By Jacob Aron It has taken 140 years, but only now do we have mathematical proof that a seminal equation describing the behaviour of gases is correct. Robert Strain and Philip Gressman at the University of Pennsylvania employed modern mathematical tools to find solutions for the Boltzmann equation, which predicts the motion of gas molecules. The existence of molecules was still being disputed when Ludwig Boltzmann formulated his equation in 1872, but its ability to accurately predict gaseous behaviour won out over any philosophical objections. Physicists today use the equation to model gases in everything from nuclear power stations to galaxies, yet until now there was no guarantee that it would work correctly in every possible situation. A formal proof has eluded mathematicians for so long because Boltzmann’s work was ahead of his time. “All the pieces have only been in place for about five years,” says Strain. The difficulty was due to a concept known as fractional derivatives. Most of the fundamental equations in physics are differential equations, which means they describe how the rate at which one variable changes is related to another variable. The rates of change are called derivatives. The first derivative of distance with respect to time is speed, for example, while the second derivative is acceleration. But what if you tried to identify the half or two-thirds derivative? They exist mathematically, but don’t make any sense in the real world, so Strain was surprised to see them while working with the Boltzmann equation. “It’s very rare to find fractional derivatives appearing in a physical model,” he says. What’s more, the mathematical theories behind them were developed many years after the equation. Strain and Gressman show that Boltzmann’s equation will always produce the right answer for gases that are close to equilibrium, such as the air in a building close to room temperature. However, they still can’t say whether the equation holds in more complex situations, such as a storm. Journal reference: Proceedings of the National Academy of Sciences, DOI: