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We study the evolution under their self-gravity of infinite ``shuffledlattice'' particle distributions, focussing specifically on the comparison ofthis evolution with that of ``daughter'' particle distributions, defined by asimple coarse-graining procedure. We consider both the case that suchcoarse-grainings are performed (i) on the initial conditions, and (ii) at afinite time with a specific additional prescription. In numerical simulationswe observe that, to a first approximation, these coarse-grainings representwell the evolution of the two-point correlation properties over a significantrange of scales. We note, in particular, that the form of the two-pointcorrelation function in the original system, when it is evolving in theasymptotic ``self-similar'' regime, may be reproduced well in a daughtercoarse-grained system in which the dynamics are still dominated by two-body(nearest neighbor) interactions. Using analytical results on the early timeevolution of these systems, however, we show that small observed differencesbetween the evolved system and its coarse-grainings at the initial time will infact diverge as the ratio of the coarse-graining scale to the originalinter-particle distance increases. The second coarse-graining studied,performed at a finite time in a specified manner, circumvents this problem. Italso makes more physically transparent why gravitational dynamics from theseinitial conditions tends toward a ``self-similar'' evolution. We finallydiscuss the precise definition of a limit in which a continuum (specificallyVlasov-like) description of the observed linear and non-linear evolution shouldbe applicable.

Gravitational dynamics of an infinite shuffled lattice: Particle coarse-graining, nonlinear clustering, and the continuum limit