  In N-body simulations of collisionless stellar systems, the forces are
softened to reduce the shot noise. Softening modifies gravity at r=|x-y|
smaller than softening length epsilon and the softened forces are increasingly
biased for ever larger epsilon. There is, thus, some optimum between reducing
the fluctuations and introducing a bias. Here, analytical relations are derived
for the amplitudes of the bias and the fluctuations in the limit of small
epsilon and large N. It is shown that the fluctuations of the force are
generated locally, in contrast to the variations of the potential, which
originate from noise in the whole system. Based on the asymptotic relations and
using numerical experiments, I study the dependence of the resulting force
error on N, epsilon, and on the functional form by which Newtonian gravity is
replaced. The Plummer softening, where each body is replaced by a Plummer
sphere of scale radius epsilon, yields significantly larger force errors than
do methods in which the bodies are replaced by density kernels of finite
extent. I also give special kernels, which reduce the errors even further.
These kernels largely compensate the errors made with too small inter-particle
forces at r<epsilon by exceeding Newtonian forces at r epsilon. Additionally,
the possibilities of locally adapting epsilon and of using unequal weights for
the bodies are investigated. These various techniques allow, without increasing
N, to reduce the rms force error by a factor 2 compared to Plummer softening
with constant epsilon. The results of this study are directly relevant to
N-body simulations using direct summation techniques or the tree code.
(abridged)
