The maths of corn starch and traffic jams

michael-cates

On 4 November 2015 I went to a special lecture by Mike Cates, the new Lucasian Professor of Mathematics at Cambridge. As university professors often do when first appointed to their jobs, he gave an “inaugural lecture” designed to explain his work to a non-specialist audience.

Cates specialises in the maths of soft matter, and his lecture was about how substances made up from a collection of small particles can flow easily under normal conditions but then “jam” in certain conditions.

An easy-to-demonstrate example, which Cates showed us in his lecture, is cornstarch. A thick mixture of cornstarch with water will flow easily in normal conditions, but “jam”, and act like a resilient solid, if a higher pressure is applied – if you hit the mixture with a hammer, or walk on it.

The experimental demonstration is well-known – http://littleshop.physics.colostate.edu/Try%20At%20Home/goorecipeone.htm – but Cates explained that a proper theoretical explanation of this behaviour has been worked out only within the last year.

The theoretical explanation depends on the shape of the graph of pressure on the cornstarch versus density. The theory builds on ideas first used by James Lighthill, a previous Lucasian Professor, to explain traffic jams, via the shape of the graph of traffic flow versus density of traffic.

Lighthill_1955

You can read Lighthill’s paper (it doesn’t use any very technical maths) here.

The “jamming transition” with traffic is substantially different in detail from with cornstarch, because traffic is an example of a system made up of lots of self-propelled particles, and with cornstarch the particles move only under external pressure.

Those “jammings” in systems made up of lots of self-propelled particles, once understood, can be consciously designed, and, so Cates explained, “used to create a wide range of new materials”.

Those appointed as Lucasian Professor of Mathematics are generally applied mathematicians, mathematicians working directly on practical applications, rather than pure mathematicians. Previous Lucasian Professors include Stephen Hawking, Paul Dirac, George Gabriel Stokes, and Isaac Newton.