Finally, 140-year-old Boltzmann Equation solved

Washington, May 14 (ANI): Two mathematicians from University of Pennsylvania have solved a 140-year-old, 7-dimensional Boltzmann equation, which remained a puzzle for more than a century despite its widespread use in modelling the behaviour of gases.

Philip T. Gressman and Robert M. Strain of Penn's Department of athematics conducted the study

The solutions of the Boltzmann equation problem, beyond current computational capabilities, describe the location of gas molecules probabilistically and predict the likelihood that a molecule will reside at any particular location and have a particular momentum at any given time in the future.

During the late 1860s and 1870s, physicists James Clerk Maxwell and Ludwig Boltzmann developed this equation to predict how gaseous material distributes itself in space and how it responds to changes in things like temperature, pressure or velocity.

Gressman and Strain were intrigued by this mysterious equation that illustrated the behavior of the physical world, yet for which its discoverers could only find solutions for gasses in perfect equilibrium.

Using modern mathematical techniques from the fields of partial differential equations and harmonic analysis - many of which were developed during the last five to 50 years, and thus relatively new to mathematics - the mathematicians proved the global existence of classical solutions and rapid time decay to equilibrium for the Boltzmann equation with long-range interactions.

Global existence and rapid decay imply that the equation correctly predicts that the solutions will continue to fit the system's behaviour and not undergo any mathematical catastrophes such as a breakdown of the equation's integrity caused by a minor change within the equation.

Rapid decay to equilibrium means that the effect of an initial small disturbance in the gas is short-lived and quickly becomes unnoticeable.

"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," said Gressman.

The study also provides a new understanding of the effects due to grazing collisions, when neighbouring molecules just glance off one another rather than collide head on.

These glancing collisions turn out to be dominant type of collision for the full Boltzmann equation with long-range interactions.

"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," said Strain.

The study was published in the Proceedings of the National Academy of Sciences. (ANI)