# Critical Slowing Down

I realize that it’s been a long while since I’ve written a post, so the topic of this one, while unintentionally so, is quite apt.

Among the more universal themes in studying phase transitions is the notion of critical slowing down. Most students are introduced to this idea in the context of second order phase transitions, but it has turned out to be a useful concept in a wide range of systems beyond this narrow framework and into subjects well outside the purview of the average condensed matter physicist.

Stated simply, critical slowing down refers to the phenomenon observed near phase transitions where a slight perturbation or disturbance away from equilibrium takes a really long time to decay back to equilibrium. Why is this the case?

The main idea can be explained within the Landau theory of phase transitions, and I’ll take that approach here since it’s quite intuitive.  As you can see in the images below, when the Landau potential is far from $T_c$, the potential well can be approximated by a parabolic form. However, this is not possible for the potential near $T_c$.

Mathematically, this can be explained by considering a simple form of the Landau potential:

$V(\phi) = \alpha (T-T_c) x^2 + \beta x^4$

Near $T_c$, the parabolic term vanishes, and we are left with only the quartic one. Although it’s clear from the images why the dynamics slow down near $T_c$, it helps to spell out the math a little.

Firstly, imagine that the potential is filled with some sort of viscous fluid, something akin to honey, and that the dynamics of the ball represents that of the order parameter. This puts us in the “overdamped” limit, where the order parameter reaches the equilibrium point without executing any sort of oscillatory motion. Far from $T_c$, as aforementioned, we can approximate the dynamics with a parabolic form of the potential (using the equation for the overdamped limit, $\dot{x} = -dV/dx$):

$\dot{x} = -\gamma(T) x$

The solution to this differential equation is of exponential form, i.e. $x(t) = x(0)e^{-\gamma(T) t}$, and the relaxation back to equilibrium is therefore characterized by a temperature-dependent timescale $\tau =1/\gamma(T)$.

However, near $T_c$, the parabolic approximation breaks down, as the parabolic term gets very small, and we have to take into consideration the quartic term. The order parameter dynamics then get described by:

$\dot{x} = -\beta x^3$,

which has a solution of the form $x(t) \sim 1/\sqrt{\beta t}$. Noticeably, the dynamics of the order parameter obey a much slower power law decay near $T_c$, as illustrated below:

At this point, one would naively think, “okay, so this is some weird thing that happens near a critical point at a phase transition…so what?”

Well, it turns out that critical slowing down can actually serve as a precursor of an oncoming phase transition in all sorts of contexts, and can even be predictive! Here are a pair of illuminating papers which show that critical slowing down occurs near a population collapse in microbial communities (from the Scheffer group and from the Gore group). As an aside, the Gore group used the budding yeast Saccharomyces cerevisiae in their experiments, which is the yeast used in most beers (I wonder if their lab has tasting parties, and if so, can I get an invitation?).

Here is another recent paper showing critical slowing down in a snap-through instability of an elastic rod. I could go on and on listing the different contexts where critical slowing down has been observed, but I think it’s better that I cite this review article.

Surprisingly, critical slowing down has been observed at continuous, first-order and far-from-equilibrium phase transitions! As a consequence of this generality, the observation of critical slowing down can therefore be predictive. If the appropriate measurements could be made, one may be able to see how close the earth’s climate is to a “tipping point” from which it will be very difficult to return (due to hysteresic effects) (see this paper which shows some form of critical slowing down in previous climatic changes in the earth’s history). But for now, it’s just interesting to look for critical slowing down in other contexts that are a little easier to predict and where perhaps the consequences aren’t as dire.

*Thanks to Alfred Zong who introduced me to many of the above papers

**Also, a shout out to Brian Skinner who caught repeated noise patterns in a recent preprint on room temperature superconductivity. Great courage and good job!

### 3 responses to “Critical Slowing Down”

1. Anand Bhattacharya

Anshul – a plug for our recent paper about critical slowing down at a coupled charge/magnetic-ordering transition in La(1/3)Sr(2/3)FeO3 – https://www.nature.com/articles/s41467-018-04199-4 .

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2. D.K.

A pity that the first sentence of the abstract is grammatically incorrect: the coupling (of many things) IS incorrect…

The physics observed is nice though!

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3. Rama

Regarding the superconductivity paper which Brian Skinner has analysed. This noise aspect pointed out by B Skinner is encountered when one measures properties in thermogravimetry, for example at temp below -4 deg C. This happens in differential scanning calorimetry, The reason for this is supercooling and shift in thermodynamic equilibrium. One has to repeat the experiment several times to find out the difference between the correct reading and noise. This takes time and money since you must have a large amount of sample prepared. There is no error analysis reported in this publish and flourish , neoliberal way of doing science.

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