Mathematicians have found that materials conduct electricity when electrons follow a universal mathematical pattern.

Quote Originally Posted by Quanta Magazine
In a wire, electrons rebound off each other in such a complicated fashion that there’s no way to follow exactly what’s happening.

But over the last 50 years, mathematicians and physicists have begun to grasp that this blizzard of movement settles into elegant statistical patterns. Electron movement takes one statistical shape in a conductor and a different statistical shape in an insulator.

That, at least, has been the hunch. Over the last half-century mathematicians have been searching for mathematical models that bear it out. They’ve been trying to prove that this beautiful statistical picture really does hold absolutely.

And in a paper posted online last summer, a trio of mathematicians have come the closest yet to doing so. In that work, Paul Bourgade of New York University, Horng-Tzer Yau of Harvard University, and Jun Yin of the University of California, Los Angeles, prove the existence of a mathematical signature called “universality” that certifies that a material conducts electricity.

“What they show, which I think is a breakthrough mathematically … is that first you have conduction, and second [you have] universality,” said Tom Spencer, a mathematician at the Institute for Advanced Study in Princeton, New Jersey.

The paper is the latest validation of a grand vision for quantum physics set forward in the 1960s by the famed physicist Eugene Wigner. Wigner understood that quantum interactions are too complicated to be described exactly, but he hoped the essence of those interactions would emerge in broad statistical strokes.

This new work establishes that, to an extent that might have surprised even Wigner, his hope was well-founded.