is unique. On a sufficiently small closed coordinate box inside the positive orthant, the partial derivatives of each rational map $P_j$ are bounded; varying coordinates successively bounds $\mathrm{\parallel }P(y)-P(z)\mathrm{\parallel }$ by a fixed multiple of $\mathrm{\parallel }y-z\mathrm{\parallel }$. Thus $P$ is locally Lipschitz. A straight segment between distinct work tuples stays in the positive orthant and lifts injectively and continuously to the graph. This proves S2–S3 for simple scaled types and all finite joins.

Step 1.1 shows that joined entropy is the maximum of separated entropy at its total energy and work tuple. Thus every split compares upward to the join, and the maximizing split compares back, proving T1 and the existence clause of T2. For each simple scaled or finite joined type $T$ let $C_T>0$ be its coefficient of $\log U$ and define ${\theta }_T(U,V):=U\mathrm{/}{C}_T$. For a pair of such types $T,R$, maximizing over their two energies gives $U_T\mathrm{/}{C}_T=U_R\mathrm{/}{C}_R$, by the same strict-concavity argument of step 1.1. The split is reversible with its join exactly when its entropy reaches this maximum, so $x{\sim }_{T}y$ if and only if ${\theta }_T(x)={\theta }_R(y)$. Equality of real labels is transitive, proving T3. Positive scaling sends $(U,C_T)$ to $(sU,s{C}_T)$ and preserves $\theta$; therefore an equal-state split into any positive scaled copies has equal labels and is reversible with its join, completing T2.

Fix a state $x=(U,V)$ in any simple scaled or finite joined type $T$, and any positive work tuple $W$ in any second such type $R$. Choose $Y=(C_{R}U\mathrm{/}{C}_T,W)$. It belongs to the second chart and has ${\theta }_R(Y)={\theta }_T(x)$, so step 2.2 gives $x{\sim }_{T}Y$, proving every ordered-pair instance of T5. For T4 keep $U$ fixed and replace one positive work coordinate $V_j$ by $V_j\mathrm{/}2$ and $2V_j$, leaving the others fixed. Its positive coefficient ${\beta }_j$ makes the resulting states $x_0\prec \prec x\prec \prec x_1$, while their equal energy gives the same $\theta$ and hence $x_0{\sim }_T{x}_1$ by step 2.2. Every type has at least one work coordinate, so this construction applies throughout the join-closed family.

If states of any two simple types $T,R$ are thermally equivalent, step 2.2 gives their common label $\theta$ and energies $U_T=C_T\theta$, $U_R=C_R\theta$. Their join has coefficient $C_T+C_R$ and energy $U_T+U_R$, so its label is also $\theta$. It is therefore thermally equivalent to each constituent by step 2.2, proving C1. Finite grouping preserves this identity by step 1.1. Every logarithm argument is a ratio of like positive quantities, hence dimensionless. The label $\theta$ is mathematical; this example asserts no empirical temperature law.