I'm aware of the fact that first order and second order phase transitions are distinguished by at what order their free energy suffers a discontinuity as a function of a state variable, and I know Landau's theory deals, vaguely at least, expressing the free energy as a Taylor expansion in the order parameter (I don't actually know what an order parameter is), but I can't work out from looking at it online what it actually is, and why it's valid within only a region close to the transition temperature.


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Loosely speaking, the order parameter is a physical quantity pertaining to the system, whose value allows you to distinguish between a state at $T>T_c$ and a state at $T<T_c$, ($T_c$ is the critical temperature or whatever quantity you're using to define the critical point), since it is $0$ for the former case and $\neq0$ (finite) for the latter. This "binary" behaviour makes it likewise useful to express other quantities of interest, such as the free energy, in terms of the order parameter.

For instance, in the Ising model - a lattice of up and down spins/magnets - the magnetization per spin of the system, $m$ (i.e., total magnetic moment divided by the number of spins), is typically used as an order parameter, as it is $0$ at $T>T_c$ (disordered phase - spins randomly oriented) and finite $\neq0$ at $T<T_c$ (ordered phase - all spins aligned). Another example of an order parameter is the difference between the liquid and vapor densities for a fluid - it is $\neq0$ for $T<T_c$, as both phases are present, and $0$ for $T>T_c$, as they "blend". Let's consider the spins example.

Landau theory kind of tries to "see the bigger picture". Upon realizing that symmetries of the system (which correspond to invariance of the Hamiltonian, $H$, for certain coordinate transformations) that are present when $T>T_c$ are no longer present when $T<T_c$, it considers a generalized free energy, $F$ that doesn't take into account the microscopic degrees of freedom of the system (i.e., it doesn't care if they are spins, magnets, etc), but only its symmetries. The way these symmetries enter $F$ is via the Boltzmann factor in the partition function, by means of $H$.

An example of a symmetry for the spins system would be the invariance of $H$ when switching up/down spins for down/up spins ($H$ has the same form when changing the values of the spins $s_i$ from $-1$ to $1$ and vice versa). Now, on one hand, we know how to get $F$ from $H$ (assuming we know how to calculate the partition function) and so $F$ will have the invariances $H$ has. On the other hand, we'd also like to express the free energy in terms of $m$. The problem is $m=\sum_{i=1}^N s_i$ is positive if the spins are all up, and negative if they're all down - that is, $m$ is not invariant the same way $H$ is -, so we must be careful and expand $F$ in terms of $m$ retaining only those which keep this invariance intact. Thus, in this particular case, $F$ contains only even powers of $m$. Then, by analyzing $F(m)$ as we vary $T$, we're able to predict the behaviour of the system and all just by looking at its symmetries!

The validity of the theory being near the critical temperature has to do with the fact the order parameter is $\sim0$ at $T_c$, and we want to expand $F$ at $m\sim0$, because $F(m)$ will tell us what kind of transition it will be, and will allow us to calculate some quantities characterizing the transition / the critical point (e.g., critical exponents). Check this for more details.


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