# Thermodynamic and statistical mechanical definitions of critical point

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How can I see these two definitions coincide? In statistical mechanics, critical point is where correlation length diverges; in thermodynamics, it is defined to be the point where a phase boundary terminates. It is intuitive from the statistical mechanical definition that critical point represents some sort of phase transition point, but the thermodynamic definition clearly seems more demanding. How can I connect them?

EDIT: Thanks to suresh's clarification, the question should be slightly reformed: Critical point is the point where first order phase transition meet second order phase transition, i.e. transitions with and without latent heat phenomenon, respectively. How is this definition the same as "correlation length diverges" definition of critical point?

edited Oct 1, 2014

What do you mean by critical point? In the liquid-gas transition, to me it is the point on the liquid-gas phase bounder where the first-order transition becomes second-order. It is same definition in statistical mechanics and thermodynamics. The correlation length always diverges when ever the order is greater than one.

@suresh, I'm confused by Wikipedia's definition, which explicitly says, I quote:

critical point, also known as a critical state, occurs under conditions (such as specific values of temperature, pressure or composition) at which no phase boundaries exist.

Also on the P-T diagram it shows, after critical point there are only one horizontal dashed line and a vertical dashed line to define the "supercritical fluid" region, which looks quite artificial.

The key point is that the order parameter (density in the liquid-gas transition) doesn't distinguish the two phases. So the dotted lines don't mean a lot to me. However, it is known that the line can be extended and is called the Widom line. The Widom line is hard to see experimentally but the claim is that it has been seen. This is discussed in this Nature physics paper that appeared in 2010 and this theoretical paper from 2011.

@suresh, I see, thank you. One more question, you said critical point is the interface between first and second order phase transition, how do I see it is the same as "correlation length diverges" definition? I apologize if this is standard textbook material.

There is no simple answer. At a second-order phase transition, the interpretation is that various response functions diverge due to long-range correlations and leads to scale invariance. Scale invariances implies that two-point functions are power law -- this implies that the correlation length diverges and the exponential decay is replaced by the power law term. I think looking at the introductory chapter in Kardar's  book "Statistical Physics of Fields" (which I like a lot) might help. See also Ron's answer below.

@suresh, thanks!

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The connection is determined by the statistical field theory at the critical point.

At the critical point, there is, by modern definition, a statistical field (or "order parameter", the name is misleading), and this statistical field has fluctuations which are completely described by some massless (scale invariant) statistical Lagrangian. The Lagrangian tells you how likely any configuration of the field is. The standard example is

$$\int |\nabla \phi|^2 + V(\phi)$$

With V a quartic polynomial. The correlation functions are a power law at the critical point, and the behavior of the statistical theory in a neighborhood of the critical point can be understood completely from the allowed relevant perturbations.

Moving away from the critical point corresponds to modifying the microscopic parameters, adding terms to the critical Lagrangian. When you add a negative mass term, you separate phases, since there is more than one minimum. In the other direction, past the critical point you are adding a positive mass term, which does not separate phases, but extinguishes the long-range power-law correlations in the statistical field, turning them into exponentially decaying correlations.

So the critical theory is, in the usual case, a boundary to a line where you have phase separation. The phase separation is in the direction where $m^2$ is made negative.

The other direction in standard phase diagram picture of the liquid-gas transition corresponds to tilting the statistical field potential with a linear term, which picks out one phase or the other as the stable one. The analog in a Ferromagnet phase diagram is adding a magnetic field. If you are restricted to zero magnetic field, you are on the line where there is symmetry between $\phi$ and $-\phi$, and in this case, you can't bias one minimum over the other.

There are two perturbation directions because the general quartic potential can be written:

$$V(\phi) = \lambda \phi^4 + m^2\phi^2 + a\phi$$

where the constant term is unimportant, as it is absorbed into normalizing the statistical path integral, and the cubic term is absent because you can always recenter $\phi$ by shifting by a constant to get rid of it. The $\lambda$ part is the most relevant operator in the action, and its coefficient flows to a fixed point. The tunable parameters are the lower-order ones, in this case $m^2$ and $a$, or for a ferromagnet at $B=0$, just $m^2$ and that's it.

The two definitions are really equivalent, because tuning to a negative $m^2$ produces a phase coexistence line, terminating at the critical point, while moving $a$ around moves you back and forth through this line, making one phase stable, then the other. This takes care of the standard critical points which are normally experimentally realized.

