This is a special topics course in probability theory, devoted to the mathematical study of polymer chains.
The focus is on understanding the properties of long polymer chains in a variety of different physical and chemical situations.
Polymer chains can be viewed as random processes interacting with themselves and with their environment. The simplest models are random walks and self-avoiding walks. The latter are used to account for the “excluded-volume effect” (i.e., for the fact that different monomers cannot occupy the same space), and have been the object of intense study since 50 years.
Polymer chains can exhibit interesting phase transitions. For instance, a polymer can change from an “expanded coil” to a “compact ball” when the temperature is decreased, due
to self-attraction. Similarly, a polymer can adsorb onto a surface when the temperature is decreased, due to an attractive interaction with the surface. Furthermore, the polymer can consist of different types of monomers arranged in a random order. Such “random copolymers” exhibit collapse and adsorption transitions as well, and there are challenging questions about how the randomness affects the nature of the phase transition. Random copolymers can also localise at an interface between two immiscible liquids, distributing their monomers between the two liquids so as to optimise their energy.
During the course, various stochastic models will be described, each dealing with a particular physical or chemical situation of interest.
The course is part of the national MasterMath program of the academic year 2011-2012.
F. den Hollander, Random Polymers, Lecture Notes in Mathematics 1974, Springer, Berlin, 2009, ISBN 978-3-642-00332-5
A basic knowledge of probability theory is required to be able to follow the main ideas and concepts presented during the course. Some more advanced tools will be needed as well (e.g. large deviation theory, graph theory), but these will all be explained along the way.