Mean-field kinetic equations for stochastic particle systems

The derivation of effective single-particle dynamics from interacting many-particle systems has a long history in the context of kinetic theory, and can pose challenging mathematical problems with the Boltzmann equation as a classical example. While effective dynamics are often used as a starting point to study stochastic particle systems in the theoretical physics literature, their rigorous derivation in this context has attracted attention only recently. We focus on the dynamics of cluster aggregation driven by the exchange of single particles, for which we derive effective rate equations in a mean-field scaling limit from an underlying particle system on a complete graph. We establish the propagation of chaos under generic growth conditions on particle jump rates, and the limit equation provides a Master equation for the single-site dynamics of the particle system, which is a non-linear birth-death chain. Conservation of mass in the particle system leads to conservation of the first moment for the limit dynamics, and to non-uniqueness of stationary measures. Our findings are consistent with recent results on exchange driven growth, and provide an interesting connection between well studied phenomena of gelation and condensation. This is joint work with Watthanan Jatuviriyapornchai.