One of the most fascinating aspects of our universe is the hierarchy of complexity that can be found in nature. Particles get together to create atoms, atoms get together to create molecules, molecules get together to create substances and the ladder goes all the way up to neurons getting together to create the human brain and the apex of complexity – conscience.

Complexity science was the name recently given to a whole set of techniques coming from different areas which can be applied to the study of systems composed by a large number of interacting parts, how they form and the several unique phenomena that they create.

From all the areas, the one that most contribute with theoretical and computational techniques is Statistical Physics. The study of phases of matter is nothing but trying to understand the emergent behaviour of a large assembly of atoms and molecules.

Complexity, however, seems to depend on the eye of the beholder. Throughout the years of research, many different definitions and measures of complexity have been proposed and are used for different applications. All these measures seem to have some relation to information and are somehow related to Shannon’s entropy.


Another important concept that underlies all our modern understanding of nature’s inner workings is that of symmetry. A symmetry of a system is a change that leaves some aspect of that system invariant. For instance, rotating an object leaves all distances between its points the same.  Reflecting a square through a line cut it into an exact half seems to lead to the same square.

Three of the four known forces of nature are described by symmetry ideas. They are described by fields called gauge fields, the most common example being the electromagnetic field which mediates electricity and magnetism. These fields seem to exist in order to guarantee that some symmetry rules are obeyed in nature.

In some situations, though, symmetries are prevented to be realised. When this happens we say that the symmetry has been broken. Several phenomena, including the existence of mass in elementary particles, are attributed to symmetry breaking.

Up to this point, though, the two concepts of symmetry and complexity have not been explicitly connected. This is what this project is all about.

The Project

In this project we introduced the idea that complexity can be measured by the amount of symmetry broken on average by an object.

This comes from the observation that systems that we tend to call complex are somewhere in the middle between being very well organised and completely random. Consider a computer screen filled with only one colour. It is completely symmetric as whatever you do with it (rotate, translate, reflect…) it remains the same. It is not difficult to recognise it as a very simple system.

On the other hand, consider a pool full of plastic balls of different colours. It is not symmetric at all as, whatever you do, it will always be a little different from the initial configuration. However, in a broader way, these small differences seem to matter very little. A shuffle version of it is, in some sense, still the same.

Life is the epitome of complexity. A totally symmetric system like the filled computer screen can hardly be imagined to be related to anything alive as its full simplicity cannot give rise to the processes necessary for life to thrive. On the other side of the spectrum, a completely random system lacks a certain organisation that also allows that. In fact, death is a return to a totally random configuration of materials.

Still, there is a sense in which randomness is symmetric. It is symmetric on average. Although the details are different, a pool of plastic balls seems roughly the same from all points. The important thing is that the disorder introduced by the random variations is itself the same at every point.

Our objective in this project is then to make full use of techniques developed in statistical mechanics to analyse and explore the consequences of associating complexity to average symmetry breaking. After introducing the main idea 1Alamino, R.C., Measuring complexity through average symmetry, J. Phys. A: Math. Theor. 48, 275101 (2015) , we are now exploring the analytical properties and trying to find physical applications for it.


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1. Alamino, R.C., Measuring complexity through average symmetry, J. Phys. A: Math. Theor. 48, 275101 (2015)