Particle number dependence in the non-linear evolution of N-body
  self-gravitating systems
  Simulations of purely self-gravitating N-body systems are often used in
astrophysics and cosmology to study the collisionless limit of such systems.
Their results for macroscopic quantities should then converge well for
sufficiently large N. Using a study of the evolution from a simple space of
spherical initial conditions - including a region characterised by so-called
"radial orbit instability" - we illustrate that the values of N at which such
convergence is obtained can vary enormously. In the family of initial
conditions we study, good convergence can be obtained up to a few dynamical
times with N $ \sim 10^3$ - just large enough to suppress two body relaxation -
for certain initial conditions, while in other cases such convergence is not
attained at this time even in our largest simulations with N $\sim 10^5$. The
qualitative difference is due to the stability properties of fluctuations
introduced by the N-body discretisation, of which the initial amplitude depends
on N. We discuss briefly why the crucial role which such fluctuations can
potentially play in the evolution of the N-body system could, in particular,
constitute a serious problem in cosmological simulations of dark matter.
