The Unreasonable Effectiveness of the Ising Model – Part 2
This is part 2 of 2 of a series on a mathematical model that refuses to stop showing up in my life. See the previous installment here.
“I won’t say [Onsager] was the world’s worst lecturer, but he was certainly in contention.”
Before I came to Berkeley, I had actually never heard of Lars Onsager, which truly is a shame. His life overflows with intellectual achievement. I would say it culminated in his Nobel Prize in 1968, but the award was for work he did essentially in graduate school (except he didn’t formally ever go to graduate school—the story of his Ph.D. is worth reading), 37 years previous. In the intervening years, his mastery of mathematics, physics, and chemistry brought about foundational breakthroughs in the study of statistical mechanics.
Part of his relative obscurity to non-scientists probably owes to his particularly eccentric nature. He didn’t have the flashiness and genius-everyman aesthetic of a Richard Feynman, nor did he have the iconography and mystery of an Albert Einstein. In fact, his eccentricity derives largely from an almost pathological refusal to present his ideas clearly—he famously wrote down a formula for the solution of an outstanding problem on the board at a conference, leaving the derivation of his formula for the scientific community to figure out themselves (which took a year). He eventually released a sketch of his ideas…twenty years later. Quoting Sir Alfred Pippard:
“This is perhaps the first important example of what has become a marked characteristic of Onsager in the last ten years—a reluctance to publish anything except fully-polished work, combined with the habit of dropping valuable hints couched in gnomic terms. The obscurity of his utterances is not due to a desire to mislead; rather it is a result of an inability to appreciate the limitations of his hearers. To those who have been able to appreciate what he tries to say, he has been a source of deep stimulation.”
Now, in the previous installment, we had just run in to the Ising Model, and considered a sort of paradoxical feature of the model: at high temperatures, our magnets seem to flit up and down every which-way, averaging out to no preferred direction over time whereas at low temperatures, our intuition suggests the magnets all align. I claim this is paradoxical, or at least interesting, because the mathematics describing our system have a certain property: symmetry under exchange. And for some reason, as we lower the temperature of our system of magnets, this symmetry is broken.
I’ve shoved some assumptions under the rug. For one, if we truly only have a one-dimensional chain of magnets, each of which is only interacting with its nearest neighbor, then such a model doesn’t exhibit this breaking of symmetry. This model is unsurprisingly called the 1-Dimensional Ising Model, and was solved by Ising himself in his thesis. In other words, at all temperatures, the 1-Dimensional Ising Model has no net “preferred direction” to the little magnets. On average, all of the magnetics flip back and forth, cancelling each other out.
The 2-Dimensional Ising Model is another monster entirely, and remained unsolved for about twenty years after Ising’s thesis. This model does exhibits spontaneous symmetry breaking—namely, if one could continuously tune the temperature of a two dimensional array of magnets, from hot to cold, one would find at a very specific temperature, the magnets would spontaneously begin to align. If you repeated this process over and over—raising and lowering the temperature repeatedly—the preferred direction would, of course, be random—half the time, the magnets would prefer to point “up”, and half the time “down” (if you have the Java browser plugin installed, Berkeley’s own Gavin Crooks wrote a particularly nice web applet to see this in action)
Onsager did, in fact, solve the 2-Dimensional Ising Model—a solution which is much too lengthy to discuss in a meager blogpost. His solution is widely considered to be an outstanding piece of mathematics, and a central achievement in the study of statistical mechanics.
For some intuition, imagine sitting on one of the magnets on this grid of magnets, again envisaged as boats which are either capsized or not capsized.
Say we begin at a very cold temperature—a relatively calm sea—and, so as not to torture our deckhands, let’s require that all of the boats are initially upright. Additionally, because the deckhands have been drinking too much, they don’t really have an absolute sense of up or down, so their only reference of “up” is whichever direction their own boat is oriented relative to their neighbors.
Every so often, a boat still capsizes, but the boats immediately surrounding that boat are able to help and flip it upright relatively quickly—despite the protestations of the upside-down boat crew, which momentarily suspects that every other boat has just capsized.
As the waves grow larger, there reaches a point where boats begin capsizing so often that upside-down boats are fighting nearly as hard to right rightside-up boats as vice versa, for if you’re on a capsized boat, and every boat around you is capsized, no one is fighting to “right” you because, again, the deckhand’s only sense of “up”-ness is set by their neighbors.
In this way, domains of rightside-up and upside-down boats begin fluctuating across the seascape until an absolute sense of “up-ness” is lost.
Likewise, as the seas calm down from the frothing madness, there reaches a point where either the upside-down boat’s deckhands or the rightside-up boat’s deckhands can help their neighbors faster than the sea can capsize them, and the order is restored.
But while the model might seem innocuous—concerning the tumultuous and frenetic lives of little magnets doomed to the whimsies of their neighbors—its applications are astonishing. It, of course, gives a fantastic description of ferromagnetism (and inebriated deckhands), but it can also be used to describe things as varied as (in no particular order) the stock market, genetic regulatory networks, surface adsorption, quantum mechanical spin networks, and certain quantum field theories.
As this whole venture was motivated by a desire to understand things like solid-liquid phase transitions, the Ising model can, in a rudimentary way, even describe the sharpness of certain liquid-vapor transitions. Jello, it seems, will remain a mystery, for now.