Do you know how to interpret “maximum divergence” in this context?
Hm, this is a good question.
In writing my original reply, I figured “maximum divergence” was a meter. You start with two trajectories an angstrom apart, and they slowly diverge, but they can’t diverge more than 1 meter.
I think this is true if you’re just looking at the atom that’s shifted, but not true if you look at all the other atoms as well. Then maybe we actually have a 10^24-dimensional state space, and we’ve perturbed the state space by 1 angstrom in 1 dimension, and “maximum divergence” is actually more like the size of state space (√12+12...1024 times=1012 meters).
In which case it actually takes two tenths of a second for exponential chaos to go from 10^-10 to 10^12.
Also, IIRC aren’t there higher-order exponents that might decay slower.
Nah, I don’t think that’s super relevant here. All the degrees of freedom of the gas are coupled to each other, so the biggest source of chaos can scramble everything just fine.
Nah, I don’t think that’s super relevant here. All the degrees of freedom of the gas are coupled to each other, so the biggest source of chaos can scramble everything just fine.
Hmm, I don’t super buy this. For example, this model predicts no standing wave would survive for multiple seconds, but this is trivial to disprove by experiment. So clearly there are degrees of freedom that remain coupled. No waves of substantial magnitude are present in the initialization here, but your argument clearly implies a decay rate for any kind of wave that is too substantial.
Yeah, good point (the examples, not necessarily any jargon-ful explanation of them). Sound waves, or even better, slow-moving vortices, or also better and different, diffusion of a cloud of one gas through a room filled with a different gas, show that you don’t get total mixing of a room on one-second timescale.
I think most likely, I’ve mangled something in the process of extrapolating a paper on a tiny toy model of a few hundred gas atoms to the meter scale.
Hm, this is a good question.
In writing my original reply, I figured “maximum divergence” was a meter. You start with two trajectories an angstrom apart, and they slowly diverge, but they can’t diverge more than 1 meter.
I think this is true if you’re just looking at the atom that’s shifted, but not true if you look at all the other atoms as well. Then maybe we actually have a 10^24-dimensional state space, and we’ve perturbed the state space by 1 angstrom in 1 dimension, and “maximum divergence” is actually more like the size of state space (√12+12...1024 times=1012 meters).
In which case it actually takes two tenths of a second for exponential chaos to go from 10^-10 to 10^12.
Nah, I don’t think that’s super relevant here. All the degrees of freedom of the gas are coupled to each other, so the biggest source of chaos can scramble everything just fine.
Hmm, I don’t super buy this. For example, this model predicts no standing wave would survive for multiple seconds, but this is trivial to disprove by experiment. So clearly there are degrees of freedom that remain coupled. No waves of substantial magnitude are present in the initialization here, but your argument clearly implies a decay rate for any kind of wave that is too substantial.
Yeah, good point (the examples, not necessarily any jargon-ful explanation of them). Sound waves, or even better, slow-moving vortices, or also better and different, diffusion of a cloud of one gas through a room filled with a different gas, show that you don’t get total mixing of a room on one-second timescale.
I think most likely, I’ve mangled something in the process of extrapolating a paper on a tiny toy model of a few hundred gas atoms to the meter scale.