Ergodicity & the Merits of a Good Kick in the Ass

In statistical physics the hypotheses of ergodicity say, in essence, that everything that can happen will eventually happen. Given enough time and a system with different states, each of those states will eventually happen with equal overall frequency.
So, is a bee that buzzes around the back seat of your car all the way from San Diego to Santa Barbara ergodic? No, because the bee doesn’t visit the whole car; it just buzzes in the back seat. A truly ergodic bee traveling for a long enough period in said car would eventually check out the car’s entire interior, assuming you don’t swat it and your kids didn’t smear honey in the back under the booster seats.
Fine, but so what? Well, while ergodicity has long been a useful assumption in complex systems, one that explains complex systems like fluids, magnets, and the like, it has never really panned out in the lab. Like economics, in other words, the ergodicity hypotheses have been something that worked in practice but not in theory.
That was first demonstrated in some famous experiments done by Stan Ulam and Enrico Fermi back in the 1950s. Using an early computer (real early), they simulated the dynamics of exciting a long chain of masses joined by springs and then seeing what happens. Their discovery? The chain never lapsed into a chaotic equilibrium; instead it got trapped, flitting between a small number of states.
The real world demonstrably didn’t work that way, so what was different? Mark Buchanan explains all in this month’s Nature:

As it turns out, ergodicity seems to come into play when the energy given to the chain is about ten times greater than that applied by Fermi and his colleagues in their original study. At this energy, rather than remaining locked in a semi-repetitive state, the vibrating chain begins to explore its possible states ergodically and relaxes slowly into equilibrium.

In other words, systems need a larger kick in the ass than Fermi and Ulam applied before they break out of non-ergodic states and fully explore their state-space. Not to trivialize or turn to pop sociology a fascinating finding, but it sounds like some people I know.


  1. Interestingly, if you take one of the simplest stochastic systems, a discrete first-order Markov chain (think of just a bee lying on some non-regular grid where it can only fly to a neighboring node with some probability based on where it was last), and then add a small dampening factor so that a uniformly small amount of weight is added between each pair of states, what you get when you measure the asymptotic percentage of time spent in each state (the ergodic distribution) is precisely PageRank. Although many physical systems may not display ergodic properties, leveraging the property of ergodicity has been key to computationally modeling many different phenomena, and in some cases enabling a succesful IPO.