Quickly cool a piece of superheated liquid glass and a strange thing happens. The glass becomes hard but brittle—so brittle that it may abruptly shatter without warning. This is because the bonds between the molecules are under strain, and the cool temperature and low velocity means they cannot escape, as if caught in a negative equity trap with the neighbors from Hell. And, like warring neighbors, eventually something gives way and the strain relieves itself catastrophically. Glass makers avoid such catastrophes by annealing the glass, which means holding it for a long time at a high enough temperature that the molecules can move past each other but not too fast. In this way, the glass can find its way into a minimum-energy state, in which each molecule has had a chance to settle itself comfortably next to its neighbors with as little strain as possible, after which it can be completely cooled without problems.
Systems in which elements interact with their neighbors and settle into stable states are called attractors, and the stable states they settle into are called attractor states, or local minima. The term “attractor” arises from the system’s tendency, when it finds itself near one of these states, to be attracted toward it, like a marble rolling downhill into a hollow. If there are multiple hollows—multiple local minima—then the marble may settle into a nearby one that’s not necessarily the lowest point it can reach. To find the global minimum, the whole thing may need to be shaken up so the marble can jiggle itself out of its suboptimal local minimum and find a better one, including (one hopes) eventually the global one. This jiggling, or injection of energy, is what annealing accomplishes, and the process of moving into progressively lower energy states is called gradient descent.
Many natural systems show attractor-like dynamics. A murmuration of starlings, for example, produces aerial performances of such extraordinary, balletic synchrony that it seems like a vast, amorphous, purposeful organism, and yet the synchronized movements arise simply from the interactions between each bird and its nearest neighbors. Each flow of the flock in a given direction is a transient stable state, and periodic perturbations cause the flock to ruffle up and re-form in a new state, swooping and swirling across the sky. At a finer scale, brain scientists frequently recruit attractor dynamics to explain stable states in brain activity, such as the persistent firing of the neurons that signal which way you’re facing or where you are. Unlike glass particles or starlings, neurons don’t physically move, but they express states of activity that influence the activity of their “neighbors”—neurons they’re connected to—such that the activity of the whole network eventually stabilizes. Some theoreticians even think that memories might be attractor states—presenting a reminder of a memory is akin to placing the network near a local minimum, and the evolution of the system’s activity toward that minimum, via gradient descent, is analogous to retrieving the memory.
Attractors also characterize aspects of human social organization. The problem of pairing everybody off so that the species can reproduce successfully is a problem of annealing. Each individual is trying to optimize constraints: They want the most attractive, productive partner, but so do all their competitors, thus compromises must be made. Bonds are made and broken, made again and broken again, until each person (approximately speaking) has found a mate. Matching people to jobs is another annealing problem and one we haven’t solved yet: How to find a low-strain social organization in which each individual is matched to their ideal job? If this is done badly, and society settles into a strained local minimum in which some people are happy but large numbers are trapped in jobs they dislike with little chance of escape, then the only solution may be an annealing one—to inject energy into the system and shake it up so it can find a better local minimum. This need to destabilize a system in order to obtain a more stable one might be why populations sometimes vote for seemingly destructive social change. The alternative is to maintain a strained status quo in which tensions fail to dissipate and society eventually ruptures, like shattered glass.
Attractors are all around us, and we should pay more attention to them.