Of particular interest is understanding how microscopic interactions (e.g., polar attraction between individual water molecules) connect to the phase behavior. A classic toy model of this is used to examine magnetism. It's a comparatively simple statistical physics problem to understand how a single magnetic spin (in real life, something like one of the d electrons in iron) interacts with an external magnetic field. The energy of a magnetic moment is lowered if the magnetic moment aligns with a magnetic field - this is why it's energetically favorable for a compass needle to point north. So, one does the statistical physics problem of a single spin in a magnetic field, and there's a competition between this alignment energy on the one hand, and thermal fluctuations on the other. At large enough fields and low enough temperatures, the spin is highly likely to align with the field. Now, in a ferromagnet (think for now about a magnetic insulator, where the electrons aren't free to move around), there is some temperature, the Curie temperature, below which the spins spontaneously decide to align with each other, even without an external field. Going from the nonmagnetic to the aligned (ferromagnetic) state is a phase transition. A toy model for this is to go back to the single spin treatment, and instead of thinking about the spin interacting with an externally applied magnetic field, say that the spin is interacting with an average (or "mean") magnetic field that is generated by its neighbors. This is an example of a "mean field theory", and may be solved self-consistently to find out, in this model, the Curie temperature and how the magnetization behaves near there.

Mean field theories are nice, but it is comparatively rare that real systems are well described in detail by mean field treatments. For example, in the magnetism example the magnetization (spontaneous alignment of the spins, in appropriate units) goes like (1-T/T

_{C})

^{1/2}at temperatures just below T

_{C}. This is not the case for real ferromagnets - the exponent is different. Because of the nature of the approximations made in mean field theory, it is expected to be best in higher dimensionality (that is, when there are lots of neighbors!). Here's a question for experts: what real phase transitions are well described by mean field theory? I can only think of two examples: superconductivity (where the superconducting gap scales like (1-T/T

_{C})

^{1/2}near the transition, just as mean field theory predicts) and a transition between liquid crystal phases. Any others?

## 16 comments:

I've always thought the most impressive thing about Landau theory is that it even works at all, let alone the few places where it has quantitative power.

An attempt to deal with the problems that arise due to the finite dimensionality of real systems is self-consistent renormalization (SCR) [Moriya,

Spin Fluctuations in Itinerant Electron Magnetism, Springer, Berlin, 1985]. Depends, however, if one still considers the MF+SCR combination a mean field theory.QuasiNewton - I had in mind pure MF. In fact, for various reasons, I'm particularly interested in systems where some order parameter really goes like (1-T/Tc)^{1/2} for

T < Tc.

Nice blog, btw. I'll add that to the blogroll.

If the interactions are long ranged, then the exponents will be mean field --- it looks like Phys. Rev. Lett. 29, 917 - 920 (1972) "Critical Exponents for Long-Range Interactions" has details. Btw, is this why the liquid crystal transition is mean-field-like?

Would you consider a Yukawa potential (and other plasma treatments that put electrons in a background field resulting from the ion mass >> electron mass) a mean field theory?

A canonical example of a real phase transition well described by mean field theory is the first order liquid-solid freezing transition in three dimensions. Here, the classical density functional theory of freezing provides remarkably accurate results for the quasi-universal properties of the freezing transition, such as justifications for the validity of the Verlet and Lindemann criteria for freezing, as well as a host of other properties.

For Anonymous: The isotropic-nematic transition in liquid crystals is first order for the reason that a third-order invariant in the Landau expansion for this problem is always allowed, unlike for spin systems in the absence of a magnetic field. This means that, within mean-field theory, a first-order transition should be the norm, rather than the exception, and it is hard to circumvent this argument. This is for symmetry reasons which have nothing to do with the range of interactions, which, in typical nematics, is usually very short ranged.

So Doug, tell me why any normal person would care about the precise numerical value of the exponent? Of course it gives a window into interesting microscopics, and the scale-free behavior is fun and surprising, but I'd be happy if more people understood mean-field behavior in the first place. Before that, I'd wish that they understood the difference between first- and second-order transitions, which you treat very casually in your entry.

Don, you're right that no normal person would care about the precise value of the exponent.

Icare because I've got data that I'm trying to interpret. You're also right that I was more than casual in my treatment of first and second order transitions. If I can think of the right angle, I'll come back to that. One challenge is that the transitions that are most familiar from every day life are first order. It's challenging to explain the comparatively subtle difference between 1st and 2nd order without resorting to examples beyond most people's experience.Anything above its critical dimension will exhibit mean-field like behavior. So this is...ummm.... lots and lots of things in 4 dimensional space... But this not what you were asking.

By the same taken there are also presumably lots of quantum transitions in which zero temperature gives you at least one effective extra dimension in the time. But this is also not what you were asking...

Other than those already mentioned, I know that some ferroelectric transitions are above their critical dimension and mean-field.

Doug,

you're not the first one to resort to using a first-order transition to convey the idea, although to my mind the difference is not subtle at all.

In his otherwise excellent PBS series on string theory a few years ago, Brian Greene illustrated the separation of the electromagnetic and weak forces by holding up ice cubes in water. Now I'm not sure what, if anything, a normal viewer got out of that metaphor. For me it was irksome, because the whole point is that the two forces are

the sameat high temperatures and become different as the temperature is lowered--a classic second-order transition. A first-order transition like water freezing is just a competition between completely different arrangements. I found it particularly annoying because the idea of spontaneous symmetry breaking is one rather clear case in which condensed matter ideas have had a profound influence on high-energy theory.Thanks, Doug.

Could you strain your system (different substrate?) in order to investigate the influence of (broken) crystal symmetries (e.g. inversion) on the phase transition you have observed?

QN - Long story, in progress. If you want to chat more about it, feel free to drop me an email.

Dear Don,

I think that the electroweak transition you mentioned above is generally believed to be actually first-order. So the analogy Brian Greene may not be totally misleading. For example, see this paper:

http://prola.aps.org/abstract/PRD/v45/i8/p2685_1

I think spontaneous symmetry breaking could always occur through a first-order transition.

Partially addressing the question of why we care about the precise value of the exponent. One thing that is crucial here is that the value is universal. ONLY the symmetry of the system matters and the properties of the phase transition do not care at all about the microscopics of the system at all -- and many many things may fall into the same universality class (Ex: melting transitions can be in the Ising class). So maybe we dont' care what the precise value is, but it is interesting that there are a finite set of possibilities in the first place.

An attempt to deal with the problems that arise due to the finite dimensionality of real systems is self-consistent renormalization .....

--

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