I'm not sure I understand what you mean when you say that the scaling is not a mean-field effect...
Mean-field is probably not a good way to describe what I was trying to say (since that implies some kind of averaging). I was thinking along these lines in using the term mean field: Once the system winds so tightly that there are order 1ish particles per wind, then the coherence is lost, and the impact of the various winds are somehow "averaged" or "mixed" together.
What I was trying to say is that the scaling behavior shows up well before the coherence disappears. In the energy plots, we saw there was a short period of violent relaxation followed by a long equilibrium followed by mixing and the gradual deterioration of the well ordered nature of the particle energies. I am saying that the $r^{-1/2}$ scaling shows up in the earliest phase of the evolution of this system during the violent relaxation, so this is the portion that we need to understand better analytically. Possibly understanding functionally how the energy is redistributed along the wind will be key to a better understanding.
Am I correct that you are finding that the particle with the k^{th} largest energy at t=0 still has the k^{th}largest energy at the end of the simulation (or close to it)?
Yes, this appears to be the case. Interestingly, the same is true for the interior particles, its the particles in the middle energies that tend to get mixed up if the simulation is run too long. Incientally, The energy gaps start to open pretty early, you can see the tracks diverging even as low as 50ATUs. Also, mixing among the ordering of the energies of the middle particles correlates with larger fluctuation on the energies of the interior particles. It may be that the phase relationship between the middle and interior particles has cycled all the way around until it has reached pi/4 again.
How are you defining energy? (1/2)v^2 + the sum of the potential
energies for that particle?
Yes.
It would be interesting to plot initial rank in energy vs. final rank in
energy. This would show the effect quite dramatically.
Sure, here is a plot for the same six timesteps. This is for the particles that started life with initial position > 0.
You can't really see it above, but departures from monotonicity begin early, especially near the center. Here are the same six snapshots, plotting final rank-initial rank.
I think that's it for Scott's questions, now on to Walter's questions.
what is the "ISW effect"?
I was referring to the integrated Saches-Wolf effect, that we see in the CMB anisotropies. Basically, when photons fall into gravitational potential wells they blueshift, and when they climb out again they redshift. In our universe, potential wells on the very largest scales are decaying at late times, and more so if there is more dark energy. Photons from the CMB traverse these decaying wells on their journey to us, and get a boost in energy because they don't redshift as much as they blueshifted. We see this as slightly elevated power in the CMB anisotropy spectrum on very large scales.
On p13, sigma and rho are power laws only out to about r~1 Is Q a power law over a significantly larger range in r? If so, for which exponent eps in Q=rho/sigma^eps?
I will look into this question further... just to clarify, in the plot I mistakenly plotted $\rho/\sigma^3$. Obviously when both $\rho$ and $\sigma$ are power laws, then Q will be for any eps, but you could be right, there may be a special eps for which the departures in power law in $\rho$ and $\sigma$ cancel each other out and Q remains powerlaw farther out in r. I'll try to find out. Actuallt eps=3 is looking pretty good!
on p15, another effect is the linear density along the "winds". The winding stretches the initial curve and the linear density along the stretched curve is reduced, the more so the more it is wound up. Thus the inner "winds" contribute each less to the density. I wonder whether one can not quantify this better (you say "hard to model" on p21), via mass conservation along the phase curve.
You touch on a point that I have been struggling with for weeks. I really wish I had a variable for the linear density along the winds. Every time I try and go down that route, Jerry is telling me that there is no metric in phase space, so the length of the arc has no meaning, therefore I can't divide the number of particles by the arclength as I would dearly love to do. This is actually how I discovered the energy thing, I was looking for a monotonic quantity along the arc, and hoping that the particles would sample this quantity in a predictable way. I will think about mass conservation... I actually think that mass scrunches up along the arc, sampling region of high velocity and small position more densely and caustics more sparsely.
I have plots of the contribution of each wind to the density (and the energy) but they don't help me much, after a while each wind is contributing 1 or 0 particles to a particular radial bin, so I can only reliably trace the contribution of the few outermost winds. Here is an example plot:
Here each "layer" is a wind where the layer level is switched as the phase angle goes through zero. Red is outermost. You might worry that yellow and pink are overlapping, but I think it's just an artifact of finite sampling, you'll note that there are just a few particles contributing to each bin in that region. I wish I'd kept initial positions for the runs that were 80k particles, but I hadn't implemented it yet. I guess I could run another, it's only 4 days. I'll think about that. I have these layer plots for energy too, but they are confusing.
As phase-space has no well defined metric, your definition of the phase angle is somewhat arbitrary. One may use the phase angle of the instantaneous orbit (the orbit of the particle if the potential were frozen at its current form).
Indeed, a fact that frustrates me daily. My interpretation was that the phase angle I plot means exactly that, the phase angle of the instantaneous orbit. Have I goofed that up? I got that plot by accident, because I was using the phase angle to define the winds. There is an even weirder plot that I made, that I totally don't understand but it could be significant. Take a look at this:
This shows the final phase angle as a function of the initial positions (which for a while are a proxy for energy except in the innermost particles, where they start to mix up in energies). You see the winds building up, but way before the point where the winds would be poorly sampled by the finite number of particles, you see the inner ones falling off the bandwagon, and hanging around at small position and high velocity. (Red and blue are particles left and right of center. T=23 should have been recentered or else was recentered poorly, sorry). Anyhow, I have no idea what this plot means.
Possibly, certainly at late times that seems likely. The plots scott asked for, however, indicate that energy mixing actually commences well before coherant spiraling disappears.
There is some funny effect (nicely visible on p33 at t=160) when the winding up is perturbed and a large gap appears. I also found this myself and have not yet found the reason for this behaviour (it occurs for both of my numerical algorithms and does not disappear when using "better" initial conditions). Perhaps there is some numerical instability in my code? Anyway, this effect seems to coincide with the onset of the energy mixing.
Yes, we wondered about the gaps. I think it would be nice to answer whether this is an artifact or real.
Okay, that's it for now. Long post. Sorry.
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