  We present a perturbative treatment of the evolution under their mutual
self-gravity of particles displaced off an infinite perfect lattice, both for a
static space and for a homogeneously expanding space as in cosmological N-body
simulations. The treatment, analogous to that of perturbations to a crystal in
solid state physics, can be seen as a discrete (i.e. particle) generalization
of the perturbative solution in the Lagrangian formalism of a self-gravitating
fluid. Working to linear order, we show explicitly that this fluid evolution is
recovered in the limit that the initial perturbations are restricted to modes
of wavelength much larger than the lattice spacing. The full spectrum of
eigenvalues of the simple cubic lattice contains both oscillatory modes and
unstable modes which grow slightly faster than in the fluid limit. A detailed
comparison of our perturbative treatment, at linear order, with full numerical
simulations is presented, for two very different classes of initial
perturbation spectra. We find that the range of validity is similar to that of
the perturbative fluid approximation (i.e. up to close to ``shell-crossing''),
but that the accuracy in tracing the evolution is superior. The formalism
provides a powerful tool to systematically calculate discreteness effects at
early times in cosmological N-body simulations.
