The Frailty of Equilibrium

Hard to build, easy to break.

From an interview with Amie Wilkinson:

Do you find that dynamical systems provide a language that you use to think about other non-math areas of your life?

“Yeah, definitely. You become much more aware of how fragile equilibrium is. Things can look stable for a long time and then suddenly fly off into some other territory. You understand how precarious being close to equilibrium is, for the most part, because most equilibriums are not stable. Things can be almost imperceptibly changing, and eventually those changes start to add up, then things change really quickly. Because that’s the nature of change in typical dynamical systems. Things can seem stable for a very long time, and then they go exponentially wrong. I do think about a lot of things — things involving human nature, historical trends, and very serious things like climate. It informs my way of thinking about life. There’s essentially no such thing as being in equilibrium.”


From Algorithms to Live By, by Christian and Griffiths:

In a game-theory context, knowing that an equilibrium exists doesn’t actually tell us what it is—or how to get there. As UC Berkeley computer scientist Christos Papadimitriou writes, game theory ‘predicts the agents’ equilibrium behavior typically with no regard to the ways in which such a state will be reached—a consideration that would be a computer scientist’s foremost concern.” Stanford’s Tim Roughgarden echoes the sentiment of being unsatisfied with Nash’s proof that equilibria always exist. “Okay,” he says, “but we’re computer scientists, right? Give us something we can use. Don’t just tell me that it’s there; tell me how to find it.” And so, the original field of game theory begat algorithmic game theory—that is, the study of theoretically ideal strategies for games became the study of how machines (and people) come up with strategies for games.

As it turns out, asking too many questions about Nash equilibria gets you into computational trouble in a hurry. By the end of the twentieth century, determining whether a game has more than one equilibrium, or an equilibrium that gives a player a certain payoff, or an equilibrium that involves taking a particular action, had all been proved to be intractable problems. Then, from 2005 to 2008, Papadimitriou and his colleagues proved that simply finding Nash equilibria is intractable as well. (Narrator: Newer findings are even more forbidding.)

Simple games like rock-paper-scissors may have equilibria visible at a glance, but in games of real-world complexity it’s now clear we cannot take for granted that the participants will be able to discover or reach the game’s equilibrium.

Viewpoint: Equilibration in Quantum Systems

Quantum systems also have unique behaviors that have no classical counterparts, quantum entanglement being one example. This lack of directly translatable behaviors makes it very hard to determine if, when, and how a quantum system reaches a state of thermal equilibrium. But that difficulty hasn’t stopped physicists from trying.

The first attempt goes back to John von Neumann, generally regarded as the foremost mathematician of his time, who, 90 years ago, formulated the “quantum ergodic hypothesis”. In his theory, entropy monotonically increases in an equilibrating quantum system. We now know, however, that many quantum systems don’t follow this behavior. Even today the problem of equilibration in quantum systems remains unsolved.

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