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Condensation is a special class of phase transition which has been observed throughout the natural and social sciences. The understanding of dynamics towards condensation on a mathematically rigorous level is currently a major research topic. Starting the system from homogeneous initial conditions, the time evolution of the condensed phase often exhibits an interesting coarsening phenomenon of mass transport between cluster sites. In this thesis, we study the coarsening dynamics in several condensing stochastic particle systems. First, we consider the single site dynamics in general stochastic particle systems of misanthrope type with bounded rates on a complete graph. In the limit of diverging system size, we establish convergence to a Markovian non-linear birth death chain, described by a mean-field equation also known from exchange-driven growth processes. Conservation of mass in the particle system leads to conservation of the first moment for the limiting dynamics, and to non-uniqueness of stationary measures. The proof is based on a coupling to branching processes via the graphical construction and establishing uniqueness of the solution for the limit dynamics. As particularly interesting examples we discuss the dynamics of two models that exhibit a condensation transition and their connection to exchange-driven growth processes. The first model is the zero-range process with bounded jump rates. It is well known that zero-range processes with decreasing jump rates exhibit a condensation transition under certain conditions. The mean-field limit of the single site dynamics leads to a non-linear birth death chain describing the coarsening behaviour. We introduce a size-biased version of the single site process, which provides an effective tool to analyse the dynamics of the condensed phase without finite size effects. The second model is the inclusion process, which has unbounded jump rates and also exhibits the condensation phenomenon. However, in this case, the mean-field equation is derived differently, and the single site process is in the form of a standard birth death chain. In addition to the site and size-biased processes, we derive some exact results on the system through duality. We compute the time dependent covariance using the self-duality of inclusion processes and a two-particle dual process. Our results are based on exact computations and are corroborated by detailed simulation data, which contribute to a rigorous understanding of the approach to stationarity in the thermodynamic limit of diverging system size and particle number. |
