Superficially, this course is about phase transitions. This is the name given to the abrupt, discontinuous changes that occur when matter is deformed in some way, whether through heating or squeezing or something else. The familiar example is the violent shaking of a pot on a stove as water approaches its boiling point, and bubbles of steam erupt from within.
Despite their familiarity, phase transitions are striking, and even a little disconcerting. Usually in physics, things happens gradually. This fact is sewn into the heart of classical physics where the positions and momenta of particles are described by smooth, differentiable functions. Indeed, historically, the idea that change happens only infinitesimally resulted in the discovery of calculus. Yet, somehow, what holds on the micro level fails at the macro. Phase transitions tell us that a large number of particles can behave collectively in a way that any individual particle cannot, with the macroscopic properties of a system changing discontinuously.
A closer look at what happens at phase transitions – in particular at so-called critical points – reveals something startling. Many different substances, regardless of their microscopic composition, exhibit identical behaviour at a phase transition. This is not just a qualitative statement, but a quantitative one. For example, as a liquid changes into a gas at the critical temperature , the heat capacity diverges as
The exponent is not known precisely. It is thought not to be a rational number, but should instead be viewed as a universal mathematical constant, similar to or , but more subtle. Remarkably, the same exponent occurs for all gases. It also occurs in other systems, including a certain class of magnets. It’s as if all knowledge of the microscopic physics has been washed away, leaving us with something pure, that carries only a vague memory of what lies underneath. This phenomenon is known as universality.
All of this makes phase transitions interesting. They involve violence, universal truths and competition between rival states. The story of phase transitions is, quite literally, the song of fire and ice.
And yet these are not the only reasons to study phase transitions. In our attempt to understand what happens as water boils, we will need to develop new tools and a new way of thinking about the world. This leads us to a paradigm which now underlies huge swathes of physics, far removed from its humble origin of a pot on a stove. This paradigm revolves around two deep facts about the Universe we inhabit: Nature is organised by symmetry. And Nature is organised by scale.
When I was a kid, I was told that there are three phases of matter: solid, liquid and gas. (Actually, this isn’t quite true. Knowing that I was interested in this kind of stuff, the teacher conspiratorially let on that there was a fourth phase of matter, “plasma”. To this day, I have no idea why. My best guess is that this fitted better with some old view of the basic elements as earth, water, air and fire.)
It won’t be any surprise to learn that the real world is much more interesting than the one we’re introduced to as kids. There are not three phases of matter, nor four: there are many. A key insight, due to Landau, is that these different phases are characterised by symmetry.
In this scheme a solid differs from a liquid because its crystal structure breaks the translational and rotational symmetries of space. Moreover, solids with different crystal structures should be viewed as different phases of matter because they break these symmetries in different ways. Perhaps more surprisingly, liquids and gases break no such symmetries and so should be viewed as the same phase. When you include further symmetries, such as rotations of spins in a magnet or more subtle quantum counterparts, this classification opens up a wide range of possibilities that allows us to understand almost all the known forms of matter.
This characterisation has its advantages. First, we can be sure that any attempt to change a material from one symmetry class to another will necessarily involve a violent phase transition. Second, it turns out that understanding the symmetries of a system will immediately determine many of its properties, especially at low temperature.
Moreover, the classification of matter in terms of symmetry has a power that goes far beyond its initial regime of application. The vacuum of space is, in many ways, like a complicated material, with quantum effects playing the role of thermal fluctuations. The vacuum can sit in different phases and is thought to have undergone several phase transitions as the Universe cooled after the big bang, each of which can be understood in terms of symmetries. All the ideas that we will develop here carry directly to theories of particle physics, cosmology and beyond.
There is an order to the Universe we live in. Roughly speaking, little things affect big things. Not the other way round.
This is something you already know: particle physics underlies nuclear and atomic physics; atomic physics underlies condensed matter and chemistry; and so on up the chain. It’s certainly true that it can be difficult to make the leap from one level to the next, and new creative ideas are needed at each step, but this doesn’t change the fact that there is an ordering. Big things don’t affect little things. This is the reason there are no astrology departments in universities.
But there is another aspect to this story, one which is not often stressed. Little things affect big things, but they rarely affect very big things. Instead, little things affect slightly bigger things. And these, in turn, affect slightly bigger things too. But as you go up the chain, you lose the information about what came long before.
This again is something that you know. A zoologist who is interested in the way that starlings flock has little reason to study the dynamics of the Higgs boson. It’s also the reason that science is possible in the first place: neither Newton nor Einstein needed to understand how quantum gravity works on microscopic distance scales to write down theories that work extraordinarily well on larger scales.
In the 1970s a mathematical formalism was developed that makes these ideas concrete. This formalism is called the renormalisation group and provides a framework to describe physics at different scales. The renormalisation group gets little coverage in popular science articles, yet is arguably the single most important advance in theoretical physics in the past 50 years. While zoologists may have little need to talk to particle physicists, the right way to understand both the Higgs boson and the flocking of starlings is through the language of the renormalisation group.
These two ideas – symmetry and scale – now dominate the way we think about physics. Yet both have their origins in the simple question: what happens when you boil water? The purpose of this course is to find out.