Response to questions II

Hi, thanks again for your prompt responses!  I'll try and provide answers/plots here.

How did you define the phase angle you are plotting? Note that getting the phase angle of the instantaneous orbit in a frozen potential requires to obtain the map from (x,v) to (J,phi) (action-angle variables), though J is not required. Thus phi is not just some polar angle in (x,v) space.

I did not do such a transformation... the angle in the plots is defined as the arctan(v/x), so in this case it is just some polar angle in phase space.  Philosophically I understand that there is an arbitrary scaling in V, so that this angle is not well defined, but on the other hand, if you look at the plot I made of angle vs. x, the density at each x is the projection through all the layers, so the density does somehow depend on this non-meaningful angle.  Another way to say this:  if all the velocities were scaled, then the windup proceeds at a different rate as a function of time... so a given snapshot would have 1) different angles yes but also 2) a different total number of winds.  Therefore I am confused.

what do you think about this: 2010arXiv1010.3008V ?

Whilst this paper is rather confuse, I think in a way they also claim that
mixing (of energies for example) is not relevant or does hardly happen,
similar to our findings. However, I don't think they provide any explanation
for the final profile.

I read this paper, thanks for pointing it out.  I had not thought of Scott's objection, but I agree with him.  Specifically, I think the energy transfer between particles on various orbits depends a lot on their exact phase relationship as they bounce through the middle (or I guess periapse), and this is not evolving dynamically in their approach, which may make a difference in the final density distribution.

I am not sure I agree with your assessment that they are finding that the energies don't mix.  Maybe I am interpreting this wrong, but if you compare the 1st and 2nd panels of figure six, the different colored curves seem to me to be roughly proxies for initial and final energies (for a snapshot at the final time).   The first panel shows that while there is no mixing in the outskirts, it's definitely particles that started life in the middle that contribute most to the inner radii.  We are finding this too.   The fact that in the second panel the ordering of the colored curves (even in the outskirts) is not the same implies that the turnaround radius of particles at the end of the day does not seem to be a monotonic function of the initial radius.   Caveat: since I am thinking the initial potential energy as the dominating energy, it's possible that whatever they used for initial velocities messes up this logic.

I think I understand what you mean by mean-field effect. The way I would say it is that the scaling behavior only shows up as long as the system looks cold, i.e. as long as there are well defined streams in phase space.

Well, I think the surprise is that the scaling behavior DOES show up while the system is still cold and there are well defined streams in phase space...   and apparently stays that way after the well defined stream disappear.

I think the ISW effect is not a useful analogy for non-cosmologists (!)

Agreed, I won't use it in the paper  :)

The situation in three dimensions is that rho/sigma^eps is a power-law over a much larger range than either rho or sigma, if and only if eps=3.  Various people claim that this is because eps=3 makes rho/sigma^eps look like a phase space density, but this is a weak argument. It's not at all obvious what value of eps works best in one dimension but if its not eps=1 then we have a counter-example to the arguments about phase space.  

Here is a plot of $\rho/\sigma^{eps}$.  The curves have been offset by a factor of 100, so their curvature at high r is easier to compare.  Eps=1 is indeed not the exponent that maximizes radial range of the power law, but I am not sure that is significant, as I will explain below:

You'll note that eps=3,4,8 all seem to give equally good power law behavior, and there is a reason for this.  Jerry pointed out to me that the behavior of particles in the outskirts of the system look to be behaving isothermally, and if that is the case, $\sigma$ will go toward a constant.  Here is a plot of $\rho$ and $\sigma$ individually, and I have also plotted $r^{-1/2}$ truncated with something that looks like $e^{-\Phi}$, which indeed seems to fit well (in a chi by eye sense).   

The velocity dispersion $\sigma$ seems to go to a constant (if you ignore the last data point), and is furthermore fairly noisy.  If you raise a constant (with noise) to a high power, it amplifies the noise. Dividing by this noisy thing makes it harder to detect any rolling off of Q, but if sigma is actually FLAT all powers of sigma are doing the same thing to the curvature of Q... nothing save hiding it in the noise. 

Of course $\sigma$ is rolling slowly toward flat, and may be rolling in concert with $\rho$ to keep Q scaling the same way in the rollover region... we might be able to argue from the plot above that $eps=3$ is a legitimately a power law farther out than $eps=1$.  But I am not sure.

You say that you have plots of the contribution of each wind to the density and the energy. A related question is this: is the potential gradient near the center dominated by the "small" winds near the center of phase space, or the "large" winds?

This question is confusing, can you clarify?  The force from particles outside r is canceling, so only the potential due to particles inside r influence the dynamics of a particle at r, or am I misunderstanding something?  Sorry...

Response to questions

Hi, Here are some responses to questions you both sent.  Scott's in red and Walter's in blue.

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.

Actually, I thought it was interesting to look at a few other times between 40 and 200.   I don't know anything about resonances, but these departures don't exactly look randomly placed. 

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.

The "mixing of energies" seen at late times (p30) may be due to loss of numerical resolution: instead of the coherently time-varying potential of the winding spiral, we have inidividual particles scattering about.

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.

The talk slides and a few extra things

Hi Scott and Walter,

    Sorry to decouple quite a bit in the last month or two.  I've done some work on our project since I last posted, there are a number of new plots.  I think the easiest way to catch up might be if you look through the talk I gave at our informal seminar here.   Here is a link:

http://public.lanl.gov/aschulz/talks/LA-astro23Feb.pdf

I had intended to send it to you for comments before giving the talk, but inevitably life intervened and I was working on it until the late the night before.  I think the main new thing I've found is that the r^-1/2 scaling is not really a mean field thing that arises after the system equilibrates, in a sense it is there from the beginning, which I suspect was not known before. 

I've recognized a few mistakes in the talk since giving it, Q in 1D should likely be rho/sigma,  just on dimensional grounds, and I don't think the equations I wrote down for N(w) make any kind of sense.  It was late at night.  

Also, I've added a bunch of slides after the conclusions, dealing with Energy evolution of the system.  These were not in the talk, but I thought you would like to see them.  The main point is that the particles behave like beads on a string, no matter how long the string gets, the beads do not pass one another, in that they keep their initial ordering along the wind.  Energy is a proxy for this ordering, because it remains a monotonic function of the initial position for an extremely long time.  This is significant because unlike the x and v, it evolves very slowly in time.  In a sense it governs how the "beads" are sampling different portions of the "string."   Energy might therefore be a good way to quantify how particles on each branch of the winding are contributing to the density at any given position.  I am still trying to work out how to do it, however.  One problem is that I don't have an expression for the potential energy, so my initial plan of solving to eliminate v is not working so well.