‘Einstein Was Wrong About Diffusion,’ Says This Professor

For a long time, long ago, scientists believed that willpower drove particle diffusion: Dropped into water or sprayed into the air, particles–like pollen, tea, ink, or perfume–decided where to move.

Then, in the early 19th century, the botanist Robert Brown disproved that theory by baking pollen, sticking it in water, and showing that it diffused anyway. Since the pollen was dead, he reasoned, it couldn’t be choosing how and where to diffuse.

It took almost 80 years and the discovery of microscopic molecules, previously unobservable, for scientists to realize that a roiling chaos of water molecules was the source of the observable diffusion in Brown’s experiments. Hopeless at the prospect of capturing the movement of all of these molecules, Einstein put forward a statistical model for diffusion: When suspended in water, each piece of pollen, he posited, took a series of random steps that he termed a “random walk,” independent of each other particle. As part of his theory, he developed the Einstein diffusion coefficient, which captured how quickly these particles spread out over time. This random walk model has become pervasive in its acceptance and application.

But Ivan Corwin has a different theory. Earlier this year, Corwin, a professor of mathematics, was named a Blavatnik National Awards for Young Scientists finalist for his work on “extreme diffusion.” The basic premise of that work, Corwin explained in a recent interview, is that “Einstein was wrong about diffusion,” at least when it comes to the behavior of extreme particles–those that move the farthest and fastest among a system of many particles diffusing together.

Columbia News spoke to Corwin about what this work means, its possible implications for wide-ranging fields like cancer treatment and public health, and the overlaps he sees between math and gardening.

What does your work on extreme diffusion theory posit?

Einstein’s theory of diffusion says that, in a system where particles diffuse, they’ll behave collectively in predictable ways and individually like random walkers independent of each other. The Einstein diffusion coefficient determines how quickly this happens and depends on the type of particle and the media in which the diffusion occurs. The assumption of independence is fraught, though, since all particles diffuse in a common environment and hence may be subject to similar forces if nearby at a given time. Remarkably, despite this assumption, Einstein’s theory works extremely well at describing the motion of the collective in a many-particle diffusion, and also that of a typical individual particle.

Real diffusion often involves billions of particles or even many more, and a theory that describes most particles may fall very short at describing special particles, such as those that move the farthest and fastest—the extreme particles. I have developed a new model to describe the motion of many-particle diffusion in which, unlike in the Einstein theory, the effect of the environment is taken into account. Our theory still predicts the same collective or typical behavior as Einstein’s, yet when we study the extreme particles, we find rather different behavior. In particular, we find a new Extreme diffusion coefficient that governs the behavior and tells us something new about the nature of the environment in which the diffusion occurs. In this way, extreme diffusion is a microscope through which to observe a new property of the hidden environment in which diffusion occurs.

How could this help us understand phenomena like the spread of tumors and pandemics, (which were mentioned in your Blavatnik citation)?

The short answer is they can’t, yet, but they may be able to in the future.

If you look at the spread of a pandemic, like the beginning of COVID, you see that, at the beginning, the events were quite localized, but they quickly started to spread like wildfire. And what was happening was that the outliers–the people who traveled from the initial epicenter of the pandemic the farthest and the fastest–drove its initial evolution.

When operating on a tumor, doctors strive to remove all cancerous cells to avoid relapse. The size of the margin is controlled by how far extreme cells can be expected to deviate from the main mass of the tumor. Anything that helps reduce margins while maintaining effectiveness of surgery could be very valuable.

Another example is getting poked by a needle. If you poke your finger with a pin, your nerve cells pick up the signal of pain. You don’t need 99% of the nerve cells, or even 1% of them, to do it, you need just a fraction of that, in order for the signal of pain to get to the brain. The speed at which you feel things is controlled by the extreme travelers.

So, my hope is our theory of extreme diffusion will help researchers understand the behavior of extreme particles in many real-world contexts where those extremes are very meaningful.

What is the project that you and your brother are working on?

My brother, University of Oregon physicist Eric Corwin, and I received a W.M. Keck Foundation Grant to work on these questions in a way that would bring together both of our expertise. The idea was to see if we could design experiments that would realize the predictions that I had started to make about extreme diffusion experimentally, using his physics expertise. We’re still working on it.

Had you worked with your brother before?

We shared a bunk bed when we were little. He’s five years older than me.

As adults, we hadn’t really worked together much until 2016, when he visited a workshop I was running at the Kavli Institute for Theoretical Physics, and we started to talk about our work and ideas in a bit more depth.

He was always explaining things to me when we were kids, so this time, it was fun to explain my work to him. The idea for this particular project came together at the beginning of COVID, when everything felt a bit different, and we were kind of like, “well, why don’t we do something different ourselves?”

What are your favorite hobbies?

I have four kids all born within four years, so that takes up a big amount of my hobby time.

I grow vegetables. Tomatoes, cucumbers, peppers, peas, that sort of thing. Nothing illicit or too exotic.

I like to plant seeds because it’s kind of like math: You start from zero. You have to build it up from nothing, so that everything is on solid ground. And then you understand the whole process. I worked for a year at a hedge fund after college, and I didn’t like that you were just a piece of a machine. I wanted to understand the whole machine. And that’s what you can get from gardening and math; you’re building something from scratch, and you can understand the whole system.