A motorcycle engine. The force on a piston doesn’t depend on the nature of the particles involved but only on their temperature and pressure.

What qualifies as “fundamental” physics? The answer might seem straightforward. It is science that investigates the universal properties of nature, like the interactions between quantum fields studied at CERN or the shape of the cosmos observed by astronomers. In the other camp, science that is not fundamental includes things like strength testing of new materials, or sports science to optimize the trajectory of a javelin. Both types of research use scientific methods to design good experiments that remove any potential observer-bias.

Straightforward as it seems, I posed the question because I want to tell you about research on non-equilibrium statistical physics. Because it contains the word “statistical”, you might put this subject into the non-fundamental category. To see why that would be wrong, you need to know some little-understood facts about how nature works – about which aspects of nature are in fact universal.

Some of the fundamental properties of nature can be found by studying matter in ever finer detail. By doing so, our predecessors have discovered that matter is made of atoms that are made of electrons and nuclei that are made of quantum fields constituting quarks, gluons, leptons and so on. These are clearly fundamental properties of nature. The universe has, for some as-yet-unknown reason, decided to allow these particular fields to exist, so we must study them to find out, fundamentally, how the universe works.

But nature has other fundamental properties that evade detection by this microscopic exploration. They are properties that emerge only when a large number of particles interact. Of course, once you know the behaviours of elementary particles, you can use a computer to simulate a large number of them and predict what they will do en masse. The particles’ motions become more complicated the more of them you have. For instance, below is a picture of the paths of two classical particles attracted by gravity, or they could be charged particles interacting via electrostatic attraction. They follow simple elliptical paths. Introduce a third particle and their paths become chaotic.

Two particles attracted to each other following simple ellipitcal paths.
Three mutually attracting particles create more complicated, chaotic paths.
Credit: Daniel Piker

Say you wanted to know how much force will be applied to the piston of an engine by a large collection of particles that constitute some gas. The pressure felt by the piston fluctuates wildly in the presence of three or more molecules that sporadically collide with it. But it turns out that, for a a large enough collection of particles, although their individual trajectories become effectively random, their collective motion becomes completely predictable. Here’s the crucial point. The pressure averages out as the number of molecules approaches infinity and, amazingly, the pressure ceases to depend on what the molecules are made of or how they interact. Its value is given by the ideal gas law that you probably learned at school. It only depends on how densely the molecules are packed into the container and on the temperature.

As Ludwig Boltzmann, the founding father of statistical physics discovered, even the temperature doesn’t depend on any detailed features of the gas molecules, only on the amount of energy that they have been given and on their symmetries – the number of ways you can rotate one of the molecules without changing its appearance.

This is a very important fact. It means that the detailed features of elementary particles do not determine how the universe works. Large-scale physics is not controlled by those details, but only by a few of their symmetries and by the statistical properties of large numbers.

Specifically, most of the macroscopic properties of matter depend only on how many different ways there are to rearrange its particles and energy. The preceding sentence sounds too bizarre to be true, and needs to be read several times. All that we experience around us; the wetness of water, the clarity of glass, the viscosity of treacle and the conductivity of silicon, are manifestations of combinatorics. They result from the mathematics of large numbers of interactions, not from the microscopic properties of those interactions.

If you still doubt that a theory based on statistics (on counting arrangements of particles) can constitute fundamental physics, you need look no further than the second law of thermodynamics, that most intriguing of laws, which states that the amount of entropy (disorder) in the universe can only increase with time. It is a consequence of statistical physics. Boltzmann realised that entropy is simply a way of counting large numbers of configurations, and yet it is responsible for our sense of the direction of time, a fundamental concept.

As a crude example of the unifying effect of large numbers, look at the flowing fluids depicted below. Their macroscopic properties, such as flow rate, diffusivity, compressibility, viscosity… are affected by a few basic features like the hardness of their particles, the packing density of those particles and the presence of gravity. At this scale, we don’t notice what the constituent particles are made of, or their detailed features like the fact that the fluid on the right is made of red blood cells in a capillary, whereas that on the left is composed of athletes in the Engadin Ski Marathon.

Athletes in the Engadin ski marathon behaving like particles of a fluid.
Red blood cells flowing through a capillary.

A more precise (but rather complicated) correspondence exists between the behaviour of a fluid, such as water, at its critical point (where the pressure is just high enough to make the gas identical to the liquid, thus removing the boiling transition) and certain types of magnet, known as Ising magnets, at their Curie point (the temperature where they lose their permanent magnetism). These very different systems behave in identical ways as they approach the critical/Curie temperature. Fundamentally, the reason for this is that they both comprise large numbers of interacting particles that can each be found in one of two states: for Ising magnets, each atom’s magnetic north pole can point either up or down while, within a sample of water, at any given location, a molecule can be either present or absent. No other microscopic features of these radically disparate sets of particles govern their collective behaviour. As a result, water and Ising magnets share identical large-scale physics and are said to belong to the same universality class.

Another material sharing the same universal features as these two is the “surfactant sponge phase”. It is an arrangement of molecules that often forms spontaneously when you mix washingup liquid with water. The detergent molecules group together, forming thin membranes that curve and connect into a sponge-like arrangement of pipes, depicted here.

A sponge-like arrangement of molecules, such as that which forms when mixing
detergent and water. Source: Physical Review Focus

This elaborate surface divides the water into two interpenetrating labyrinths. So, once again, within the sponge phase, each point in space can be characterised by a single binary digit, specifying which of the two labyrinths it belongs to. It also undergoes a transition at a critical temperature – not from gas to liquid, or from non-magnetic to magnetic, but from a “symmetric” state in which the labyrinths have equal volumes, to a state in which their symmetry is broken, with one region bigger than the other. [Roux D, Coulon C and Cates ME, “Sponge phases in surfactant solutions”, J. Phys. Chem 96, 4174-4187 (1992)].

Since Kenneth Wilson’s Nobel-prize-winning development of “Renormalization Group” theory in the 1970s – the mathematical tool for dealing with universality – many different “universality classes” have been discovered amongst a mid-boggling variety of interacting systems. Wilson and Boltzmann’s theories apply to systems of particles at thermodynamic equilibrium, meaning that they are not flowing. More recently, there have been hints that the behaviour of flowing systems may be governed by a unifying statistical theory of “Large Deviations”, giving them universal features that are not obvious in the microscopic laws of nature. I honestly cannot image a more captivating subject to research.

Mike Evans

Mike Evans

Dr Mike Evans is a lecturer in Applied Mathematics at Leeds University. His research is on non-equilibrium statistical mechanics, soft matter and complex systems
Mike Evans

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