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.

Algebra

I have repeated the calculation a couple of times. I have not used anything in the notes except equation 2 and the postulated form of x(mu,t). I am finding a different power of t than the two of you. I bringing it up because I think it affects the conditions for which equation 3 is solvable. Here is my calculation, you can click on it to enlarge. Can either of you spot a goof?

Looking at poisson error bars

So far, the fits to a power law have all been using error bars that are obtained by computing the dispersion among the 6 runs for each set of initial conditions. I have now used Poisson error bars to do the same thing. Here I present a comparison. For one of the cold runs I have studied, here are the fits to a power law using the two different sets of error bars:



These look visually equivalent, though the reduced chi squared value is much smaller for the Poisson error bars. In order to compare the two sets, I made the following plot

Here it's clearly apparent that the Poisson error bars are systematically larger than those obtained using the dispersion among the runs. Just to be very specific, here is a snippet of code to illustrate how these two error bars were computed:

#this part is inside a loop ii over the 6 runs
d1[:,ii] =dens1 #these are the densities
d2[:,ii] =dens2
ep1[:,ii]=dens1/N.sqrt(h1[0]) #Poisson errors for the individual run
ep2[:,ii]=dens2/N.sqrt(h2[0]) #h1[0] is the number of particles in the bin

#Some statistics and output file
avgd1=d1.mean(axis=1) #This is the average density of the 6 runs
avgd2=d2.mean(axis=1)
if poisson==0:
stdd1=d1.std(axis=1) #This is the std. dev. of the 6 densities for each bin
stdd2=d2.std(axis=1)
else:
stdd1=N.sum(ep1*ep1,axis=1) #This step combines the Poisson error
stdd2=N.sum(ep2*ep2,axis=1) #bars from each run to get the error
stdd1=N.sqrt(stdd1)/N.sqrt(6) #on the mean.
stdd2=N.sqrt(stdd2)/N.sqrt(6)


I have also done the k-s test for these two cases, here are the plots.



In conclusion, it looks to me that the discrepancy between the chi square statistic and the outcome of the k-s test is not resolved by using Poisson errors instead of the run-run dispersion, indeed the Poisson errors are in fact larger than the dispersion errors, which causes the chisq statistic to improve. At the percent level, the choice of error bar affects the fit value of the exponent, but this is not an especially huge effect, obviously.