The transition from orderly progress to chaotic wandering, as happens when a herd of sheep walk together into a pasture then disperse unpredictably, has been given a detailed mathematical description for the first time.
The equations, figured out by a team of engineering researchers at Washington University in St Louis, Missouri, and published in a paper in Physical Review E, describe a universal process that occurs in a huge range of phenomena from nanoparticle scattering and bacterial migration to gas diffusion and stock-price fluctuations.
“We have shown a new starting point to investigate randomness,” says Rajan Chakrabarty, who led the research. “We can see the prelude to chaos, so that people might have the ability to intervene and reverse a trend.”
The classic example of random movement is the behaviour of a tiny particle such as a grain of pollen suspended in water. Buffeted from place to place by collisions with the water molecules, the particle goes on a jittery random walk known as Brownian motion. This movement, which is also what makes a droplet of dye steadily diffuse through a still glass of water or the smell of a baking cake spread through a whole house, was first explained by Albert Einstein in 1905.
Due to the enormous number of particles involved, the details of Brownian motion are unpredictable and it can only be described statistically, in terms of the average movements of the particles. Einstein speculated that in the very first instants of Brownian motion, particles would move in a predictable, linear way, but this “ballistic” motion was only observed in 2011, using optical traps and other technology undreamt of in Einstein’s day.
Even then, after the transition from ballistic to diffusive behaviour had been seen in the lab, it wasn’t clear exactly when it would occur and under what circumstances.
Using detailed mathematical analysis and computer simulations, Chakrabarty’s team found relationships between the concentration of particles (or bacteria, or sheep), their effective shapes (which determine the angles at which they bounce off one another), and the amount of time it takes for orderly ballistic motion to collapse into diffusive chaos.
“We’ve come up with mathematical formulations that can be applied to any random motion to find the critical time at which the transition from ballistic to diffusive takes place,” says co-author Pai Liu.
“We hope to apply this to various systems and see how general our predictions are,” adds Chakrabarty. Missouri sheep farmers might be advised to keep an eye out for mathematicians taking notes.