  N-body simulations of the hierarchical formation of cosmic structures suffer
from the problem that the first objects to form always contain just a few
particles. Although relaxation is not an issue for virialised objects
containing millions of particles, collisional processes will always dominate
within the first structures that collapse. First we quantify how the relaxation
varies with resolution, softening, and radius within isolated equilibrium and
non-equilibrium cuspy haloes. We then attempt to determine how this numerical
effect propagates through a merging hierarchy by measuring the local relaxation
rates of each particle throughout the hierarchical formation of a dark matter
halo. The central few percent of the final structures - a region which one
might naively think is well resolved at the final time since the haloes
contains about 10^6 particles - suffer from high degrees of relaxation. It is
not clear how to interpret the effects of the accumulated relaxation rate, but
we argue that it describes a region within which one should be careful about
trusting the numerical results. Substructure haloes are most affected by
relaxation since they contain few particles at a constant energy for the entire
simulation. We show that relaxation will flatten a cusp in just a few mean
relaxation times of a halo. We explore the effect of resolution on the degree
of relaxation and we find that increasing N slowly reduces the degree of
relaxation proportional to N^{-0.25} rather than proportional to N. Simulated
with the same relative mass resolution (i.e. equal numbers of particles)
cluster mass objects suffer significantly more relaxation than galaxy mass
objects since they form relatively late and therefore more of the particles
spend more time in small N haloes.
