  We introduce a mean-field framework for the study of systems of interacting
particles sharing a conserved quantity. The work generalises and unites the
existing fields of asset-exchange models, often applied to socio-economic
systems, and aggregation-fragmentation models, typically used in modelling the
dynamics of clusters. An initial model includes only two-body collisions, which
is then extended to include many-body collisions and spontaneous fragmentation.
We derive self-consistency equations for the steady-state distribution, which
can be solved using a population dynamics algorithm, as well as a full solution
for the time evolution of the moments, corroborated with numerical simulations.
The generality of the model makes it applicable to many problems and allows for
the study of systems exhibiting more complex interactions that those typically
considered. The work is relevant to the modelling of barchan dune fields in
which interactions between the bedforms and spontaneous fragmentation due to
changes in the wind are thought to lead to size-selection. Our work could also
be applied in finding wealth distributions when agents can both combine assets
as well as split into multiple subsidiaries.

Physics and Society;Geophysics;Applications