College of Arts & Sciences The Graduate Division Professional and Liberal Education

# Math Mystery

During the late 1860s and 1870s, when steam-powered technology was at its height, scientists were eager to understand the mechanisms underlying their empirical observations of how the temperature, volume and pressure of gaseous material interacted. In 1872, physicists Ludwig Boltzmann and James Clerk Maxwell developed a foundational equation predicting the motion of gas molecules. At that time, the very existence of molecules had yet to gain public acceptance, but because the Boltzmann equation modeled the behavior of gases so well, it was fully adopted. Major insights of the equation were that gases naturally settle into an equilibrium state when they are not subject to any external influence, and that even when a gas appears to be macroscopically at rest, it comprises a frenzy of molecular activity in the form of collisions, which account for gas temperature.

*Phillip Gressman (top) and Robert Strain*

Boltzmann's formula was also mathematically ahead of its time. While the predictions it made were backed up by subsequent experimentation, a mathematical solution proving that the equation would work correctly in every possible situation was elusive until mathematicians Phillip Gressman and Robert Strain turned to it nearly a century-and-a-half later.

"The mathematical objects that are essential to understanding this equation didn't exist until close to 100 years after the equation was derived," says Strain, who is the Calabi Assistant Professor of Mathematics. He and Gressman, who is an Assistant Professor of Mathematics, used modern mathematical techniques from the areas of partial differential equations and functional analysis to derive their solution.

The mathematical objects that are essential to understanding this equation didn't exist until close to 100 years after the equation was derived. – Robert Strain

The difficulty in pursuing a solution to the Boltzmann equation lay in not knowing whether mathematical proof for it actually existed. "All along, physicists have been making the assumption that since Boltzmann was a smart guy and his model is reasonably accurate, there must be a solution to his equation," says Dennis DeTurck, Robert A. Fox Leadership Professor of Mathematics and Dean of the College of Arts and Sciences. "Once they assume that, they infer properties about the solution and draw conclusions. However, the conclusions are always a tiny bit suspect because you don't know if the solution actually exists in the first place. Rob and Phil have changed that."

Gressman and Strain's findings, published in the Proceedings of the National Academy of Sciences, show that Boltzmann's equation will always produce the right answer for gases that are close to equilibrium, and that the effect of an initial small disturbance in a gas is short-lived and quickly becomes unnoticeable. It is still unknown whether the equation holds mathematically in a situation in which a gas is dramatically disrupted.

The qualitative properties of Gressman and Strain's solution can potentially be built into computer models of how gases work to make them more accurate. Additionally, the methodology they employed to solve the Boltzmann equation could lead to better mathematical understanding of more complex interactions among gas molecules.

"These mathematical advances will in turn feed back into the physics," DeTurck says. "That's the remarkable relationship between math and physics — when somebody makes progress in one field, people who work in the other immediately look for ways to exploit it."

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