k-s clarifications

Here are a few clarifications and responses to Walter's email, I wrote my post latish last night, and was perhaps neglecting a few of the details:

we best take a simulation with equal-mass particles, such that the
hypothesis is equivalent to the number density is proportional to $r^{-\alpha}$ for $r$ less than $R$.

Yes, this is what I did.

Then the K-S distance is just

D = max_{r < R} | N(< r)/N(< R) - (r/R)^{1-alpha} |

Indeed. This is exactly what I calculated. I used a canned k-s routine, but I also computed the D statistic by hand and checked that it agreed with what was coming out of the kstest function.

From your plots it looks like alpha is less than 1/2

What you call alpha is what I called expt in the previous post. The answer to this is that alpha is rolling from less than 1/2 for some of the range of the data over to greater than 1/2 for other parts. A better way to put it might be that the data is simply not particularly well described by a power law (a result that is seemingly in conflict with the chi-squared result). More on this below.

Also it seems to me that you performed the K-S test for data at r>R, but as you did not explain what exactly you did (no mentioning of R or a similar parameter), I can only guess.

I am not sure if this is what you were concerned about, but I truncated the data I used in the k-s test to only include the range where the power-law scaling behavior is seen. I did the analysis at two different choices of truncation, 0.1 and 0.3, and included only data between 0 and the truncation radius in the analysis. The results were qualitatively similar for the two choices. The plots in the previous post were for 0.3. What you call R was included in my previous post as $x_max$, in the first version of the cdf I used. The second version was done as a sanity check, because the results came out so poorly, I wanted to double check with a number density function that I was sure was a good fit to the density data (with error bars). It surprised me how much the two disagreed.

I had a chat with Scott about this, he thought that potentially the run-to-run scatter may be substantially greater than just the shot noise. So I will repeat the chisq. fit using only the shot noise, rather than the variance between runs, and see if the value of reduced chi square increases dramatically (suggesting that a power law is not an especially good fit). That would make the results more consistent.

I'll also have a look at the k-s test between two separate runs to see if the profile is at least something like "universal" even if it is not particularly well described by $r^{-1/2}$.