  We present a rigorous thermodynamic treatment of irreversible binary
aggregation. We construct the Smoluchowski ensemble as the set of discrete
finite distributions generated from the same initial state of all monomers upon
fixed number merging events and define a probability measure on this ensemble
such that the mean distribution in the mean-field approximation is governed by
the Smoluchowski equation. In the scaling limit this ensemble gives rise to a
set of relationships completely analogous to those of familiar statistical
thermodynamics. The central element of the thermodynamic treatment is the
selection functional, a functional of feasible distributions that connects the
probability of distribution to the specific details of the aggregation model.
We obtain scaling expressions for general kernels and closed-form results for
the special case of the constant, sum and product kernel. We study the
stability of the most probable distribution, provide criteria for the sol-gel
transition and obtain the distribution in the post-gel region by simple
thermodynamic arguments.
