# Mathematicians Solve 140-Year-Old Boltzmann Equation

##### May 13, 2010

(PhysOrg.com) -- Two University of Pennsylvania mathematicians have found solutions to a 140-year-old, 7-dimensional equation that were not known to exist for more than a century despite its widespread use in modeling the behavior of gases.

The study, part historical journey but mostly mathematical proof, was conducted by Philip T. Gressman and Robert M. Strain of Penn’s Department of Mathematics. The solution of the Boltzmann equation problem was published in the . Solutions of this , beyond current computational capabilities, describe the location of 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.

The equation maintains a significant place in history because it modeled gaseous behavior well, and the predictions it led to were backed up by experimentation. Despite its notable leap of faith -- the assumption that gases are made of molecules, a theory yet to achieve public acceptance at the time — it was fully adopted. It provided important predictions, the most fundamental and intuitively natural of which was that gasses naturally settle to an equilibrium state when they are not subject to any sort of external influence. One of the most important physical insights of the equation is that even when a gas appears to be macroscopically at rest, there is a frenzy of molecular activity in the form of collisions. While these collisions cannot be observed, they account for .

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 Penn 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 behavior 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,” Gressman said.

The study also provides a new understanding of the effects due to grazing collisions, when neighboring 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,” Strain said. “The mathematical techniques needed to study such phenomena were only developed in the modern era.”

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##### hemitite
2 / 5 (4) May 13, 2010
Molecular frenzy,
Unnoticed it the still air,
The wind must really rock!
##### drel
not rated yet May 13, 2010
I have heard it said that it is possible in principal for a glass of 70deg F water in a 70F room to freeze if all the molecular collisions canceled each other out. However as far as I know it has never been observed even at microscopic volumes.
I have also heard talk that it is not possible to freeze water this way and that the extremely small probability actually does not exist and is an artifact of an imprecise mathematical model. I know we are talking gases and liquids but... How does this solution impact on this question? Does it resolve the question once and for all?
##### Au-Pu
1.5 / 5 (2) May 14, 2010
What I think this proves is that great minds and by this I mean the really great minds appear to have a capacity to visualise their solutions and then as with Einstein et al, their problem becomes detailing or describing it in a manner that others can utilise.
It often takes decades or longer for lesser minds to properly comprehend the reality they have visualised and explained (discovered).
##### frajo
not rated yet May 15, 2010
It often takes decades or longer for lesser minds to properly comprehend the reality they have visualised and explained (discovered).
The notion of "lesser minds" is misleading and disgusting. The human mind is not mappable to a set with a well-ordering. Is Kim Peek's mind a lesser one?
##### weewilly
not rated yet May 17, 2010
My Algebra for Dummies book has verified that I am indeed a certified page turner. I can percieve things in concept even though I might not understand its true underlying principals. Things like this are always interesting.