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 ...
Mark for mathematical reference
you lost me after ‘seven dimensional equation’.
Hate it when that happens. And it’s not covered by my homeowner’s policy, either.
Hey if they can figure this out, they can figure that out!
I can see that sometimes, Moe Howard had the right idea! ;’)
Nyuk, nyuk, nyuk...
My understanding is the “4-color map” question was answered in a brute force way, and not so much as proof.
I think it was done via computer.
(Sound of grey_whiskers purring *and* kneading biscuits with his velvet paws.)
Cheers!
Cheers!
>> When they solve Fermat’s last theorem, call me.
Are theorems solved, or proven not to be false?
Their job was to prove that the set of test cases they created was complete, and that the computer algorithm exhaustively examined those cases.
There were several small corrections and refinements of the proof in subsequent years.
Have you ever met anyone who didn’t hate Transport? I enjoyed wisdom teeth pulling more.
Thanks for the ping!
Transport. Been there, done that. I’m not sure which I liked less, Transport or Thermodynamics.
Tough call.
(BS in Chemical Engineering from NC State)
NC State has good engineering as does my school - Georgia Tech.
When I was in school, the Process Control (equivalent) class in the Mechanical Engineering school was an acceptable substitute for ChE majors.
Most, like me, took it because it was easier (or the professor made it seem that way).
The labs in ChE were a PITA, because of the time it took. Looking back I don’t know why I was in such a hurry. I should have gone slower and appreciated the experience more.
Transport. Been there, done that. I’m not sure which I liked less, Transport or Thermodynamics.
Tough call.
(BS in Chemical Engineering from NC State)
Transport hands down. All theory, no pratical application. That came with Thermo, Mass Transfer and the labs. (BS ChE - UF; MS ChE - Lamar University)
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.