Monday 08 Feb 2016: Dynamics seminar: Instantaneous gelation and explosive condensation in non-equilibrium cluster growth
Colm Connaughton - University of Warwick
The kinetics of various mechanisms of non-equilibrium cluster growth such as aggregation or exchange-driven growth are characterised by an interaction kernel, K(x,y), which specifies the average rate of interaction of particles having sizes x and y respectively. If the kernel increases quickly enough as a function of cluster size, then the second moment of the cluster size distribution can diverge in a finite time. This singularity, known as the gelation transition, is interpreted as signifying the formation of clusters of infinite size within a finite time. It has been known for some time that there exists a subclass of kernels for which the gelation transition occurs instantaneously. It is not the case that such behaviour is a mathematical pathology since there exist physically reasonable models which exhibit this behaviour such as coagulation driven by differential settling of liquid droplets in the Stokes regime. It was considered unlikely however that such behaviour could survive in spatially extended systems. A counter example was given by Waclaw and Evans in 2012 in which a total asymmetric version of a mass transport model on a one-dimensional lattice in the spirit of the zero-range process was shown to exhibit condensation of all of the particles onto a single site in a time which vanishes as the system size grows, a phenomenon known as "explosive condensation". In this talk I will discuss the relationship between instantaneous gelation and explosive condensation in the light of what is known about cluster growth models such as aggregation and exchange-driven growth. I will also show that a symmetric variant of Waclaw and Evans' model can exhibit the same behaviour provided that the rate of particle exchange is high enough. The fact that the model is spatially extended allows a second regime to exist for lower rates of particle exchange in which clusters grow algebraically by clustering, an regime which is absent at mean-field level.
Joint work with S. Grosskinsky and Y.-X. Chau