The AMS sponsored an exhibit at the 19^{th} annual Coalition for National Science Funding (CNSF) Exhibition & Reception on Capitol Hill held on May 7, 2013. Philip T. Gressman, University of Pennsylvania, presented work on “The Boltzmann Equation: Where Mathematics and Science Collide.” The Exhibition drew over 285 people, including ten Members of Congress, to view 35 exhibits on research funded by the National Science Foundation.

Gressman and his colleague Robert M. Strain 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 Boltzmann equation was developed to predict how gaseous material distributes itself in space and how it responds to changes in things like temperature, pressure or velocity. Using modern mathematical techniques from the fields of partial differential equations and harmonic analysis, Gressman and Strain 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.

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.

*CNSF is an alliance of over 130 scientific and professional societies and universities united by a concern for the future viability of the national science, mathematics and engineering enterprise. *

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