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.

翻译：我们引入了一个用于研究互动粒子系统共享节量的平均值框架。工作概括并整合了资产交换模型的现有领域,通常适用于社会经济系统,以及集成分化模型,通常用于模拟集群动态。初始模型仅包括两体碰撞,然后扩大到包括多体碰撞和自发碎裂。我们为稳定状态分布得出了自相一致的方程式,稳定状态分布可以使用人口动态算法解决,以及时间演变的完整解决方案,并辅以数字模拟。模型的笼统性使其适用于许多问题,并允许研究显示通常考虑的更为复杂的相互作用的系统。工作与铺垫和风的变化造成的自发碎裂相互作用被认为导致大小选择的条体元模型有关。我们的工作还可以用于寻找财富分配,因为代理人既可以将资产合并,也可以分成多个子公司。