This is a short survey of the Boson Random Point Processes method and its application to the mean-field interacting boson gas trapped by a weak harmonic potential.
A model of the mean-field interacting boson gas trapped by a weak harmonic potential is considered by the boson random point field methods. We prove that in the weak harmonic trap limit there are two phases distinguished by the boson condensation and by a different behavior of the local particle density. For chemical potentials less than a certain critical value, the resulting random point field (RPF) coincides with the usual boson RPF, which corresponds to a noninteracting (ideal) boson gas. For the chemical potentials greater than the critical value, the boson RPF describes a divergent (local) density, which is due to localization of the macroscopic number of condensed particles. Notice that it is this kind of transition that observed in experiments producing the Bose–Einstein condensation in traps.
A model of a mean-field interacting boson gas trapped by a weak harmonic potential is considered by means of the theory of Random Point Field [RPF].In the previous work, the weak potential limit of the RPF which describes the position distribution of the constituent particles of the gas has been obtained.The limiting RPF concerns the distribution of the gas near the bottom of the harmonic potentials.In this note, we deal with the same model in the different scale to consider the large scale distribution of the system. §1. Introduction and Main ResultsWe consider the quantum statistical mechanical models of boson gas in \mathbb{R}^{d}(d> 2) equipped with a κ -parameterized family of one-particle Hamiltonians of harmonic oscillators:(which are self-adjoint operators in the Hilbert space \mathfrak{H}:=L^{2}() .In this model, the limit κ\rightarrow\infty (weak harmonic limit [WHL]) can be considered as a kind of thermodynamic limit.In the previous paper [15], the position distribution of the constituent bosons of the model and its WHL is studied by means of random point