Thermal-contact axioms T1–T4

Use thermal joins and ${\sim }_T$ exactly as defined in Thermal joins and thermal equivalence. For every ordered pair of simple systems and every admitted finite join, adopt:

T1 (joining). $(x,y)\prec J_{ij}(x,y)$.

T2 (reversible splitting). Every join state has at least one split into component states at its stated total energy and work coordinates that is reversible to the join state. Also, for every simple-system state $x=(u,v)$ and every $0<t<1$, adopt

$J_{ti,(1-t)i}(tx,(1-t)x)=(u,tv,(1-t)v)\sim (tx,(1-t)x)\mathrm{.}$

The displayed equality specifies coordinates in the tagged join of the two scaled types; it does not identify that join with the original one-work state $x$. A5 of Accessibility axioms A1–A6 separately gives $(tx,(1-t)x)\sim x$, hence transitivity also gives equivalence of the join and $x$.

T3 (thermal-equilibrium transitivity). If $x{\sim }_{T}y$ and $y{\sim }_{T}z$ in the model’s declared contact family, then $x{\sim }_{T}z$. This requires the model to provide the relevant joined types and identifications.

T4 (transversality). For each simple-system state $x$, there are $x_0,x_1$ in that system with $x_0\prec \prec x\prec \prec x_1$ and $x_0{\sim }_T{x}_1$.

The axioms apply to admitted types and their joins only. T4 is an assumption about the contact relation; it is not a consequence of an equation of state or of a thermometer reading. The stronger range condition T5 is stated separately in Universal thermal-equilibrium range T5, and join coherence C1 is separately stated in Equilibrium-join coherence C1.