Finding Order in Chaos

UBC Math Prof. Gordon Slade - photo by Martin Dee
UBC Math Prof. Gordon Slade – photo by Martin Dee

UBC Reports | Vol. 51 | No. 11 | Nov. 3, 2005

Prof finds new clues in puzzle that has stymied mathematicians

By Brian Lin (with files from Mari-Lou Rowley)

Water freezes at zero degrees Celsius and vapourizes at 100. These are just two of the most common examples of phase transition, where changes in a single parameter cause physical properties to metamorphose.

By studying and mathematically modeling the fine line between two completely different physical properties — the temperature where ice melts or water vapourizes, for example — math professor Gordon Slade has proven there is order in chaos.

“Phase transitions are observed in nature, studied by physicists and seen numerically on the computer, but from a mathematical perspective, they are still very mysterious,” says Slade.

The heart of the mystery lies in the critical point that divides two completely different but stable states of being — as zero degree Celsius is to water and ice. While typically characterized as sporadic and random, Slade is convinced there is underlying mathematical structure.

“During phase transition, varying a particular parameter such as temperature changes the physical properties of the entire system,” says Slade. “Right at the critical point, the system is poised between two very different possibilities, and appears to exhibit utter irregularity.

“One way of looking at it is that the molecules can’t decide which form of behaviour they want to adopt.”

By applying probability theory to this randomness, however, Slade has found order in the form of fractals, a geometric pattern that, on a large scale, appears irregular but when divided and magnified, repeats the original pattern.

“Imagine the molecules moving randomly along a path. If you take a section of that path and blow it up mathematically, it bears amazing resemblance to the original path. The similarity appears again when you take a little piece of that section and magnify it, and so on.”

Slade has also found that Brownian motion — a mathematical model discovered by botanist Robert Brown in 1827 — proves extremely accurate in modeling this self-similarity.

Often described as a “drunkard’s walk,” Brownian motion has been used to describe random movements in a variety of areas, from sub-atomic physics to stock market fluctuations. The application of Brownian motion to critical phenomena, Slade says, greatly contributes to the understanding of critical phenomena and helps enable scientists to accurately predict behaviour at the critical point.

“One of the main goals in mathematics is finding elegant and short solutions that cut to the heart of things. Intriguing clues are now being found towards solving a puzzle that has stymied mathematicians for more than half a century.”

Read more about Prof. Slade’s work in probability theory in the latest issue of Synergy, the Faculty of Science newsletter, at