Landau functions for non-interacting bosons

We discuss the statistics of Bose-Einstein condensation (BEC) in a canonical ensemble of N non-interacting bosons in terms of a Landau function L_N^{BEC} (q) defined by the logarithm of the probability distribution of the order parameter q for BEC. We also discuss the corresponding Landau function for spontaneous symmetry breaking (SSB), which for finite N should be distinguished from L_N^{BEC}. Only for intinite N BEC and SSB can be described by the same Landau function which depends on the dimensionality and on the form of the external potential in a surprisingly complex manner. For bosons confined by a three-dimensional harmonic trap the Landau function exhibits the usual behavior expected for continuous phase transitions.