The strange maths behind freezing water, pandemics and rumours

We tend to imagine that science has solved pretty much all the mysteries of everyday physics. But evidence of how wrong this is can be found right in your kitchen. For example, when you make ice cubes. 

Ice forms when the water molecules, vigorously jiggling around at higher temperatures, slow down and arrange themselves into a hexagonal crystal lattice as the liquid freezes.
 

Tipping points and the bewildering power of mathematics

This transition, as mundane as it may seem, happens to be exemplary for a tantalising mystery, says Nathanaël Berestycki, mathematician and professor of stochastics at the University of Vienna: "You might think that ice forms gradually: the more you lower the temperature, the more ice you get. But this is not the case." As the water cools down, it reaches a critical temperature after which its overall behaviour abruptly flips.

This is what puzzles mathematicians like Nathanaël Berestycki about phase transitions: "How does a small change in a system lead to such a dramatic effect? You need sophisticated mathematics to describe it, and we do not have a complete picture as to what exactly happens as the systems flips at the tipping point." 

Despite the many question marks surrounding them, our world is filled to the brim with phase transitions. "As soon as you understood the concept of a phase transition, you cannot unsee that they are everywhere", says Berestycki. For example, popcorn suddenly pops once a critical temperature is reached. Or take a pandemic: if each infected person spreads the virus to more than one other person on average, the outbreak jumps from controllable to explosive. "Similarly, you can think of a rumour as an idea that spreads like a virus: once it hits a critical pass-on rate, it spreads rapidly through the population," he adds. 

These examples may seem wildly different, yet they are all mathematically analogous to the phase transition between liquid and solid states of water. “This showcases the power of mathematics to describe a whole host of physical phenomena, says Berestycki, who co-leads the Special Research Area 'Discrete Random Structures' at the University of Vienna. "It makes me think of what physicist Eugene Wigner called the 'unreasonable effectiveness of mathematics': it is so good at describing reality, it is almost uncanny." 

To describe physical phenomena, social networks or any other system, mathematicians try to establish a mathematical model, that is "a more or less accurate mathematical description of something ," explains Berestycki. A good model allows us to understand the properties of a system and to predict how they change when certain parameters are tweaked. 

For example, a model of contagion helps epidemiologists predict whether a disease will die out or explode. Albert Einstein's theory of general relativity includes a model of gravity that he used to predict the existence of black holes. Models are the theoretical tools that can lead us to novel insights into the world around us.

From heated magnets to protein folding

One of the most important models in mathematical physics saw the light of day about a hundred years ago, when German physicist Ernst Ising worked out a model of ferromagnetism (the magnetism of some materials containing certain metals such as iron) – one that explains a peculiar transition. Pierre Curie had previously observed that a ferromagnet would lose its magnetism when heated past a critical temperature, the Curie point. As the magnet cools down below the Curie point, it suddenly regains its attractive property. 

The reason, in a nutshell: a magnet consists of countless microscopic magnets, because atoms are intrinsically magnetic. When these 'magnetic moments' of the atoms point in the same direction, the material as a whole becomes magnetic. When they are not aligned, the magnetic effects cancel each other out. At high temperatures, the atoms in a ferromagnet will jostle around enough to prevent alignment, leaving it with zero net magnetism.

Ising's trick was to imagine a magnet as a random arrangement of arrows on a grid that point either up or down, representing the magnetic orientation of atoms. Arrows influence their neighbours in the grid to make them match their own orientation, until eventually most point into the same direction. Globally, the system then has an overall magnetic orientation. Disorder wins, however, when heat is added, which disrupts the tendency of the arrows to align.

This strikingly simple framework is called the Ising Model, and – after modifications by scientists who came after its namesake – still stands as one of the most useful approaches to describe the emergence of magnetism from the complex collective behaviour of elementary magnets. Today, it is relevant way beyond mathematics, serving as a model of earthquake dynamics, protein folding, the brain, or even social segregation. 

"The Ising Model fits many phenomena where global patterns emerge from the local interactions of the system's individual components," explains Berestycki. This is what makes the model a great framework for studying phase transitions, he adds: "It gives us a chance to have an insight into how phase transitions occur and what exactly is happening at the transition. Without it, these processes would be impossibly hard to compute."

But there is a catch: The Ising Model is far from finished business. Ising and his successors could only provide exact solutions of the model in one and two dimensions. Both versions are useful for many applications, but still: "We cannot fully describe how a phase transition happens in 3D. This means we don't truly understand some of the most basic workings of our three-dimensional reality", says the mathematician. To this day, solving the model in 3D remains one of the biggest challenges in mathematical physics.

The language of randomness

A key ingredient for a deeper understanding of complex systems is randomness, Berestycki reveals. "You have to think probabilistically, because phase transitions are fundamentally caused by randomness", he says, again touching on the example of freezing water: "A single water molecule is subjected to a lot of chaos as it is being randomly pushed around. But when you have a lot of molecules, a collective behaviour emerges from this randomness."

The good part about this is that randomness is not as, well, random, as we might think, he adds: "The ancient Greeks thought randomness is chaos, so it would be pointless to grapple with it. Today, however, we know while the path of a single particle might be impossible to track, the behaviour of millions of them creates an inevitable, predictable structure."

He illustrates this with a coin toss: The outcome of one single coin flip is random and cannot be predicted. Flip a coin once and you are left with heads or tails, flip it 10 times, it still seems random. However, once you flip it a million times, the outcome will be close to 50/50.

"The same principle applies to more complex problems like the Ising model", continues Berestycki. "A global pattern emerges from the randomness at the level of the individual arrows. And you can harness these patterns to understand the global properties of the system." 

(In)describable world

While Nathanaël Berestycki is not working on the 3D Ising Model, he is currently studying the so-called 'dimer model', a related framework that is similarly powerful to describe how a complex physical system responds to changes in the interactions between its individual components. 

Mathematicians can work on the same problem for many years, says Berestycki. What keeps him going is, as he puts it, "a fascination with reality and how it is reflected in maths". The most powerful moments, he says, come when deep connections between concepts begin to reveal themselves. At such times, he says, "you feel like standing on a mountaintop and seeing the whole landscape spread out below, getting a rare sense of clarity."

The hard part, however, often is finding the proof. "Showing that things actually are the way your intuition suggests can be very, very difficult", he explains. "You might not even get anywhere in the end, and you could absolutely go insane." It is all the more rewarding when a complicated proof is finished as all arguments finally fall into place, he adds. 

The art of mathematics is to find the needles in the haystack, Berestycki closes, "the few ideas that can be proven to be true amongst an ocean of potentially true ideas. While we can unearth some of these truths, others will probably remain forever beyond our grasp."