Zeroth Law of Thermodynamics - Zeroth Law As Equivalence Relation

Zeroth Law As Equivalence Relation

A system is said to be in thermal equilibrium when it experiences no net change in thermal energy. If A, B, and C are distinct thermodynamic systems, the zeroth law of thermodynamics can be expressed as:

If A and C are each in thermal equilibrium with B, A is also in equilibrium with C.

This statement asserts that thermal equilibrium is a Euclidean relation between thermodynamic systems. If we also grant that all thermodynamic systems are in thermal equilibrium with themselves, then thermal equilibrium is also a reflexive relation. Relations that are both reflexive and Euclidean are equivalence relations. One consequence of this reasoning is that thermal equilibrium is a transitive relationship: If A is in thermal equilibrium with B and B is in thermal equilibrium with C, then A is in thermal equilibrium with C. Another consequence is that the equilibrium relationship is symmetric: If A is in thermal equilibrium with B, then B is in thermal equilibrium with A. Thus we may say that two systems are in thermal equilibrium with each other, or that they are in mutual equilibrium. Implicitly assuming both reflexivity and symmetry, the zeroth law is therefore often expressed as:

If two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

Again, implicitly assuming both reflexivity and symmetry, the zeroth law is occasionally expressed as the transitive relationship:

If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C.