# Order parameters

A very important feature of a thermodynamic system that, as we will see later on, is very important if we want to study its behaviour near a phase transition or a critical point is that of order parameter. By order parameter we mean a characteristic quantity of the system which is nonzero for $T and that vanishes when $T\geq T_{c}$ . This quantity can be taken as a sort of "measure" of the order of the system: in fact when the temperature of a system is lower than its critical one the system will exhibit some kind of "order" that was not present at higher temperatures (due to the strong thermal fluctuations), and accordingly the value of the order parameter will keep track of that.

Let us see a couple of examples in order to make these statements more comprehensible. Let us consider a fluid and the projection of its phase diagram in $(T,\rho )$ space (which is of course equivalent to $(T,V)$ space): Projection of the phase diagram of a $PVT$ system in $(T,\rho )$ space

As we can see the difference between the densities of the system in its liquid and gaseous phases $\rho _{\text{l}}-\rho _{\text{g}}$ vanishes when $T\geq T_{c}$ (because for those temperature there is actually no distinction between a liquid and a gas) and grows as the temperature drops under the critical temperature. Thus, this difference of densities can be taken as the order parameter of the system (the "order" consisting in the less chaotic motion that the particles have in the liquid phase). Let us now consider, on the other hand, a magnetic system and the projection of its phase diagram in $(M,T)$ space when $H=0$ , as shown in the following figure: Projection of the phase diagram of a magnetic system in $(T,M)$ space

(where the symmetry of the curve is due to the fact that, as we have seen at the end of Phase transitions and phase diagrams, the magnetization of the system can point upwards or downwards).

In this case, we see that the magnetization $M$ itself satisfies the right requirements in order to be the order parameter of the system (and in this case the "order" is quantified in the amount of spins that point in the same direction).

A very important feature of an order parameter, as we shall see when we will be discussing the statistical mechanics of critical phenomena, is its nature: the properties of the system and its behaviour near a critical point will depend strongly on the fact that the order parameter of the system is either a scalar, a vector, a tensor etc.

Sometimes, order parameters can be distinguished in conserved ones (like in the case of a fluid, where the average density of the system remains constant during the transition) and not conserved ones (like the magnetization of a magnetic system).