a.k.a. collective motion, swarm dynamics, herd behaviour, particle systems…
A type of agent-based model where the emphasis is on motion in space.
The simplest one I know is due to Vicsek. Vicsek’s simulation is a special case of the older but less-analytically-tractable “Boids” model, invented by Reynolds in the 80s. Reynolds wanted to model the flight of birds simply but realistically, and his model had several rules to ensure, e.g., that his simulated birds did not collide in mid air.
Vicsek and colleagues flensed that already gangly model to the bone. Soon, circling physicists has joined him at the carcass (ahem), and now there’s a regular mini-research-field built around it. (This is the classical origin of the symposium.)
The model in the end about as simple as a classic Ising model, but covers moving things, which, motile creatures as most of us are, we might find easier to empathise with than abstract magnetic spins and such.
The formula,  in Vicsek’s words:
The only rule of the model is: at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius \(r\) with some random perturbation added.
Of key interest to the excitable creature who tends to become a working physicist, this model demonstrates classic “phase transition” behaviour. Sweep that “noise” slider from nought to maximum and you’ll see some radical changes in the behaviours of those little moving particles as they go from orderly marching through to fidgety jiggling. In the middle, somewhere, you’ll find a graceful, evolving, lifelike (if you squint) flocking behaviour, like birds or fish, if birds or fish had happened to evolve as translucent rust-coloured bricks. This is the “phase transition” region, where statmech folks can get overstimulated and use phrases like “self organised criticality”, or possibly “edge of chaos”, and might need to have a bit of a lie-down. And that’s what I’m investigating at the moment, the phase transition and the lie-down both.
One thought this model provokes, which is frankly the main appeal for me, is that it would make a really nice algorithm to drive a granular sampler. Has anyone done that yet? Turns out they have, e.g. Nicholas Mariette
Why not read….
- Boids inventor Craig Reynolds has a page dedicated to this algorithm with more than you could possibly need to know about his classic model, including other implementations, animations, and trainspotting of the algorithm at work in Hollywood blockbusters.
- Thomas Lux. 1995. Herd Behaviour, Bubbles and Crashes. The Economic Journal. (Online) —- Nice paper on the relevance of this to human behaviour.
- Dion Harmon, Marcus A M de Aguiar, David D Chinellato, Dan Braha, Irving R Epstein, Yaneer Bar-Yam. 2010. Predicting economic market crises using measures of collective panic. (Online) (cf Wired’s heart-warmingly enthusiastic coverage of same.)
- Iain Couzin and colleagues have created some neat hacks in this area including exotica like mixing real animals with simulated ones in lab environments, and racetracks for locusts, and thereby actually qualify as doing real science.
- Gama sutra’s diverting article on the intersection of the classic Reynolds Boids algorithm and human opinion dynamics. (Warning - human opinion dynamics are likely topologically different to this kind of bloody-minded literalism. But yay! video gamers like it; this is an Established Field now, then.)
|||To nitpick - I’ve not checked that our angular perturbation models are exactly equivalent, and I use a lot fewer particles than they recommend.|