Different flavours of the mean-field theory
Seminar Room 1, Newton Institute
Since the proposal for a remarkably simple theory of ferromagnetism made by Weiss in 1906, under the assumption that each molecule suffered an effective magnetic field (le champ intÚrieur, in Weiss' words) mimicking the average action of all other molecules, the notion of mean field has grown and acquired a life of its own. The most striking application to liquid crystal science of the mean-field formalism is perhaps the Maier-Saupe theory for the nematic phase. Many other models and approximations are comprised under the general heading of mean-field theory, though often one may hardly find any trace of an average, collective field there, its place being taken instead by a generalized order field. Some theories in this ample catalogue are variational, while others are not. All feature a key self-consistency condition, which may involve a probability distribution density as well as an order field. The lecture will attempt to justify the key self-consistency equatio n in a rigorous way for the different flavours that theory has taken.