Active Matter

walkers 15 A simple random walk is the least interesting-looking process that turns out to be the most load-bearing idea in this whole series. A particle sits at some position, and at every tick of a clock it takes a step in a random direction, completely independent of every step that came before it. There is no memory, no preferred direction, and no notion of “trying” to go anywhere. And yet, out of this featureless rule comes one of the most robust results in statistical physics: the particle’s typical distance from its starting point grows with the square root of time, not linearly with it. That single scaling law is the fingerprint of diffusion, and it shows up whether the walker is a pollen grain being kicked by water molecules, a photon bouncing through the sun’s interior, or a foraging animal with no memory of where it has already searched. ...

tumbling rate λ 0.05 Run-and-tumble motion is the first model in this series that is trying to describe something a real organism actually does, rather than an idealized particle. It’s the motility strategy of E. coli and many other flagellated bacteria: swim in a roughly straight line (“run”) for about a second, then abruptly reorient in a random direction (“tumble”) over a much shorter timescale, and repeat. Unlike the plain random walk, a run-and-tumble particle has a well-defined instantaneous velocity and moves ballistically over short times — it’s only once you zoom out past several run lengths that the path starts looking diffusive again. The trick that makes this more than “a random walk with extra steps” is what the bacterium does with the tumbling itself: by biasing how long it runs (not which direction it tumbles into) based on whether the local chemical concentration is improving or worsening, it converts a directionless reorientation process into directed drift up a nutrient gradient, without ever computing a gradient outright. ...

rotational diffusion Dr 0.010 Active Brownian Particles (ABPs) are the model you reach for when you want the simplest possible description of a self-propelled particle, without committing to any particular biological or chemical mechanism. Instead of alternating between discrete runs and tumbles, an ABP moves continuously at a fixed speed along a body orientation that itself performs an ordinary random walk — not in space, but in angle. Picture a swimmer that never fully changes direction in one sharp event; it just slowly, continuously forgets which way it was pointing, the way a compass needle would if it were being nudged by random noise instead of a magnetic field. This is a reasonably good caricature of synthetic active colloids (Janus particles powered by a chemical reaction on one face) and is close enough to real microswimmer behavior that it shows up constantly in the active matter literature, including work adjacent to the kind of Chlamydomonas tracking this site’s about-me page mentions. ...

noise η 1.00 order parameter va = 0.00 Every model earlier in this section describes a single particle, on its own, ignoring everyone else. The Vicsek model is the point where this series stops being about individuals and starts being about crowds. The rule is almost insultingly simple: each particle moves at a fixed speed, and at every time step it adopts the average heading of every other particle within some fixed radius, plus a bit of random noise thrown in to keep things honest. There’s no leader, no global coordination, and no particle that’s aware of anything beyond its own local neighborhood. And yet, tune the amount of noise down (or the density up) past a critical point, and the whole system spontaneously locks into a common direction of travel — a flock, a school, a swarm — indistinguishable in spirit from the murmurations of starlings or the coordinated streaming of dense bacterial colonies. ...