Gressman and Strain were intrigued by this mysterious [Boltzman] equation that illustrated the behavior of the physical world, yet for which its discoverers could only find solutions for gasses in perfect equilibrium.
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.
I wonder how this might affect Polywell modelling.
ObsessiveMathsFreak (from other site) wrote:
It's worth noting that someone says that an equation has been "solved" in modern mathematics, they typically don't mean that you plug in the initial conditions and then get a formulae for your answer. Generally what they mean is that you can apply some other--probably numerical or approximate--techniques in an effort to solve the equation, and as long as you are careful, use enough computational resources, and don't go to far out, your solutions will be reasonably accurate.
This appears to be more or less what the team has done. They've proven the "the global existence of classical solutions and rapid time decay to equilibrium for the Boltzmann equation with long-range interactions". In other words, they've proven that the equation has "well behaved" solutions and not solutions for which something goes horribly wrong at some distance from your starting point.
While it doesn't sound like much, this is actually a very big deal. If the proof had gone the other way, it would mean that the equation would produce something akin to "ultraviolet catastrophes" under certain conditions, which means that the equation did not properly describe physical systems. With this proof, that's not an issue anymore and we now know that the equation will always produce reasonable solutions when given reasonable (i.e. physical) initial conditions.
Perhaps they've gone farther than just existence proofs and also provided a formula or technique for obtaining or approximating solutions. However, the Proceedings of the National Academy of Sciences journal is a closed publisher and the article is locked behind a paywall, so I guess the vast majority of us will never know.
In theory there is no difference between theory and practice, but in practice there is.
I don't think this treatment effectively covers local non-linearity where the kinetic factor is extremely high. It may be a reasonable presumption to disregard this, but not sure if it is as reasonable an assumption for [non-reversible] nuclear interactions as it is for [fully reversible] gas interactions. Just as well, I suppose, else it'd be saying that Polywell will simply thermalise, down an exponential decay.
It's not clear to me whether the Littlewood-Paley decomposition favours, or hinders, a thermalisation process for a scenario with non-reversible nuclear interactions [like a fusion reaction]. I rather suspect it might lead to a similar conclusion (but different method) I have seen before in which fusion plasmas tend towards an inverse 8th power distribution, rather than a Maxwellian distribution.