When employing any particular method to study the reliability of large
*N*-body simulations, it is essential that the method follow the
*N*-body system for a typical duration that an astronomer is likely
to simulate the system.
Thus the short simulations
in the previous chapter
tabulating speedups for simulations lasting one crossing time, though
good enough for measuring speedups, are not appropriate for studying
the reliability of long running large *N*-body simulations.
Although the number of large *N*-body systems that I have studied is
still small, some interesting trends have already been seen.

The shadowing attempts documented in this chapter were produced in
the same fashion as the previous chapter, except that longer
shadowing times were attempted, up to a maximum of 256 shadow steps
(25.6 standard Heggie and Mathieu time units). Shadowing was attempted
first on 1 shadow-step, then doubling the number of steps until two
successive failures occured on noisy orbits 2*S* and 4*S* steps long,
where *S* was the longest shadow that was successfully constructed.
In all the simulations, the total number of particles was held
at *N*=100, while the number of moving particles *M* was varied from 1 to 25;
*N*-*M* particles remained fixed.

Figure 3.0 plots the longest shadows found in
the above systems as a function of *M*. The number samples per
value of *M* was 10 for all except *M* equal to 1,3, and 5, which had
50, 22, and 13 samples, respectively. Although the
sample sizes are small and there is much noise in this graph, it
clearly shows that the length of the longest shadow that could be found,
using the algorithms in this thesis, decreases with increasing *M*.

**Figure 3.0:** Longest shadows found in unsoftened systems of *N*=100 particles,
as a function of the number of moving particles *M*,
while *N*-*M* of the particles remain fixed.
The vertical axis measures shadow steps, which were each 0.1 standard time units.
Each diamond represents the longest shadow found for a particular orbit.
The piecewise-linear line links the averages of these systems for each *M*.
For comparison, the curve 250/*M* is also plotted.

These shadowing attempts find the glitch position to within a factor of 2. The amount of noise in this graph could be decreased significantly by attempting more accurately to pinpoint the timestep of a glitch.

Although this result does not look promising for *N*-body simulations,
there are a number of things to keep in mind. First, it is not clear
that allowing *M*>1 particles to move amongst *N*>*M* particles is
significantly more realistic, in comparison to real systems, than
having only 1 particle moving, unless *M*>>1. This is because as long
as *M*<<*N*, each moving particle acts independently of the other
moving particles. Until the number of moving particles is comparable
to the number of fixed ones, the moving particles will still encounter
fixed ones more often than their moving counterparts. It is not clear
that 25 out of 100 is many enough moving particles.

Second, perhaps the method used to scale this problem is not
realistic. Perhaps a more realistic scaling of the problem, at least from the
astronomer's point of view, would be to have *M* particles move amongst
100*M* fixed ones. This would smoothen the gravitational
potential, which we know from other studies
[21, 10] decreases the Lyapunov
exponent, and thus may lengthen the average shadow length.

Despite the above caveats, there is reason to believe that
high-dimensional shadowing may be difficult. Dawsen *et al.*
[8] show that shadowing becomes extremely
difficult in systems where a Lyapunov exponent fluctuates about zero.
They claim that such fluctuating Lyapunov exponents occur frequently in
high-dimensional systems. I do not believe the Dawson result applies
to Hamiltonian systems, because it can be shown that the number of
positive eigenvalues in a Hamiltonian system is always
equal to the number of negative ones.

However, I think there is another reason that
high-dimensional shadowing of large *N*-body systems may be difficult.
Assume that, for a fixed noise amplitude,
there exists a mean shadow length *L* for a QT-like system
of 1 particle moving amongst *N* >> 1 fixed ones. (It is possible that
no such mean exists, if the scatter in shadow lengths is great
enough.) Then, in a system in which *M*>1, *M* << *N* particles move,
each moving particle will encounter fixed particles far more often than
it encounters other moving particles. Thus each particle, if followed
individually, will have a mean shadow length comparable to *L*. Since
work in this thesis and previous work has shown that glitches seem to
occur most often near close encounters, and since close encounters occur
as a stochastic process, a shadow length of *L* is
equivalent to a mean *glitch rate* of 1/*L* -- *i.e., * a
particle encounters glitches at a rate of 1/*L* per unit time. Thus,
the system of *M* moving particles, as a whole, encounters glitches at
a rate *M*/*L* per unit time, thus resulting in shadow lengths proportional
to *L*/*M*. As *M* becomes large enough to become
comparable to *N*, the rate that moving particles encounter other
moving particles increases, perhaps offsetting the fact that each encounter
lasts a shorter period of time. This leads to the following conjecture:

**Conjecture 1**
*If a chaotic system with D
dimensions has an average shadow length of T time units, then the
equivalent system scaled appropriately to MD dimensions will have
an average shadow length of T/M time units, if everything else is
held constant. (Especially the integration accuracy.) In other words,
shadow length is inversely proportional to dimensionality.*

The graph in Figure 3.0 seems consistent with this
conjecture, as the curve 250/*M* indicates.

Sun Dec 29 23:43:59 EST 1996

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