Boltzmann Equation Finally Solved

Two experts at the University of Pennsylvania Department of Mathematics announce that they were able to solve a 140-year-old equation. The two professors, Philip T. Gressman and Robert M. Strain, explain in the latest issue of the esteemed publication Proceedings of the National Academy of Sciences (PNAS) that they were finally able to uncover the solution of the Boltzmann equation. This set of calculations, which is also known as Boltzmann transport equation, deals with predicting the statistical distribution of particles in a gas at any given time, even in the future. It has been widely accepted and used ever since it was first proposed.

The calculus was developed by physicists James Clerk Maxwell and Ludwig Boltzmann during the 1860s and 1870s. Their effort was aimed at discovering a method of determining how gas fills a given space, hence the “distribution” part in the name. Additionally, their equation also factored in changes in variables such as velocity, pressure and temperature, which meant that scientists had a new and powerful tool at their disposal in conducting gas-related studies. Even if their theory was based on a then-unproven assumption – that gas is made up of particles – the scientific community recognized its value, and accepted it from the start.

Over the years, the equation was demonstrated over and over again in carefully-controlled experiments, which supported its predictions. Its main advantage is that it established the location of gas molecules in a probabilistic manner, at any given time. It also determines what chances a certain particle has of being at a certain location at a given time in the future. It's also interesting to note that the set of calculations which establishes these parameters is so complex, and covers so many aspects of gas behaviors, that it even exceeds the current capabilities of supercomputers. The advanced instruments we have today cannot rival the mind of scientists who lived 140 years ago.

“Even if one assumes that the equation has solutions, it is possible that the solutions lead to a catastrophe, like how it’s theoretically possible to balance a needle on its tip, but in practice even infinitesimal imperfections cause it to fall over,” says Gressman. He adds that the solution he and Strain developed was reached using partial differential equations and harmonic analysis, which are both modern mathematical techniques. “We consider it remarkable that this equation, derived by Boltzmann and Maxwell in 1867 and 1872, grants a fundamental example where a range of geometric fractional derivatives occur in a physical model of the natural world. The mathematical techniques needed to study such phenomena were only developed in the modern era,” Strain concludes.