Path Dependence and Tipping Points

[This is a transcript for the video embedded below. Part of the text may not make sense without the graphics in the video.]



Most of the physics we learn about in school is, let’s be honest, a little dull. It’s balls rolling down slopes, resistance proportional to voltage, pendula going back and forth and back and forth... wait don’t fall asleep, that’s what I will not talk about today. Today I will talk about weird things that can happen in physics: path dependence and tipping points.

I want to start with chocolate. What’s chocolate got to do with physics? Chocolate is a crystal. No, really. A complicated crystal, alright, but a crystal, and a truly fascinating one. If you buy chocolate in a store you get it in this neat smooth and shiny form. It melts at a temperature between thirty-three and thirty-four degrees Celsius, or about ninety-two degrees Fahrenheit. That’s just below body temperature, so the chocolate will melt if you stuff it into your mouth but not too much earlier. Exactly what you want.

But suppose your chocolate melts for some other reason, maybe you left it sitting in the sun, or you totally accidentally held a hair drier above it. Now you have a mush. The physicist would say the crystal has undergone a phase transition from solid to liquid. But no problem, you think, you will just put it into the fridge. And sure enough, as you lower the temperature, the chocolate undergoes another phase transition and turns back into a solid.

Here’s the interesting thing. The chocolate now looks different. It’s not only that it has lost some of its original shape, it actually has a different structure now. It’s not as smooth and shiny as it previously wass. Even weirder, it now melts more easily! The melting point has dropped from about thirty-four to something like twenty-eight degrees Celsius. What the heck is going on?

What happens is that if the chocolate melts and becomes solid again, it does not form the same crystal structure that it had before. Instead, it ends up in a mixture of other crystal structures. If you want to get the crystal structure that chocolate is normally sold in, you have to cool it down very carefully and add seeds for the structure you want to get. This process is called “tempering”. The crystal structure which you get with tempering, the one that you normally buy, is actually unstable. Even if you do not let it melt, it will decay after some time. This is why chocolate gets “old” and then has this white stuff on the surface. Depending on what chocolate you have, the white stuff is sugar or fat or both, and it tells you that the crystal structure is decaying.

For our purposes the relevant point is that the chocolate can be in different states at the same temperature, depending on how you got there. In physics, we call this a “path dependence” of the state of the system. It normally means that the system has several different states of equilibrium. An equilibrium state is simply one that does not change in time. Though, as in the case of chocolate these states may merely be long-lived and not actually be eternally stable.

Chocolate is not exactly the example physicists normally use for path dependence. The go-to example for physicists is the magnetization of a ferromagnet. A ferromagnet is a metal that can be permanently magnetized. It’s what normal people call a “magnet”, period. The reason ferromagnets can be magnetized is that the electron shell structure means the atoms in the metal are tiny little magnets themselves. And these tiny magnets like to align their orientation with that of their neighbors.

Now, if you find a ferromagnetic metal somewhere out in the field, then its atomic magnets are almost certainly disordered and look somewhat like this. To make the illustration simpler, I will pretend that the atomic magnets can point in only one of two directions. If the little magnets are randomly pointing into one of these directions, then the metal has no overall magnetization.

If you apply a magnetic field to this metal, then the atoms will begin to align with the field because that’s energetically the most favorable state. At some point they’re just all aligned in the same direction, and the magnetization of the metal saturates. If you now turn off the magnetic field, some of those atoms will switch back again just because there’s some thermal motion and so on. However, at room temperature, the metal will keep most of the magnetization. That’s what makes ferromagnets special.

If you turn on the external magnetic field again but increase its strength into the other direction, then the atomic magnets will begin to line up pointing into that other direction until saturated. If you turn down the field back to zero, again most of them will continue to point there. Turn the external field back to the other side and you go back to saturating the magnetization in the first direction.

We can plot this behavior of the magnet in a graph that shows the external magnetic field and the resulting magnetization of the magnet. We started from zero, zero, saturated the magnetization pointing right, turned the external field to zero, but kept most of the magnetization. Saturated the magnetization pointing left, turned the field back to zero but kept most of the magnetization. And saturated the magnetization again to the right.

This is what is called the “hysteresis loop”. Hysteresis means the same as “path dependence”. Whether the magnetization of the metal points into one direction or the other does not merely depend on the external field. It also depends on how you got to that value of the field. In particular, if the external field is zero, the magnet has two different, stable, equilibrium states.

This path-dependence is also why magnets can be used to store information. Path-dependence basically means that the system has a memory.

Path-dependence sounds like a really peculiar physics-y thing but really it’s everywhere. Just to illustrate this I have squeezed myself into this T-shirt from my daughter. See, it has two stable equilibrium states. And they keep a memory of how you got there. That’s a path-dependence too.

Another common example of a path dependence are air conditioning units. To avoid a lot of switching on and off, they’re usually configured so that if you input a certain target temperature, they will begin to cool if the temperature rises more than a degree above the target temperature, but will stop cooling if the temperature has dropped to a degree below the target temperature. So whether or not the air condition is running at the target temperature depends on how you got to that temperature. That’s a path-dependence.

A common property of path-dependent systems is that they have multiple stable equilibrium states. As a reminder, equilibrium merely means it does not change in time. In some cases, a system can very suddenly switch between different equilibrium states. Like this parasol. It has a heavy weight at the bottom, so if the wind sways it a little, it will stay upright. That’s an equilibrium state. But if the wind blows too hard, it will suddenly top over. Also an equilibrium state. But a much more stable one. Even if the wind now blows into the other direction, the system is not going back to the first state.

Such a sudden transition between two equilibrium states is called a “tipping point”. You have probably heard the word “tipping point” in the context of climate models, where they are a particular pain. I say “pain” because by their very nature they are really hard to predict with mathematical modeling, exactly because there are so many path-dependencies in the system. A glacier that melts off at a certain level of carbondioxide will not climb back onto the mountain if carbondioxide levels fall. And that’s one of the better understood path-dependencies.

A much discussed tipping point in climate models is the Atlantic meridional overturning circulation. That’s a water cycle in the atlantic ocean. Warm surface water from the equator flows north. Along the way it cools and partly evaporates, which increases the density of salt in the water and makes the water heavy. The cool, salty water sinks down to the bottom of the ocean, comes back up where it came from, warms, and the cycle repeats. Why does it come back up in the same place? Well, if some water sinks down somewhere, then some water has to come up elsewhere. And a cycle is a stable configuration, so once the system settles in the cycle, it just continues cycling.

But. This particular cycle is not the only equilibrium configuration and the system does not have to stay there. In fact, there’s a high risk this water cycle is going to be interrupted if global temperatures continue to rise.

That’s because ice in the arctic is mostly fresh water. If it melts in large amounts, as it presently does, this reduces the salt content of the water. This can prevent the water in the atlantic overtuning circulation from sinking down and thereby shut off the cycle.

Now, this circulation is responsible for much of the warm wind that Europe gets. Did you ever look at a world map and noticed that the UK and much of middle Europe is North of Montreal? Why is the climate in these two places so dramatically different? Well, that atlantic overturning circulation is one of the major reasons. If it shuts off, we’re going to see a lot of climate changes very suddenly. Aaaand it’s a path-dependent system. Reducing carbondioxide after we’ve crossed that tipping point will not just turn the circulation back on. And some evidence suggests that this cycle is weakening already.

There are many other tipping points in climate models, that, once crossed can bring sudden changes that will stay with us for thousands of years, even if we bring carbondioxide levels back down. Like the collapse of the Greenland and West Antarctic Ice Sheet. If warming continues, the question is not whether it will happen but just when. I don’t want to go through this whole list, I just want to make clear that tipping points are not fear mongering. They are a very real risk that should not be dismissed easily.

I felt it was necessary to spell this out because I recently read an article by Michael Shellenberger who wrote: “Speculations about tipping points are unscientific because levels of uncertainty and complexity are too high, which is exactly why IPCC does not take such scenarios seriously.”

This is complete rubbish. First, tipping points are covered in the IPCC report, it’s just that they are not collected in a chapter called “tipping points,” they are called large scale singular events. I found this out by googling “tipping points IPCC”, so not like it would have taken Shellenberger much of an effort to get this right. Here is a figure from the summary for policy makers about the weakening of the atlantic overturning circulation, that’s the tipping point that we just talked about. And here they are going on about the collapse of ice sheets, another tipping point.

Having said that, tipping points are not emphasized much by the IPCC, but that’s not because they do not take them seriously, but because the existing climate models simply are not good enough to make reliable predictions for exactly when and how tipping points will be crossed. That does not mean tipping points are unscientific. Just because no one can presently put a number to the risk posed by tipping points does not mean the risk does not exist. It does mean, however, that we need better climate models.

Path-dependence and tipping points are cases where naïve extrapolations can badly fail and they are common occurrences in non-linear systems, like the global climate. Just because we’ve been coping okay with climate change so far does not mean it will remain that way.

I want to thank Michael Mann for checking parts of this transcript.