MVEX01-17-16 Fusion-disaggregation processes

​​The origin of this model is in colloidal chemistry. Imagine molecules of a substance dispersed in another substance - substrate. Theses molecules form chains of various lengths, where the atoms are bonded together in a linear way. Each of these bonds can break over time leaving two pieces of the chain. This is disaggregation processes. Reciprocally, the ends of two nearby chains can fuse together resulting in a joint chain. We will assume that the bonding only possible between the ends of the chains and that a chain cannot create a loop. The process can be modelled by a finite state Markov chain, so whatever is the initial configuration of the sizes and the number of chains, there is always an equilibrium distribution to which the process converges. The aim of the project is to study this process and the limiting distribution. It should be possible to establish it via direct computation for small total number of atoms and we will also attempt to give some characterisation of the distribution in a general case.

The model described above is autonomous since the total number of atoms does not change. Another model allows for small chains, say, single atoms, to diffuse away from the system leading eventually to all the atoms disappearing. Can we say something about the average time to this empty state given the initial configuration of the chains?

Finally, we may consider an open system, where chains of molecules of different sizes drift into the system and small chains drift out of it. The main question here is stability: will the mass accumulate indefinitely or will an equilibrium distribution exists? And what is it in this latter case?

The project requires from students basic knowledge of probability, matrix calculus and some programming proficiency since a large part of it assumes computer simulations.  The students will study (or revise) basic techniques of discrete Markov chains, stochastic convergence and stability. The project is open-ended in the sense that it can be tailored to particular expertise of the participants and also could become a part of a larger Master project in the f