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.
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?
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
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 future.
Gruppstorlek 3-4 studenter
Målgrupp GU- och Chalmersstudenter. För GU-studenter räknas projektet som ett projekt i Matematisk statistik (MSG900/MSG910).
Projektspecifika förkunskapskrav Basic knowledge of probability and stochastic process and some programming proficiency.
respektive kursplan för allmänna förkunskapskrav. Utöver de allmänna
förkunskapskraven i MVEX01 ska Chalmersstudenter ha avklarat kurser i
en- och flervariabelanalys, linjär algebra och matematisk statistik.
Handledare Sergei Zuyev, email@example.com
Examinator Maria Roginskaya, Ulla Dinger
Institution Matematiska Vetenskaper