# Introduction to Landau theory

Landau theory is a phenomenological mean field theory that aims at describing the occurence of phase transitions in a unitary framework.

We have already stated that even very different many-particle systems exhibit universality: we have seen this explicitly in Critical exponents and universality, but also in the study of mean field theories we have that two very different systems, namely an Ising magnet in the Weiss approximation and a fluid in the Van der Waals theory, have the very same critical exponents. The reason of this fact can be understood if we expand the state equations of these systems in terms of their order parameters. In fact, considering the Van der Waals theory for fluids we have seen in Van der Waals equation that:

${\displaystyle \pi =1+4t-6\eta t-{\frac {3}{2}}\eta ^{3}+O(\eta ^{4},\eta ^{2}t)}$
where we have called ${\displaystyle \eta :=\varrho }$ the order parameter. In a similar way we can expand the state equation of an Ising model in the Weiss approximation. In this case we get:
${\displaystyle {\frac {H}{k_{B}T}}=\eta t+\eta ^{3}+O(t\eta ^{3})}$
where ${\displaystyle \eta :=m}$ is again the order parameter. Therefore we can see how the two equations of state, when expanded around a critical point, behave in the same way, i.e. they involve the same powers of the order parameter ${\displaystyle \eta }$: this is why these two very different systems exhibit the same critical exponents.

From these very simple examples we can prefigure that the properties and the possible occurrence of phase transitions of every universality class can be described once we have expanded the state equation of the system in terms of its order parameter (whatever it is, and independently of its nature). These kinds of observations led Landau to suggest that such considerations can be done for all phase transitions, at least qualitatively. The resulting theory, which we now proceed to study, is the so called Landau theory for phase transitions.