Mathematical Proof for Universal Tracy-Widom Distribution in Particle Mixtures

It’s considered one of the great mysteries of science and mathematics – universality.

Universality is when certain mathematical expressions resurface time and again to describe many phenomena, even when the specific details of those phenomena are very different.

Probably the best known example is the Gaussian (or normal) distribution, because of its well-known bell-shaped distribution. The bell curve often is used to describe the probability of values of a set of variables such as test scores, the heights of students in a classroom, or the sum total when you roll more than one dice.

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Prof Jan de Gier, ACEMS Chief Investigator, The University of Melbourne

But in the 1990s, another universal distribution was uncovered with important contributions from ACEMS Chief Investigator Prof Peter Forrester from The University of Melbourne. It is now called the Tracy-Widom distribution, and it has received a lot of attention from researchers in mathematics and physics, including Prof Forrester and fellow ACEMS Chief Investigator at The University of Melbourne, Prof  Jan De Gier.

“It’s a distribution function that seems ubiquitous, or universal, in nature. But it’s not the usual Bell curve,” said Jan.

Unlike the bell curve, where the variables are independent of each other, Tracy-Widom deals with dependent or “correlated” variables in complex systems.

“It describes, for example, the interface of a growing coffee stain, where the growth depends on the local interface profile,” said Jan.

But the systems Jan and his colleagues are interested in are far more complex, and often microscopic and with many constituents. Examples include growth of droplets in liquid crystals, biological transport and traffic flow. But the Tracy-Widom distribution is also important to describe quantum mechanical energy level statistics for atoms, the Riemann zeta function and the prime number.

Until now, there has been a lot of literature in mathematics and physics on the Tracy-Widom distribution for a system with one type of particle, like a single gas or fluid.

But for the first ever, Jan and three colleagues from Melbourne and Tokyo are now able to describe a mixed system where you have two families of particles. For example, a system that involves a gas or fluid mixture with two different types of molecules. Their solution was just published as a highlight in Physical Review Letters, one of the most prestigious journals in Physics.

“The TW distribution has extremely rich underlying mathematics and to derive it you need to use a whole arsenal of mathematical tools from advanced algebra, combinatorics, probability, integrable models, random matrix theory and complex analysis. It is the convergence of all these different subareas of mathematics that makes it so interesting and rewarding,” said Prof De Gier.

The curve in the TW distribution is not symmetrical like a perfect bell curve. It’s steeper on the left side than the right, and its standard deviation is proportional to the sixth root of the number of variables instead of the Gaussian square root. Jan and his team are interested in developing a theory called “nonlinear fluctuating hydrodynamics”, which is assumed to describe complex flows in fluids.

“In the physics literature, researchers test their ideas usually on numerical simulations, but a rigorous mathematical foundation for nonlinear fluctuating hydrodynamics is an important topic of research in mathematics,” said Jan.

The research team includes Jan and one of his ACEMS PhD students from The University of Melbourne, Zeying Chen. It also includes Professor Tomohiro Sasamoto and his student Iori Hiki from the Toyko Institute of Technology. Professor Sasamoto is a leading expert on non-equilibrium statistical physics, and for Jan, it was his first chance to collaborate with him, even though he has known him for a long time.

“I have known Sasamoto since we were early career postdocs. We worked on similar problems, talked to each other at conferences and knew of each other’s work, but we never collaborated,” said Jan.

Until now.

In 2016, Sasamoto organised a research program on nonlinear growth models at the Kavli Institute for Theoretical Physics in Santa Barbara, California, and invited Jan to spend a few weeks. It is there that they started to work on this problem.

After some preliminary calculations, they convinced themselves their ideas could work, so the collaboration would move forward. One of their collaboration get-togethers occurred at MATRIX, Australia’s  international and residential mathematical research institute that Jan helped form in 2015 with the help of ACEMS and a partnership between The University of Melbourne and Monash University.

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Prof Peter Forrester

“We made some significant progress when he visited the MATRIX program ‘Non-equilibrium systems and special functions’ earlier this year.  Tomohiro visited Melbourne once and I visited Tokyo twice for productive one week visits. I think the two visits at Kavli and MATRIX were instrumental in achieving this breakthrough, as at these institutes we had ample, un-fragmented time to delve deeply into the problem,” said Jan.

The team wants to continue to push their research forward to other models of two families of particles, as well as to more complex mixes of fluids with more than two families.

What makes the research so exciting, though, is the concept of ‘universality.’ Jan believes it is critical that researchers continue to develop the mathematics that underlie these diverse and complex systems.

“A precise mathematical derivation tells you a lot about the underlying processes leading to certain emerging behaviour. Such a derivation also illuminates how universal behaviour can occur in widely different systems, be they biological, economical, mathematical or physical. It really boils down to saying that the underlying mathematics is the same, it is just that the symbols can be interpreted to mean different things in different contexts,” said Jan.

ACEMS researchers such as Prof Forrester and Prof de Gier keep exploring and developing new mathematical tools.

“Mathematics is an active field of research that constantly sharpens its tool and invents completely new ones without which we would not be able to fully understand the world we live in,” said Jan.

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