Landau theory

[2] It can also be adapted to systems under externally-applied fields, and used as a quantitative model for discontinuous (i.e., first-order) transitions.

Although the theory has now been superseded by the renormalization group and scaling theory formulations, it remains an exceptionally broad and powerful framework for phase transitions, and the associated concept of the order parameter as a descriptor of the essential character of the transition has proven transformative.

Landau was motivated to suggest that the free energy of any system should obey two conditions: Given these two conditions, one can write down (in the vicinity of the critical temperature, Tc) a phenomenological expression for the free energy as a Taylor expansion in the order parameter.

Consider a system that breaks some symmetry below a phase transition, which is characterized by an order parameter

In a simple ferromagnetic system like the Ising model, the order parameter is characterized by the net magnetization

For the system to be thermodynamically stable (that is, the system does not seek an infinite order parameter to minimize the energy), the coefficient of the highest even power of the order parameter must be positive, so

This finite jump is therefore not associated with a discontinuity that would occur if the system absorbed latent heat, since

Landau expanded his theory to consider the restraints that it imposes on the symmetries before and after a transition of second order.

They need to comply with a number of requirements: In the latter case more than one daughter structure should be reacheable through a continuous transition.

, and the Landau free energy will change as a result: In this case, the minimization condition is One immediate consequence of this equation and its solution is that, if the applied field is non-zero, then the magnetization is non-zero at any temperature.

This implies there is no longer a spontaneous symmetry breaking that occurs at any temperature.

, one can find the dependence of the order parameter on the external field: indicating a critical exponent

There are two different formulations, depending on whether or not the system is symmetric under a change in sign of the order parameter.

Here we consider the case where the system has a symmetry and the energy is invariant when the order parameter changes sign.

This shows the clear discontinuity associated with the order parameter as a function of the temperature.

, but its first derivative (the entropy) suffers from a discontinuity, reflecting the existence of a non-zero latent heat.

, and since the free energy is bounded below, there must be two local minima at nonzero values

That is, As in the symmetric case the order parameter suffers a discontinuous jump from

It then followed from Landau theory why these two apparently disparate systems should have the same critical exponents, despite having different microscopic parameters.

It is now known that the phenomenon of universality arises for other reasons (see Renormalization group).

In fact, Landau theory predicts the incorrect critical exponents for the Ising and liquid–gas systems.

The great virtue of Landau theory is that it makes specific predictions for what kind of non-analytic behavior one should see when the underlying free energy is analytic.

Then, all the non-analyticity at the critical point, the critical exponents, are because the equilibrium value of the order parameter changes non-analytically, as a square root, whenever the free energy loses its unique minimum.

The extension of Landau theory to include fluctuations in the order parameter shows that Landau theory is only strictly valid near the critical points of ordinary systems with spatial dimensions higher than 4.

This is the upper critical dimension, and it can be much higher than four in more finely tuned phase transition.

In Mukhamel's analysis of the isotropic Lifschitz point, the critical dimension is 8.

Now, the free energy of the system can be assumed to take the following modified form: where

From these, the Ginzburg criterion for the upper critical dimension for the validity of the Ising mean-field Landau theory (the one without long-range correlation) can be calculated as: In our current Ising model, mean-field Landau theory gives

and so, it (the Ising mean-field Landau theory) is valid only for spatial dimensionality greater than or equal to 4 (at the marginal values of

As a clarification, there is also a Ginzburg–Landau theory specific to superconductivity phase transition, which also includes fluctuations.