Decades-old mathematics might finally explain some features of the 'oddballs' of matter: granular materials that sometimes behave like a solid, and at other times flow like a liquid.

Strange as it sounds, just think of the sand in an hourglass compared to the sand on a beach. Poured slowly through a constriction, sand – or rice, or coffee – will flow freely. Funnel that same material fast enough or stamp on it with force, its particles will typically jam, snapping from a flow state into a solid-like one.

To avoid sudden blockages where a gentle flow is desired, we'd need to understand how and when this sudden shift happens. Two US-based physicists now think they've found a way to describe the behavior of granular materials nearing that 'jamming point'.

"The tendency of flowing granular matter to get 'jammed' and stop flowing at low densities is a practical problem that limits the flow rate in the industrial use of granular materials," Onuttom Narayan of the University of California, and Harsh Mathur at Case Western Reserve University in Ohio, explain in their published paper.

That problem becomes increasingly complicated when you consider it involves various materials in industries as diverse as agriculture, pharmaceuticals, and construction. We're talking compacting granules into pellets to make pills, processing cereals and, in civil engineering, predicting the behaviors of different sediments our buildings might be anchored into.

For their simulations, Narayan and Mathur used numerical data other researchers had collected from studying packs of frictionless polystyrene beads in the lab. The pair compared their simulations of beads nearing the jamming point with predictions of a branch of mathematics developed in the 1950s called random matrix theory.

Specifically, Narayan and Mathur were looking at vibrations within bead packs. Although it varies from batch to batch, beads vibrate at certain frequencies, creating a 'spectrum' of vibrational frequencies.

Put another way, a granular material only allows certain vibrational frequencies to propagate through it – a property physicists refer to as the system's density of states.

Other researchers have tried studying how the distribution of those vibrational states evolves in granular materials nearing the jamming point, where particles are jostling together before they get stuck.

This problem lends itself to random matrix theory, which can be used to describe physical systems with many random variables. But without comparing calculations to numerical data from the beads themselves, earlier studies could not distinguish between different 'flavors' of random matrix theory that might explain vibrations in granular materials.

Where those researchers fell short, Narayan and Mathur succeeded: Their comparison of numerical simulations and theoretical predictions showed a specific distribution of statistical probabilities known as a Wishart–Laguerre ensemble "correctly reproduces the universal statistical properties of jammed granular matter."

The crucial observation, they say, was recognizing that as beads bump into each other, they compress and recoil like a spring, such that a slight contact of two beads results in quite large forces.

What's more, the pair also developed a model that managed to describe the properties of the beads close to the jamming point, and far from it, when granular materials aren't moving.

"That the same model is able to reproduce both the static and vibrational properties of granular matter suggests it may be more broadly applicable to provide a unified understanding of the physics of granular matter," Narayan and Mathur conclude.

The study has been published in European Physical Journal E.