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.

我们为研究共享守恒量的相互作用粒子系统引入了平均场框架。这项工作概括并统一了资产交换模型的现有领域，通常应用于社会经济系统，以及聚合碎片模型，通常用于对集群动态进行建模。初始模型仅包括两体碰撞，然后扩展到包括多体碰撞和自发碎裂。我们推导出稳态分布的自洽方程，可以使用种群动力学算法求解，以及矩时间演化的完整解，并通过数值模拟得到证实。该模型的通用性使其适用于许多问题，并允许研究表现出通常考虑的更复杂交互作用的系统。这项工作与新月形沙丘场的建模有关，其中，由于风的变化，床形和自发碎裂之间的相互作用被认为会导致大小选择。当代理人既可以合并资产又可以拆分为多个子公司时，我们的工作也可以应用于寻找财富分配。