The Living Thing / Notebooks : Flocking

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, [1] 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….

[1]To nitpick - I’ve not checked that our angular perturbation models are exactly equivalent, and I use a lot fewer particles than they recommend.