  The usual thermodynamic limit for systems of classical self-gravitating point
particles becomes well defined, as a {\it dynamical} problem, using a simple
physical prescription for the calculation of the force, equivalent to the
so-called ``Jeans' swindle''. The relation of the resulting intrinsically out
of equilibrium problem, of particles evolving from prescribed uniform initial
conditions in an infinite space, to the one studied in current cosmological
models (in an expanding universe) is explained. We then describe results of a
numerical study of the dynamical evolution of such a system, starting from a
simple class of infinite ``shuffled lattice'' initial conditions. The
clustering, which develops in time starting from scales around the grid scale,
is qualitatively very similar to that seen in cosmological simulations, which
begin from lattices with applied correlated displacements and incorporate an
expanding spatial background. From very soon after the formation of the first
non-linear structures, a spatio-temporal scaling relation describes well the
evolution of the two-point correlations. At larger times the dynamics of these
correlations converges to what is termed ``self-similar'' evolution in
cosmology, in which the time dependence in the scaling relation is specified
entirely by that of the linearized fluid theory. We show how this statistical
mechanical ``toy model'' can be useful in addressing various questions about
these systems which are relevant in cosmology. Some of these questions are
closely analagous to those currently studied in the literature on long range
interactions, notably the relation of the evolution of the particle system to
that in the Vlasov limit and the nature of approximately quasi-stationary
states.
