Stellar systems are finite number systems by nature. Because the
characteristic relaxation time is so long in many cases of
astronomical interest, the continuum limit is a good approximation and
the collisionless Boltzmann equation (CBE) governs the evolution.
Because the CBE is difficult to solve in any generality, many
researchers have turned to n-body simulation as a primary source of
insight. For a modest number of particles, however, the naive n-body
problem is not a simulation of the CBE but of the collisional
Boltzmann equation. Enormous effort has been applied to manipulating
the interparticle force to reduce the intrinsic relaxation to produce
a near-collisionless solution. The most commonly used technique
smoothes the particle over a finite size region leading to the
softened the point mass potential: 
. As long as the smoothing size is not larger
than the mean interparticle spacing, intuitively, there should be no
significant change in the results.
Unfortunately, two-body interactions are only part of the story. Poisson fluctuations excite structure at all scales in the system. Many simulations are optimized to resolve large-scale features but the relaxation is enhanced by large-scale collective excitations on these same scales (Weinberg 1993). One can think of this excitation as the projection Poisson noise on the modal spectrum of the stellar system. This is the same modal spectrum responsible for producing a response by a perturber of interest, such as a galaxy reacting to orbiting or passing companion. Therefore, a collisionless solution is physically impossible without irrevocably changing the dynamics of the system under study. In other words, one suppresses the global part of the relaxation at the risk of throwing out the physics responsible for the evolution one is studying. There is no other recourse but large numbers of particles.
For example, this work was motivated by n-body experiments with a thin
disk in a live halo. The observed fluctuations vertical in force at
the disk plane were larger than predicted for Poisson fluctuations for
n-body simulations of halos with 
particles using the SCF scheme
(e.g. Clutton-Brock 1972, 1973, Hernquist & Ostriker
1992). However, real galactic halos
are certainly not smooth and contain gas clouds, star clusters, dwarf
galaxies, stellar streams, and possibly as of yet undetected massive
objects such as 
black holes (e.g. Lacey & Ostriker
1985). All of these can contribute to correlated
fluctuations at large scales leading to warped disks and other
possibly observable distortions (see Weinberg 1997).
In this paper, I describe the expected amplitude of noise-generated fluctuations in a spherical equilibrium stellar system including self-gravity. The main result is the power spectrum of fluctuations generated by long-range correlations of particles moving their own gravitational field. This is computed using the polarization cloud method developed by Rostoker & Rosenbluth (1960) for plasma physics. The same approach can be used for any regular system (see Nelson & Tremaine 1997 for a general discussion). The power at very small scales will be Poisson, but at large scales, it will be modified by the global gravitational response. The basic results developed in §2 show that this might have been predicted a priori and is applied to a few standard spherical models in §3 and the predictions corroborated by n-body simulation. We conclude in §4.