QnAs with J. Michael Kosterlitz

From biology to astrophysics, the transition from a disordered to an ordered state in any complex system proceeds through often nonlinear steps. The variability inherent in the transitions hampers researchers’ ability to predict where and how the systems will come to a resting state. J. Michael Kosterlitz, the Harrison E. Farnsworth Professor of Physics at Brown University, uses the mathematical field of topology to investigate these and other dynamic processes, drawing on five decades of expertise in theoretical condensed matter physics. In 2016, Kosterlitz, along with David J. Thouless and F. Duncan M. Haldane, received the Nobel Prize in Physics for work using topology to examine phases and phase transitions in matter. This research facilitates understanding of superconductivity and superfluidity and guides the development of quantum computers. PNAS recently spoke to Kosterlitz, who was elected to the National Academy of Sciences in April 2017, about his current research.

J. Michael Kosterlitz. Image credit: Willis Peter Bilderback (Brown University, Providence, RI).

PNAS: Your Inaugural Article (1) demonstrates a method to understand systems that are out of equilibrium. Why is it important to study such systems?

Kosterlitz: Almost all real-world systems are not in equilibrium; they are perturbed by random or stochastic noise. Every experiment, too, has noise associated with it. For example, experimental vibrations or equipment faults can cause noise. Or, if a truck drives by the laboratory, that’s noise you can’t control. And certainly, there is a lot of noise in biological systems. The interior of a cell is chaotic, almost unimaginably so. But cells carry out processes very reliably. And from our investigations, it turns out that noise is absolutely vital.

PNAS: How does this work fit into the broader context of your research?

Kosterlitz: My training was in critical phenomena and influenced by renormalization theory. I learned that a lot of physical effects were technically unimportant. In other words, one could write down a fairly simple and approximate model of a system and get exact results from it. Sounds crazy, but the stable, fixed points are basically the same for many different systems, which gives us a chance to investigate complicated systems that are too messy to describe in all their detail.

This modeling explains the phenomenon of universality, and I have been thinking along these universal lines for quite some time. As I became interested in trying to understand stochastic noise, I applied this concept of universality to see if there were stationary states in systems far from equilibrium that can actually be predicted. In fact, noise helps determine a system’s final, stationary state from an infinite number of possibilities.

PNAS: How did you develop the method you describe in the Inaugural Article (1), and what does it reveal about systems that are out of equilibrium?

Kosterlitz: For systems that are driven out of equilibrium, there was no obvious potential one could calculate to determine its final state, the state with a free-energy minimum, to borrow language from statistical mechanics. It wasn’t even obvious that such a state existed. After all, if you have a deterministic equation and add random noise to it, you would expect the noise to mess everything up, that there would be many final states possible, all of which were mathematically equivalent.

Such systems, far from equilibrium, I thought, were analogous to ones with stochastic noise. Our group and others have experience with noisy systems and did numerical simulations on them. It turned out that the noise helped select the stationary state, but there was no obvious reason why of all possible final states one would be preferred.

Numerical exercises are fine, but if you try to describe every molecule in a system in all of the details, you’ll get nowhere. But this initial work helped me realize that there must be some potential to construct that would get you to this final state, that would even predict this state.

We needed to extend calculations to an infinite number of degrees of freedom as in a real physical situation. So we worked to construct a potential for a particular dynamic equation and chose the stabilized Kuramoto–Sivashinsky equation. This equation is a nonlinear partial differential equation developed in the 1970s to model the dynamics of laminar flow in flames. It is known for exhibiting chaotic behavior, as well as other properties that one would expect from real systems.

Using the stabilized Kuramoto–Sivashinsky equation, our work in this paper (1) revealed that these final, stationary states are all interconnected; they form a topological web of saddle points and that stochastic noise essentially selects one of these points as the final state.

PNAS: Does this approach have applications in other disciplines?

Kosterlitz: While there are no direct physical realizations of these results, this is the simplest equation that produces stationary states seen in real systems. It was a purely theoretical investigation based on a simple model to construct a potential.

On principle, this method is very powerful when applied to a specific problem, regardless of the discipline: biology, physics, chemistry. If you can get it to work, this method basically answers all of the questions you would want to ask. But getting it to work is not easy. Any given system is fraught with its own difficulties, and there are a lot of steps from the initial problem to the answer.

However, the stabilized Kuramoto–Sivashinsky equation, like many other partial differential equations, is amenable to standard methods. So our approach paves the way to explore nonlinear phenomena under the general umbrella of statistical physics. This means that, in principle, if you can write down a dynamic equation for the system of interest, then it should be possible to construct a potential using our methods. We have laid the basis for a general method to study such dynamical systems.

PNAS: What is next for these investigations and for your research?

Kosterlitz: The stochastic noise we’ve looked at is additive. It is the simplest possible noise. There is more complicated, multiplicative noise, and we are interested in seeing if we can construct a potential for those systems. But it’s certainly not going to be easy.