# Status of the zeroth law of thermodynamics

The**zeroth law of thermodynamics**is the law which states that

*thermal equilibrium*is

*transitive*. Two systems are said to be in thermal equilibrium when 1) both of the systems are in a state of equilibrium, and 2) they remain so when they are brought into contact, where 'contact' is meant to imply the possibility of exchanging heat, but not work or particles. Thus, thermal equilibrium is a relation between thermodynamical systems. In the following we will write for 'A is in thermal equilibrium with B'.

The *zeroth law* states that this relation is transitive, which means that whenever system A is in thermal equilibrium with B, and A is in thermal equilibrium with system C, then B and C are also in thermal equilibrium. Formally:

**The zeroth law**:

## Temperature and the zeroth law

It is often claimed, for instance by Max Planck in his influential textbook on thermodynamics, that this law proves that we can define a temperature function, or more informally, that we can 'construct a thermometer'. Whether this is true is a subject in the philosophy of thermal and statistical physics. We will look at it in a formal way.

**Definition:** If we can assign to the state spaces of all thermodynamical systems functions such that , where is the state of system , we can define a temperature function.

**Claim:** The zeroth law of thermodynamics implies that we can define a temperature function.

It is easy to see that the zeroth law is a *necessary* condition for the existence of a temperature function. The '=' in is, of course, a transitive relation, so should be so as well. The '=' relation is not just transitive, it is an equivalence relation, which means that it is reflexive, symmetric and transitive. We would need two additional 'laws of thermodynamics' to express this:

**Reflexivity:**

**Symmetry:**

The relation "is in equilibrium with" is symmetric by definition. It is trivial to extend the relation "is in equilibrium with" so that A~A.

The temperature so defined may indeed not look like the centigrade temperature scale, or even be continuous, but it is a temperature function.

Particular systems may have continuous states, in which case states of constant temperature will form surfaces in the state space, and the normal provides a natural order of nearby surfaces. It is then simple to construct a global temperature function that provides an continuous ordering of states.

For example, if two systems of ideal gas are in equilibrium, then P_{1}V_{1}/N_{1} = P_{2}V_{2}/N_{2} where P_{i} is the pressure in the i^{th} system, V_{i} is the volume, and N_{i} is the 'amount' of gas.

The surface defines surfaces of equal temperature, and the obvious (but not only) way to label them is to define T so that where R is some constant. These systems can be now be used as a thermometer to calibrate other systems.

## References

- Jos Uffink, J. van Dis, S. Muijs; Grondslagen van de Thermische en Statistische Fysica; Utrecht University