But, mathematically speaking, there are much more diverse and complicated higher order phase transitions, with more complicated statistical fields, which have many more modes of behavior converging together at the critical point. These are classified as in the simplest case by the allowed relevant deformations of the critical theory. My favorite example is the Lifschitz point, which can be mathematically constructed using a scalar field starting from the next higher order rotationally invariant critical statistical Lagrangian:

$$\int |\nabla^2 \phi|^2 + Z_i |\partial_i \phi|^2 + V(\phi)$$

where $V(\phi) = \lambda \phi^4 + m^2 \phi^2 + a \phi$ is the general quartic potential as before. This description of the Lifschitz point is valid in dimensions close to eight, because a repeat of the usual dimensional analysis for $k^2$ Lagrangians reveals that eight is the critical dimension here (the field $\phi$ has canonical dimension $d-4\over 2$). In eight dimensions, the running of the quartic coupling and coefficient of the $\nabla^4$ kinetic term becomes logarithmic and above eight dimensions the behavior at the Lifschitz point is mean-field.

The $Z_i$ terms are usually considered as separate anisotropic perturbations to the critical theory in the literature, although only the case of $Z_i$ equal preserves $8-\epsilon$ dimensional rotational invariance, and this is the only case I personally thought about. The mass term is always a rotationally invariant relevant perturbation, as is "a", the bias in the field, although this breaks $\phi\to -\phi$ as in the case of the ordinary Wilson-Fisher point. This theory is a perfectly well defined statistical theory, although it is not unitary, so it has no QFT continuation, so it is not an example high-energy folks consider normally.

The $8-\epsilon$ expansion for this critical point was originally worked out by Mukhamel in 1977 (for more recent work, see, for instance, this reference: http://arxiv.org/pdf/cond-mat/0205284.pdf ) , and the diagrammatic analysis is mathematically nearly identical to the $4-\epsilon$ Wilson-Fisher expansion, because if you ignore the modified propagators, the diagram combinatorics is identical.

The anisotropic derivative terms, the ones indicated by $Z_i$, are relevant, and produce phases of the system near the critical point, even requiring that $\phi\to -\phi$ is a symmetry (so that you aren't forced to consider a whole zoo of terms). But there are lots and lots of these crazy phase-sheets in eight or seven dimensions. If your seven-dimensional system has parameters that serve to modify the $Z_i$, the critical point is a boundary between regions with positive and negative $Z_i$. The case where, say, three of the $Z_i$'s are equal and negative and the rest are zero has a critical wavenumber in the three dimensions involved, the fluctuations of the field in those three dimensions make the field prefer to go in rolling waves with a special $k$ value, think of the positive and negative values of $\phi$ making a pattern like zebra stripes. The reason is that $k$ values in an intermediate range are statistically favored over no $k$ values. The wavelength for these spontaneous statistical patterns diverges at criticality, and at zero $Z_i$, the theory is rotationally invariant.

In the seven dimensional case, assuming you can adjust anisotropic parameters individually, you have 7 $Z$ directions, an $m^2$ direction, and an $a$ direction, and the critical point is the place where various lower-dimensional nested phase-sheets with special lower-dimensional subspaces with higher rotational symmetry, all come together and terminate. The positive Z case has no critical wavenumber, the fluctuations in the positive Z case are described by an ordinary Wilson-Fisher like theory, which is mean-field in eight dimensions.

The main lesson is that the structure around the critical points is determined from the relevant perturbation to the critical Lagrangian, however crazy these might be, and this is determined by looking at the number of operators of dimension less than the dimension of space. Each such operator consistent with the imposed symmetry could theoretically be a direction you can tune in the phase diagram. On a computer simulating the higher order phase-transition point, you can tune them all.

if you have an experimental realization of this system in a hypothetical seven dimensional universe, you might be restricted by 7d rotational invariance and simultaneously $\phi\to-\phi$ symmetry, as in a ferromagnet at zero magnetic field, so that you really only have two parameters to twiddle experimentally, the overall $Z$ (negative $Z$ gives the 7d analog of zebra-stripes, positive $Z$ means you have a normal Ising-like model at long distance), the overall $m^2$ (negative $m^2$ means you have two phases with different values of $\phi$, positive $m^2$ means one phase). (which determines which phase is favored). The critical point in this case is the origin, and it is the termination of the phase-separation sheet spanning the quadrant between the negative $m^2$ axis where there is ordinary phase separation and the negative $Z$ axis where there is zebra pattern phase separation. The various directions on the sheet in between are phase-separating behavior parameterized by both m and Z, and I have no idea what they look like.

answered Oct 1, 2014 by (7,720 points)
edited Oct 1, 2014

Thanks Ron, I'll read it in detail when I get home.

+1, this is a pretty intuitive explanation. From what you said, a Lagrangian in statisitical field theory plays the role of free energy in statisitical mechanics?

The Lagrangian is the free energy of the configuration of the long-wavelength statistical field $\phi$.  The free energy (over T, the free entropy) is the (negative of the log of the) probability weight of the configuration, the number of microstates consistent with the macroscopic description, so it's the path integral weight by definition. I call it the Lagrangian because of residual high-energy thinking, but it's the same thing up to an analytic continuation (which is meaningless here, because the Lifschitz theory doesn't continue to a unitary thing).

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