Mean field theory, the Ginzburg criterion, and marginal dimensionality of phase transitions

By applying a real space version of the Ginzburg criterion, the role of fluctuations and thence the self-consistency of mean field theory are assessed in a simple fashion for a variety of phase transitions. It is shown that in using this approach the concept of ’’marginal dimensionality’’ emerges in a natural way. For example, it is shown that for many homogeneous structural transformations the marginal dimensionality is two, so that mean field theory will be valid for real three-dimensional systems. It is suggested that this simple self-consistent approach to Landau theory should be incorporated in the teaching of elementary phase transition phenomena.