Posted on 05/24/2010 5:23:52 PM PDT by SunkenCiv
Mathematicians have found solutions to a 140-year-old, seven-dimensional equation that were not known to exist for more than a century, despite its widespread use in modeling the behavior of gases. 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. Solutions of the equation, 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...
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, are 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.
(Excerpt) Read more at scientificcomputing.com ...
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