What is order parameter and how to find critical temperature?

[KFC]

I am reading some material on phase transition. In those, they use order parameter quite a lot, but what is order parameter? What's the physical significance of it?

If I got the free energy , which involving order parameter m, how can I find the critical temperature?

 

[CompuChip]

The order parameter is the number that indicates in which phase you are, briefly put.

What the order parameter is, depends on your system. For example, if you are looking at a ferromagnet, the order parameter would be the average magnetisation (above the critical temperature, the magnetisation is random so the average is zero, below the critical temperature there is a clear preferred direction and the average will be non-zero); for a phase transition between liquid and gas, the order parameter may be the density.

The name comes from the fact that a phase transition usually (dis)orders the system - depending on which phase you are going to - for example by ordering / randomising the magnetisation directions or by ordering molecules in a lattice / breaking the lattice structure.

The phase transition usually occurs when the free energy changes minima. For example, if the free energy is of the form
$$F \sim \alpha(T) T^4 + \beta(T) T^2$$
with $\alpha > 0$, then for $\beta > 0$ F only has a global minimum at T = 0. However, when beta changes sign, a new global minimum occurs at some temperature T > 0 and the system may "slip" from the now local minimum at T = 0 to the new minimum which has even lower energy.

 

[xepma]

The post of CompuChip pretty much explains it all. A small contribution:

In what is called Landau theory (= theory of second order phase transitions) one has the important property that the free energy is expressed in terms of the order parameter. On the basis of symmetries of the Hamiltonian, a proper order parameter and the fact that the free energy is an analytic function one can come up with a reasonable estimate (Taylor series) of the free energy.