For years, working on an interdisciplinary project investigating order and collective behavior among complex systems — particles, ant colonies, flocks, robot swarms and the like — the computer scientist Dana Randall had heard about “rattling.”
“Our colleagues had strong intuition,” Dr. Randall said. “The idea was vague, but it seemed like there was something to it.”
The rattling notion was first introduced in a 2021 physics paper. It suggested that rattling was analogous to energy: In nonequilibrium systems rattling might play a role similar to energy in equilibrium systems — just as equilibrium systems seek lower energy states and thereby achieve spontaneous order, nonequilibrium systems with orderly configurations tend to evolve towards states of lower rattling.
Now, a few years on, the rattling concept is becoming clearer.
“That original paper was full of breadcrumbs,” said the probabilist Jacob Calvert, a postdoctoral fellow with Dr. Randall at Georgia Tech. In its accuracy and scope, the rattling heuristic hinted at a simple underlying mathematical theory.
Dr. Calvert proposed — “boldly,” Dr. Randall said — that they try to understand rattling more rigorously.
Last fall, they published initial results in PNAS, “A local–global principle for nonequilibrium steady states.” They provided a formal mathematical theory, as well as deeper insight. The work was motivated by Dr. Calvert’s idea to translate the problem into a question about Markov chains, mathematical models used to represent and understand steady states of physical systems.
The researchers found that the accuracy of rattling as a predictive measure depended on the relationship between local and global “parts” of the steady state (in Markov chain lingo, the exit rates and jump chain stationary distribution, respectively). “The conceptual heart of rattling,” Dr. Calvert and Dr. Randall explained in the paper, “is the simple observation that many systems tend to spend more time in configurations that they exit more slowly.”
Such scenarios are described as “low rattling.” “Low rattling is when a complicated system settles down into a small rhythmic pattern where it seems calm and local for a while, before it shoots off and chaotically bounces around the whole state space again,” Dr. Randall said. “It’s capturing these moments when order seems to emerge for a period of time, even if shortlived.”
The authors also noted in the paper: “Surprisingly, we find that the core idea of rattling is so general as to apply to equilibrium and nonequilibrium systems alike.”
Dr. Randall elaborated: “Low rattling is only predicted for certain systems and cannot be true in general. With that said, the conditions can — and also might not — be met both for equilibrium and nonequilbrium systems, which is surprising.”
rattling.png519.31 KBThere is a simple recipe for making systems that exhibit order, in the sense that they spend a disproportionate fraction of their time in relatively few states. Start with a system where every state “looks the same” and so favors no state (left). Then, for each state, speed-up or slow-down the transitions out of that state by a random factor. This results in a system that spends a greater fraction of time in some states, and for a “local” reason (right). Calvert, den Hollander, and Randall proved that a large class of systems exhibit order for local reasons because they appear as though they were produced by this recipe.
CREDIT: Jacob Calvert
This result provided a way to quantify how good the rattling explanation is for a given system. “But you don’t necessarily know how typical the systems are for which the explanation is good,” Dr. Calvert said. And it is known that there is no explanation that applies to all nonequilibrium systems (that is, there is no explanation analogous to energy in all equilibrium systems). Some call this impossibility result “Landauer’s blowtorch.”
So, having shown that when certain conditions are satisfied, low rattling holds, and that local and global characteristics of a system can be connected, the next step was to validate that the required conditions actually occur in many systems.
“For which systems does rattling explain how order arises?” Dr. Calvert said.“If you want to say this theory is going to be really valuable, you need to show, rigorously, that there are huge classes of systems where the explanation is a really good one.”
At that, it is a mathematical problem — and one with many possible approaches.
Dr. Calvert and Dr. Randall made some progress during the program on “Probability and Statistics of Discrete Structures” at the Simons Laufer Mathematical Sciences Institute (January 21 to May 16, 2025; Dr. Randall was on the organizing committee).
The program prompted a pivot in their approach. Dr. Randall suggested exploring whether there were classes of random graphs for which rattling would likely hold — since a random Markov chain can be thought of as a certain kind of random graph. “Looking at random classes seemed like a good approach to start to get a handle on the problem,” she said.
They went to lots of talks about probability of discrete models and random graphs — and connected with participants from both the P.S.D.S. program, and the partner program on “Extremal Combinatorics.” Said Dr. Randall: “This question fell squarely in between both programs!”
And fortuitously Dr. Calvert, the SLMath Berlekamp postdoctoral fellow for the semester, was assigned as his mentor Frank den Hollander, a research professor in the SLMath program and a professor emeritus of probability theory at the University of Leiden. While Dr. den Hollander is an expert on random graphs, the notion of rattling in nonequilibrium statistical physics was “completely new to me.” He added: “Being able to explore new tracks is exactly the kind of thing you hope for at SLMath.”
By modelling the nonequilibrium systems with random graphs, the three of them found a way to assess how typical it is for rattling to explain order. They posed the question: Is there a large collection of graphs for which we could show rattling would almost certainly hold?
The answer was yes — and they provided a proof, which, they wrote, “constitutes the first rigorous validation of an emerging physical theory of order in nonequilibrium systems.”
Generally, Dr. Calvert said, the result is about typicality: How typical are systems for which it is easy to explain where order comes from? They haven’t fully answered the question but it’s a “humble step,” he said.
And even more broadly, he said, reflecting on the SLMath program, “It was really interesting to, for the first time, present our work to other mathematicians, and to highlight the mathematically interesting aspects. There are lots of things for mathematicians to do in this direction.”
The mathematical theory might also provide further insights to the physical theory — such as a conceptual reason why the physical theory could apply to many systems.
Already, physicists on the 2021 paper have gotten in touch with some feedback: This is very much the type of argument they were looking for. Several of the researchers have since joined forces in working on extensions.
