  The exchange-driven growth model describes the mean field kinetics of a
population of composite particles (clusters) subject to pairwise exchange
interactions. Exchange in this context means that upon interaction of two
clusters, one loses a constituent unit (monomer) and the other gains this unit.
Two variants of the exchange-driven growth model appear in applications. They
differ in whether clusters of zero size are considered active or passive. In
the active case, clusters of size zero can acquire a monomer from clusters of
positive size. In the passive case they cannot, meaning that clusters reaching
size zero are effectively removed from the system. The large time behaviour is
very different for the two variants of the model. We first consider an isolated
system. In the passive case, the cluster size distribution tends towards a
self-similar evolution and the typical cluster size grows as a power of time.
In the active case, we identify a broad class of kernels for which the the
cluster size distribution tends to a non-trivial time-independent equilibrium
in which the typical cluster size is finite. We next consider a non-isolated
system in which monomers are input at a constant rate. In the passive case, the
cluster size distribution again attains a self-similar profile in which the
typical cluster size grows as a power of time. In the active case, a surprising
new behavior is found: the cluster size distribution asymptotes to the same
equilibrium profile found in the isolated case but with an amplitude that grows
linearly in time.
