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Results for paper

Movie

Movie with a stable loop containing 4 kinks all of which have \vec p_+\cdot\vec p_- < -0.9. This loop is colored red in the movie and we zoom in on this loop for part of the movie.

Loop lengths

Length histograms per generation.
Stable loop length histograms. Shows the resolution limit. Also includes SP (Scherrer and Press) and CA (Casper and Allen) results.

Kink Counts

Number of kinks per loop. Only a few cases are plotted since they are all the same.

Stable loop velocities

Velocity of stable loops. Only a few cases are plotted since they are all the same. Here we see the importance of resolution. SP has the wrong behavior for v\sim1 which is improved by CA and even beter by us.

Eigenvalue averages

Average of the eigenvalues per generation. This is for N=10,000, M=10.

Kink sharpness

Kink sharpness histograms. This is now for the sharpness defined by (1-\vec p_-\cdot\vec p_+)/2. Also, fewer models are plotted, consistent with some of the other plots.

Old results

Daughter Loop length distributions

Distribution of lengths as above but now comparing N=10,000, N=20,000, and N=50,000. The N=50,000 currently has poor statistics. Even so, we see the the shape of the distribution is independent of N and the feature at short lengths has moved to the left, a strong indication that this is the resolution limit of the simulations.

New length histograms and weighted length histograms where the histogram is weighted by the length of the loops. These plots include better statistics and the M=20 case for N=10,000. Notice in the weighted histograms how all at the same M agree extremely well.

Comparison to Scherrer and Press

Here we compare to the results from Scherrer and Press 1989 (hereafter SP). It shows the power of higher resolution! Further, our results are not resolution dependent except for statistics related to the smallest loops formed (ie, the ones at the resolution limit). This can be seen by the secondary peak in the loop length plot.

I refer you to SP for a more detailed description of the quantities calculated for these plots, though I have included the appropriate plots from SP for convenience.

All my runs are equivalent to their case (A). This means the initial loop is chosen so that

\vec a(s) = \sum_{m=1}^M \vec a_m\cos(ms+\vec \phi_m),

where the \vec a_m are drawn from a uniform distribution. This puts equal power on all scales.

I have used M=10 to be consistent with SP but have also done runs with M=20 to get "wigglier" loops.

Loop length plots for stable loops: SP; New

Loop fragmentation fractions for all loops. This fraction is the ratio of the length of the smaller daughter loop to the length of the parent loop.: SP; New
Velocity histograms for stable loops: SP; New

Velocity scatter plots for stable loops: SP; New

Number of stable loops per generation. Note that SP got the peak of this right, and the rough shape is ok, the steepness at few generations and the tail at many generations is off, as would be expected. Also note that it doesn't depend on resolution.: SP; New

Fragmentation probability as a function of loop generation.

This could be defined in (at least) two ways: at generation g we could define it as (1) total number of loops that split / total number of loops created where the total is the sum over all loop realizations; or (2) number of loops that split / number of loops created then average over all runs. To understand the difference suppose there are 2 loop realizations. For realization (a) at generation 4 there were 6 loops formed and 4 split and for realization (b) there were 4 loops formed and all 4 split. Then the ratio using method (1) would be 8/10 = 0.8 and method (2) it would be (4/6 + 4/4)/2 = 0.833. I have used method (1) in my plots. I haven't checked method (2).
I have included error bars on my plot so you can see where the values are to be trusted and how more statistics would improve these plots.

NewThis plot is somewhat hard to read so see the next plot for a clearer view of what is going on.

New with N=10,000, M=20 The N=10,000 runs have the most statistics and the M=20 runs are "wigglier" so have more generations formed. Thus there are better statistics at higher generation number. It appears that the curve asymptotes. This is one case where better statistics is very helpful.