Thermal contact family and constraints

For each material constitution $c$, let $D_c$ be the equilibrium state domain from Equilibrium state domains. A thermal-contact family is a model-declared set $\mathcal{T}$ of tagged pairs $a=(c_A,x_A)$ with $x_A\in D_{c_A}$. Its tag and coordinate projections are $\mathrm{tag}(a)=c_A$ and $\mathrm{coord}(a)=x_A$; equal numerical coordinate tuples with different constitution tags are not identified.

Fix two members $a=(c_A,x_A)$ and $b=(c_B,x_B)$, named constituents $A,B$ of those constitutions, and a declared contact-protocol convention. A thermal-contact protocol is an admitted preparation/process record $P$ of their compound system $A\bigsqcup B$, with the configuration curve $z_P:I_P\to Z_{A\bigsqcup B}$ and equilibrium endpoint configurations specified in Thermodynamic process records and transfer conventions. The contact model supplies constituent endpoint-coordinate maps

${\xi }_A:Z_{A\bigsqcup B}^{\mathrm{e}\mathrm{q}}\to D_{c_A},{\xi }_B:Z_{A\bigsqcup B}^{\mathrm{e}\mathrm{q}}\to D_{c_B},$

and requires ${\xi }_A(z_P(t_0))=x_A$ and ${\xi }_B(z_P(t_0))=x_B$. These constituent maps are additional declared model data; they are not inferred from the compound's coordinate tuple. Write $x_A^{+}={\xi }_A(z_P(t_1))$ and $x_B^{+}={\xi }_B(z_P(t_1))$ for the constituent final coordinates. The two constituents are separated by a declared diathermal contact: matter transfer and mechanical or composition work across that contact are forbidden, while the protocol permits thermal energy transfer between the constituents. The compound's external boundary and all retained constraints are specified as in Thermodynamic systems and permitted exchanges and Thermodynamic process records and transfer conventions. No continuity or differentiability of the configuration curve is added here.

Let ${\mathcal{P}}_T(A,B)$ be the set of contact records admitted by the fixed convention. The model supplies an internal transfer map $q_{A\to B}:{\mathcal{P}}_T(A,B)\to \mathbb{R}$, whose values are numerical magnitudes in joules: $q_{A\to B}[P]>0$ means net thermal energy transferred from $A$ to $B$ across this contact during $I_P$. Set $q_{B\to A}[P]:=-q_{A\to B}[P]$. Zero net internal heat transfer means exactly $q_{A\to B}[P]=0$; it does not assert zero transfer at every intermediate time. This map is additional protocol data, distinct from the compound's external-boundary map $Q$ in Thermodynamic process records and transfer conventions. In particular, an isolated compound may have nonzero $q_{A\to B}$ while its external $Q$ is zero. Identifying this internal map with constituent heat records requires separately supplied constituent records and consistent boundary assignments.

For each tested ordered pair $(a,b)$, the convention specifies one admitted contact preparation $P_{a,b}\in {\mathcal{P}}_T(A,B)$ with the initial coordinates above, and a subset ${\mathcal{E}}_T(A,B)\subseteq {\mathcal{P}}_T(A,B)$ designated as equilibrium contact preparations. These are model inputs, not an equilibration theorem. The pair is in thermal contact equilibrium, written $a{\sim }_{T}b$, exactly when its selected record $P_{a,b}$ belongs to ${\mathcal{E}}_T(A,B)$, has $x_A^{+}=x_A$ and $x_B^{+}=x_B$, and satisfies $q_{A\to B}[P_{a,b}]=0$. Untested pairs are not in the relation. This criterion is protocol-dependent: changing the selected preparation or retained constraints can change the relation. For fixed constitutions, coordinate notation $x_A{\sim }_T{x}_B$ abbreviates $(c_A,x_A){\sim }_T(c_B,x_B)$ using the two specified tag embeddings. It does not define a relation on untagged coordinates across unspecified material types. The relation is defined only on the stated family $\mathcal{T}$ and for the declared contact protocols. This definition does not assert that ${\sim }_T$ is reflexive, symmetric, or transitive, and it does not assign a real-valued temperature to its equivalence classes. Lieb–Yngvason §IV.A instead defines thermal equilibrium using adiabatic equivalence with a thermal join; the present record-based criterion is a separately adopted convention.