The cornerstone of this work, which is partly motivated by the
characterization of the so-called eternal additive coalescents by Aldous
and Pitman, is an explicit expression for the general eternal solution
to Smoluchowski's coagulation equation with additive kernel. This
expression points at certain Lévy processes with no negative jumps
and more precisely at a stochastic model for aggregation based on such
processes, which has been recently considered by Bertoin and Miermont
and is known to bear close relations with the additive coalescence. As
an application, we show that the eternal solutions can be obtained from
some hydrodynamic limit of the stochastic model. We also present a
simple condition that ensures the existence of a smooth density for an
eternal solution.