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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
Kink Counts
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
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.
 SP
 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.