Statistical Mechanics
In the
summer of 2004 I taught a course Mathematical Physics II at the TU Berlin. the course will give an introduction to mathematical
statistical mechanics. The course continued as Mathematical Physics III in the fall
2004/2005 term. This course covered the statistical mechanics of disordered
systems. Disordered systems are a very active area of research in statistical
mechanics, both in theoretical and in mathematical physics. Physically they are
motivated by material, such as alloys, that have an irregular spatial
structure. E.g., in spin glasses, one has ferromagnetic atoms (such as iron)
embedded in a matrix of a conducting metal, (such as gold or copper). The
magnetic properties of such material are dominated by the interaction of the
magnetic moments of the ferromagnetic atoms. But the interaction is modulated
by the conductor and oscillates in sign with the distance between these atom. A reasonable way of modelling this is through random
interactions between Ising spins. The resulting model
turns out to be quite hard to analyse, mainly because of the competing
tendencies of the interactions. There has been a lot of progress in the
mathematical understanding of such systems over the last few years, and the
lecture will cover some of this.
A good part
of the material can be found in my lecture notes for the MaPhySto
course "STATISTICAL MECHANICS of
DISORDERED SYSTEMS" that was written in 2001.
These lecture notes have been expanded into a book that also contains
the material from MP2. The book appeared in 2006
at Cambridge University Press. This book should serve as a
concise introduction to statistical mechanics, and to
the field of disordered systems in particular. It contains a wealth of
references for further reading.
For the impatient, here is a short
introduction to just to the mean field theory of spin glasses intended to bring
the reader quickly towards the Parisi solution and
Guerra’s bounds.
A lot of interesting stuff is missing, of course.
A good source on material on mathematical statistical
mechanics are also the Proceedings of the 2005 Les Houches
Summer School on Mathematical Statistical Mechanics, that appeared this year at
Elsevier. They contain in depth lecture notes on many topics of current
interest.
