Image of authors courtesy Nathan Pitt, Department of Chemistry

A bewildering physics problem has apparently been solved by a team of researchers led by a group of theoretical chemists in the Department of Chemistry.

The study provides a mathematical basis for understanding issues ranging from predicting the formation of deserts, to making artificial intelligence more efficient.

The research team developed a computer program that can answer this mind-bending puzzle: Imagine that you have 128 soft spheres, a bit like tennis balls. You can pack them together in any number of ways. How many different arrangements are possible?

If we would only consider ordered (crystalline) arrangements, the number would be easy to compute. The challenge is to compute also all disordered arrangements, of which there are many more.

The answer, it turns out, is something like 10^{250} (1 followed by 250 zeros). The number, also referred to as ten unquadragintilliard, is so huge that it vastly exceeds the total number of particles in the universe.

Far more important than the solution, however, is the fact that the researchers were able to answer the question at all. The method that they came up with can help scientists to calculate something called configurational entropy – a term used to describe how structurally disordered the particles in a physical system are.

Being able to calculate configurational entropy would, in theory, eventually enable us to answer a host of seemingly impossible problems – such as predicting the movement of avalanches, or anticipating how the shifting sand dunes in a desert will reshape themselves over time.

These questions belong to a field called granular physics, which deals with the behaviour of materials such as snow, soil or sand. Different versions of the same problem, however, exist in numerous other fields, such as string theory, cosmology, machine learning, and various branches of mathematics. The research shows how questions across all of those disciplines might one day be addressed.

Stefano Martiniani, a postgraduate who is supervised by Professor Daan Frenkel and one of the paper's authors, explained: “The problem is completely general. Granular materials themselves are the second most processed kind of material in the world after water and even the shape of the surface of the Earth is defined by how they behave.”

“Obviously being able to predict how avalanches move or deserts may change is a long, long way off, but one day we would like to be able to solve such problems. This research performs the sort of calculation we would need in order to be able to do that.”

Martiniani added that the Frenkel research group’s problem-solving technique could be used to address all sorts of problems in physics and maths. He himself is, for example, currently carrying out research into machine learning, where one of the problems is knowing how many different ways a system can be wired to process information efficiently.

“Because our indirect approach relies on the observation of a small sample of all possible configurations, the answers it finds are only ever approximate, but the estimate is a very good one,” he said. “By answering the problem we are opening up uncharted territory. This methodology could be used anywhere that people are trying to work out how many possible solutions to a problem you can find.”

The paper, Turning intractable counting into sampling: computing the configurational entropy of three-dimensional jammed packings, was co-authored by Stefano Martiniani, K. Julian Schrenk, Jacob D. Stevenson, David J. Wales and Daan Frenkel, and is published in the journal, Physical Review E. It is based on a method developed by Frenkel and others, described in Direct Determination of the Size of Basins of Attraction of Jammed Solids, and first applied in Numerical Calculation of Granular Entropy.