# Why are there large fluctuations at the critical point and why does Landau theory work despite such large fluctuations?

The question is about the critical point in a second-order phase transition: Why do fluctuations become so large at the critical point?

As I understand, Landau’s theory of phase transition is some sort of truncated expansion of order parameter around the critical point. According to this theory, at or around the critical point fluctuations are large, hence any mean-field theory should not work. Sufficiently below the critical point, where the order parameter is large, the expansion and truncation at lower powers of order parameters should not hold. Then the question is why Landau theory works despite this fact?

As already mentioned in a comment by elifino, it is generally known that near a critical point, two (or several) different phases, with almost the same free energy, are competing to determine the ground-state (or low-energy states). Therefore, relatively small fluctuations in the system would lead to drastic effects. As the simplest example, in the figure$^\dagger$ below, for a 2d Ising model near criticality, the “critical fluctuations” are shown by islands of black and white (representing up and down directions of the magnetic moment):

$\hskip2in$

Why do fluctuations become so large at the critical point?

Such fluctuations are large as a consequence of the definition of such transitions; namely, the continuous change in the free energy and hence, the competing ground-states. Actually, this is the fact that “is so special about the critical point of a phase transition” (answer to the first question). The Landau-Ginzburg theory of second-order (continuous) phase transitions is, in fact, a phenomenological theory which provides a particularly good description of such a transition because it is based on such an observation. Therefore, the LG theory does not explain per se why the fluctuations are large near the critical point, but it is based on that fact; basically, it is an efficient way to formulate that observed fact.

Landau’s theory of phase transition is some sort of truncated expansion of order parameter around the critical point. According to this theory, at or around the critical point fluctuations are large, hence any mean-field theory shouldn't work. Sufficiently below the critical point, where the order parameter is large, the expansion and truncation at lower powers of order parameters shouldn't hold. So how come everything turns out to be so clean? Why does Landau theory work [so well]?

In LG theory, the statistical average of the magnitude of an “order parametre” (say, $\langle \phi \rangle$) determines the transition point; that is, below the transition, in the ordered phase, it has a finite value (with relatively small fluctuations), and above the transition, in the disordered phase, it vanishes. In between, near the critical point, the fluctuations become stronger and finally destroy the order, in the sense that $\lim_{T \rightarrow T_C} \langle \phi \rangle = 0$. More analytically, this behaviour is described by a free energy (density) which is of the form$^{*}$ $$F_{LG} = r \, \phi^2 + c \, | \nabla \phi |^2 + g_4 |\phi|^4 + \cdots ~,$$ where the coefficients $r$, $c$, $g_4$, etc. depend on the microscopic details of the physical system and are usually a function of temperature and cannot be determined by the Landau-Ginzburg theory itself. Nonetheless, LG theory provides a general and unified explanation of continuous phase transitions in terms of an order parametre and some coefficients — and that is its strength.

LG theory is not a “truncated expansion of order parameter around the critical point”. The order paramater $\langle \phi \rangle$ can actually have any mean-field value, $\phi_{MF}$. The important point is the change from (or fluctuations around) this mean-field value, $\langle \phi \rangle - \phi_{MF}$; that means only the fluctuations around the mean-field value are important – since they could destroy the order. The basic idea is that below the critical point, one devises a free energy (the Landau-Ginzburg free energy) in terms of an order parametre, which yields the possible configurations of the system in terms of some given parametres, $r$, $c$, etc. This free energy is not an expansion of order parametre; originally, the form of the Landau free energy was based on a good choice of order parametre (e.g., magnetization) and the symmetries of the system only. In this approach, the particular value of the order parametre does not matter – e.g., one can rescale it to be in $[-1 , 1]$. The crucial issue is to see how fluctuations would “smear out” this fixed value or even lead to an utterly new configuration with different properties (e.g., from a magnetically-ordered phase to a paramagnetic phase).

In this regard, the LG theory provides a good description of the system below the transition point (provided a proper order parametre is chosen and the symmetries are respected). It will ultimately yield the break-down point of the ordered phase (the transition point). This is essentially the point where the LG theory breaks down itself – due to large fluctuations. More concisely, it tells you where (in the phase-space) the fluctuations overwhelm the system so that the particular LG theory itself ceases to be a good description.

For a detailed discussion, see e.g., Huang, K. “Statistical Mechanics” (1987), chp. 17 <WCat>, or Sethna, J. P. “Statistical Mechanics: Entropy, Order Parameters, and Complexity” (2012), chp. 12 <WCat>.

$^\dagger$ The figure is adopted from the book by Sethna cited above.
$^{\ast}$ Different notations are used depending on the reference material.

• Thank you for your answer. But the answer does not solve my query "As I understand, Landau’s theory of phase transition is some sort of truncated expansion of order parameter around the critical point. According to this theory, at or around the critical point fluctuations are large, hence any mean-field theory shouldn't work. Sufficiently below the critical point, where the order parameter is large, the expansion and truncation at lower powers of order parameters shouldn't hold." Dec 15, 2015 at 20:57
• @nitin: Does the modifications to the post suffice to answer your question? Dec 17, 2015 at 12:04
• Thank you for an elaborate answer. Now I see the point of LG theory and why it works. Just one more quick question. Does studying Landau theory alone (without comparing it with exact solution for the phase transition), can one say at what point it breaks down? Dec 17, 2015 at 13:18
• Yes, you would obtain a transition point from the Landau theory -- in terms of the coefficients r, c, etc. @nitin Dec 17, 2015 at 14:03

Just an aside comment: I think Ginzburg criterion has been not mentioned in this conversation (I apologize if it's not so).

The Ginzburg criterion gives a measure for both the fluctuations $$\delta \phi$$ of the order parameter around the mean constant value $$\phi_0$$ and of the specific heat $$C$$. It turns out that $$$$\frac{<{\delta \phi^2}>}{\phi_0^2} \propto \xi^{4-d}, \quad C\propto \xi^{4-d},$$$$ where $$\xi$$ is the correlation length. This means fluctuations are singular near the transition point where $$\xi\to\infty$$ if $$d<4$$. The same happens for $$C$$ and the divergent behaviour is attributable to the growing fluctuations. In $$d<4$$ the prediction of LG theory are no longer reliable and methods of renormalization must be employed.