I have often heard that the Landau theory of phase transitions is a mean field theory. Why is this so? What is the connection between the two ideas? One stresses symmetry breaking and one averages the interactions between all the particles. Naively, they don't seem related.
"Mean field theory" is a theory which says "don't worry, in a first approximation, about where the particles are and how they are clumped together; just say the net effect of all the other particles on the energy of any one particle is given by an average over all the other particles, and you get the same answer no matter which particle you pick". This is what the Landau method does when it asserts that the free energy is a function of a single internal parameter (the order parameter) in addition to external constraints such as temperature.
The deep relation between mean field theories (MFT) and Landau Theory of phase transitions is rooted in two general features of both approaches (which are not confined to the study of second order transitions):
- both can be cast in the form of a generating functional (Landau free energy or mean field partition function) depending on the order parameter as well as on other thermodynamic variables;
- in both cases such functional is an analytic function of the order parameter. This is a postulate in Landau Theory, and a consequence of the effective one-particle approximation in the case of MFT.
As a consequence, both theories share many features, including the same values for the critical exponents in the neighborhood of a critical point.
As far as I understand, the Landay theory uses the expansion of the free energy in the order parameter near the phase transition that gives the same results as the mean-field theory (near the phase transition) (http://galileo.phys.virginia.edu/~pf7a/rg1.pdf). In real systems, free energy is typically not an analytical function of the order parameter near the continuous phase transition.