Zeroth-law equivalence

Define a relation $\sim$ on equilibrium states as follows: for two systems (or two equilibrium states) $A$ and $B$, write $A \sim B$ if placing them in contact through a diathermal contact produces no net heat flow and no macroscopic change—i.e., they are in thermal equilibrium .

The zeroth law of thermodynamics asserts the transitivity of this relation: if $A \sim B$ and $B \sim C$, then $A \sim C$. Together with the physically evident symmetry and reflexivity, this makes “being in thermal equilibrium with” an equivalence relation on the set of equilibrium states.

Physical interpretation

Zeroth-law equivalence partitions equilibrium states into equivalence classes: all states in the same class are mutually in thermal equilibrium. The temperature is then understood as a scalar label for these classes: states have the same temperature exactly when they are equivalent under $\sim$.

This is the conceptual basis of thermometry. A thermometer is a system whose equilibrium state changes monotonically across equivalence classes; the zeroth law guarantees that when the thermometer equilibrates with a system, its reading depends only on the system’s temperature class, not on what other systems were involved. With an agreed calibration, this yields a consistent temperature scale, and with additional conventions one can define an absolute temperature scale .

Key properties

Interpreting $\sim$ as “is in thermal equilibrium with,” the zeroth-law structure can be summarized as:

• Reflexive: $A \sim A$ (a system is in thermal equilibrium with itself).
• Symmetric: if $A \sim B$ then $B \sim A$ (no preferred direction once equilibrium holds).
• Transitive: if $A \sim B$ and $B \sim C$ then $A \sim C$ (the substantive content of the zeroth law ).

A standard consequence is the existence of an empirical temperature function $T$ on equilibrium states such that $A \sim B$ if and only if $T(A)=T(B)$; different monotone reparameterizations of $T$ correspond to different but equivalent temperature scales.