Random fragmentation-coalescence processes out of equilibrium

Stochastic coalescence models describe how blocks of mass randomly join together over time according certain rules of random evolution. Conversely, stochastic fragmentation models describe how blocks of mass break apart over time, again according to certain rules of random evolution. Processes with one or antoher of theses actions, have been widely investigated. Coalescence has been an active field since the seminal work of Smoluchowski 100 years ago. Fragmentation is a more recently investigated phenomenon, with the the main foundational development starting from the work of Bertoin in the early 2000s.

Little has been done, howver, in the rigorous mathematical literature regarding the combination of both actions of fragmentation and coalescence. Despite this fact, there is a strong motivation for the treatement of such models in the scientific literature thanks to applications in physical chemistry and genealogy, and more recently in group dynamics in the social sciences and biology.The purpose of this project is to thus investigate new probabilistic techniques to characterise the dynamics of tractable families of stochastic fragmentation-coalescence processes.

One of the mathematical difficulties with such models is that they do not possess so-called reversibility properties. This means that when considering such processes time reversed, they do not exibit the mathematical convenience that would allow known analytical techniques to be used. For this reason, their analysis is generally difficult.

In this proposal we will look at some special classes of fragmentation-coalescence models that were only very recently introduced into the literature (by the PI and CI as well as others) and for which some degree of tractability has already been demonstrated. We will use a mixture of techniques to analyse their stationary and quasi-stationary behaviour, exposing currently unknown behaviours and laying out a deeper understanding of how such models can be treated in general.

Many real-world systems of interest exhibit both coalescence and fragmentation processes simultaneously. The list includes e.g. social group formation, modelling of terrorist cells, protein assembly, and biological growth models. Fragmentation dynamics can be quite straightforwardly introduced into Smoluchowski-type models, and may, or may not, prevent the onset of gelation (the suddent creation of a large, dominant cluster of mass). More importantly, introducing fragmentation can change the nature of the gelation phase transition from a non-equilibrium dynamical transition, to a (better understood but less interesting) equilibrium transition in the stationary probability law.

In this respect, the reserach we propose addresses a number of fundamental questions from the mathematical stand point. Hence we look the foundational basis and behaviour of abstract models with the expectation that the results will find their way into the broader scientific literature. In order to faciliatate that process, as part of our action points to develop pathways to impact, we propose to look at a potential scientific application in bubble reactor chambers that are current in chemical engineering research. Given the stimulus of the fragmentation-coalesence models from the scientific literature real world systems, we hope in the lonter term, beyond the scope of this grant, to stimulate further reserach in the application areas.