The supposed thermodynamic limit

According to the Kiessling paper, there is an exact thermodynamic limit (computed by Rybicki) that these systems should relax to, whose density profile is given by the form $\rho(x)=a sech^2(2ax)$. I wanted to see if any part of the profile matched this form, and if so, at what scale it departed from it, and started scaling like $r^{-1/2}$. My reasoning was that since $sech^2(x)$ rolls over, then somewhere locally it looks vaguely like $r^{-1/2}$. If that happened to be in the range we were studying, and we just were not probing far in enough for it to go flat, then we would have found the explanation. Here are the fits.



For what it's worth, I have fit all the data here, not just the scaling law portion in the interior. I tried just fitting the same range as I'd used before to fit $r^{-\alpha}$, the fit is equally bad.

Just as an experiment, I tried fitting the amplitude separately from the argument of sinh $\rho(x)=a sinh^2(bx)$ but this fared no better, see here:




I noticed that this new functional form qualitatively resembled what was going on in the warm data much more. I decided to try fitting hte warm data, because it might be interesting if the warm data was reaching this so-called thermodynamic limit, but the cold data was somehow being prevented from doing so. The results were disappointing I must say, here are the 1 and 2 parameter fits for the warm data:






So, I guess it looks like none of our simulations agree with the Rybicki derivation of the eventual density profile.