AbstractPhase transitions play an important role in the field of statistical mechanics and such phenomena are manifest in everyday life. Examples include the evaporation of water, magnetisation of metals and the jamming of traffic. The purpose of this project is to examine the properties of phase transitions in the model of magnets known as the Ising model and to test a recent approach (extended scaling) to their analysis. The literature is abundant with many models which describe phase transitions; with the Ising model being the one which is most broadly studied (approximately 700-900 papers are published each year, which have some relationship to the Ising model). The first objective is to analyse the Ising model as an example of a mathematical model of phase transitions. In particular, the exact solution of the Ising model in one dimension is derived and the mean field approach is also considered. The scaling behaviour of these Ising models is examined in the context of the scaling relations which are also derived.
Since no exact solution is available, approximate methods must be used to investigate the phase transition of the Ising model in higher dimensions. In particular, one is interested in the scaling behaviour of various thermodynamic functions close to the phase transition. One such method is the high-temperature series expansion. A new method has recently been proposed to extract the scaling behaviour, which is claimed to be superior to the traditional approach. This so-called extended scaling approach has been tested only in two and three dimensions at temperatures above the phase transition.
Here, the new scheme is retested in two dimensions to gain experience in the technique. It is then applied and tested for the first time above the upper critical dimension. It is demonstrated to be successful there in the sense that it follows the critical expansion in the critical regime and follows the high temperature series expansion in the high temperature regime.
The application of this new technique in high dimensions led on to a jointly authored paper which has now been published (Appendix D) (on the DVD version only).
|Date of Award
|Ralph Kenna (Supervisor) & Robert Low (Supervisor)
- applied mathematics
- statistical mechanics
- phase transitions
- Ising model