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In Statistical mechanics textbooks it is usually purported that first order phase transitions have a finite correlation length $\xi$. Why is that and/or how can we derive that?

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3 Answers 3

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A possible answer is that the second order phase transitions have divergent correlation length "by definition".

Indeed, there is some ambiguity about how one classifies phase transitions, so the divergent correlation length does not follow from some more basic distinction, but usually serves as one of the criteria by which a transition is classified as second order. E.g., Wikipedia says:

Second-order phase transitions are also called "continuous phase transitions". They are characterized by a divergent susceptibility, an infinite correlation length, and a power law decay of correlations near criticality.

(emphasis is mine)

One could roughly classify transitions as the transitions with jumps (first order) and transitions with divergences (second order). Sometimes one introduces other (sub)classes, like infinite order phase transitions, stressing a specific characteristic of these transitions rather than from some more principled classification.

On a deeper level, phase transitions are the simplest example of the emergent behavior, where large collections of particles exhibit new properties, not inherent to their elementary constituents. Thus, we cannot predict and classify all the possible behaviors from the elementary principles (i.e., from the form of the partition function, symmetry or something like that), but rather base our classification on the cases that are known - which are obviously not all the possible cases.

See more on emergent behavior in thread Does physics explain why the laws and behaviors observed in biology are as they are?

Related: What does the correlation function look like in first-order transition?

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  • $\begingroup$ Is it possible to have a "jump" when there is a divergence in the correlation length? If not, why? $\endgroup$
    – Quillo
    Commented Nov 6, 2023 at 9:33
  • $\begingroup$ @Quillo a jump of what? If we are talking about the same quantity, then divergence means infinite jump - hence the distinction is between finite and infinite jumps. I actually think the question needs to be more specific: e.g., if we have a jump in heat capacity, can we show that the correlation length is finite. $\endgroup$
    – Roger V.
    Commented Nov 6, 2023 at 9:38
  • $\begingroup$ The term refers to "One could roughly classify transitions as the transitions with jumps...". I am asking if it's known why transitions with jumps (i.e. 1st order) imply finite correlation length. According to the old Ehrenfest classification, 1st-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable: does this imply finite correlation length? Alternatively, in the modern classification 1st-order phase transitions are those that involve a latent heat: is latent heat incompatible with infinite correlation length? $\endgroup$
    – Quillo
    Commented Nov 6, 2023 at 9:49
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    $\begingroup$ @Quillo In the second order phase transitions quantities behave as $\sim |T-T_c|^{-x}$, where $x>0$ is a critical exponent. This also applies to the heat capacity and the correlation length: $C\sim |T-T_c|^{-\alpha}$, $\xi\sim |T-T_c|^{-\nu}$, both of which follow from the second derivative of the free energy. If the first derivative has a jump, the second derivative does not exist. So there may be also a problem with defining the correlation length in the first order transitions. See Critical exponents. $\endgroup$
    – Roger V.
    Commented Nov 6, 2023 at 10:13
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    $\begingroup$ @Quillo in fact, correlation length characterizes correlation of order parameter at different points - it inherently implies some degree of coarse-graining and can be derived from the free energy functional, but not from the classical free energy that enters Ehrenfest definition (where the free energy is independent on coordinates.) So, if we adopt Ehrenfest classification, we simply cannot talk about correlation length. $\endgroup$
    – Roger V.
    Commented Nov 6, 2023 at 10:26
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The Ising model is a nice example where you can see the effect of the size of the correlation length. You should have the following picture in mind:

enter image description here

The correlation length goes from finite (upper left) to infinite ($J/k_b=2.269$) to finite again. This is a typical second order phase transition. I will first discuss the precise meaning of correlation length and then I will explain how this is related to first order phase transitions.

The correlation function basically answers the following question: at which length scale do fluctuations exist? More formally, we consider a spin $\sigma_i$ at a site $i$ and we ask how much it correlates with another spin at site $j$. $$\bigg\langle(\sigma_i-\langle\sigma\rangle)(\sigma_j-\langle\sigma\rangle)\bigg\rangle$$ Note that we have to subtract the background $\langle\sigma\rangle$. Otherwise we are not measuring fluctuations. For example, at low temperatures we would always measure correlation because of the background.

For high temperature fluctuations decay exponentially fast as a function of the distance $d_{ij}$ between lattice sites. Because of the noise, the size of a fluctuation is finite. Beyond a few correlation lengths, two distinct sites have nothing to do with each other.

For low temperatures fluctuations also decay exponentially fast. The system tries to be uniform, so the fluctuations can only get so big. Note: the picture at $T=1$ is probably out of equilibrium. If given more time it would be one color.

At $T=2.269$, the correlation length diverges. No matter how much you zoom out, you will always see islands of the same color (=fluctuations). This means that a particular spin will correlate with another spin even though it is really far away.

You can now imagine red and blue representing two phases, for example ice and water. In a first order phase transition, the 'new' phase starts appearing as bubbles which grow in size, like in the $T=2$ picture. In a second order phase transition, the new phase starts appearing everywhere at once, like the pictures $T=2.2, T=2.5$. Now I will explain that is unlikely that a first order phase transition has divergent correlation length.

A first order phase transition has a discontinuity in the energy, which means there is latent heat when going through the phase transition. What would happen if a system with latent heat has divergent correlation length? The system would phase transition everywhere at once! All the latent heat would start releasing at every length-scale.

To put differently, in first order phase transitions there is some cost associated with going between different phases. This puts a limit on the size of fluctuations.

This story is not a derivation in the slightest, but hopefully this gave some physical intuition.

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I agree with all the previous answers. Another simple way to see why it is so, is to recall that one of the distinguishing properties of a first-order phase transition is the co-existence of different phases at the transition point (e.g. ice floating in water). Since correlation length obviously cannot be larger that the size of the region with one of the phases, it is finite even in the transition point.

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