One experiment of a QT-like system was performed with 65,535 fixed particles and 1 moving particle. This shadowing experiment took 20 hours on a Fujitsu VPX240/10 vector supercomputer. A vectorization percentage of about 90% was achieved. A reasonable estimate for the time for this simulation would take on a Sun SPARCstation IPC would be about 100 times longer -- about 10 weeks. (If 99% vectorization could be achieved, it would be about 1,000 times longer -- about 2 years.)

As with all the simulations in this thesis, the particles each had mass
1/*N*, were distributed uniformly in the unit cube, and the initial
velocity of the moving particle was also generated in the uniform
unit cube in velocity space. However, the intent for this simulation
was to determine how the particle would react to a smooth potential,
so softening was set to 0.01; the average inter-particle distance is
0.024. The noisy integrator used was time-centred leapfrog with
a constant timestep of 0.001. Leapfrog is a second-order,
time-symmetric, symplectic integration method used commonly by astronomers
doing large *N*-body simulations.
Shadow steps were of size 0.1, and shadowing was attempted over
shadow step sequences of length of 1,2,4,8,.... The longest successful shadow was
512 shadow steps, or 51.2 standard time units. This is quite a long
shadow, and is encouraging for simulations of softened systems.
Considering that the unsoftened *M*=1 systems rarely had shadows longer
than 256, the one sample taken here with a shadow of length 512 would
seem to suggest a longer average shadow -- although only by a factor of 4.

In hindsight, this experiment may not be significantly more realistic than one with, say, and appropriate softening. However, it does seem to show that in a smooth potential, shadowing times are significantly longer than in more collisional systems.

Sun Dec 29 23:43:59 EST 1996

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