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数理解析研究所講究録

Boson gas mean field models in weak trapping potentials by means of random point fields (Applications of Renormalization Group Methods in Mathematical Sciences)

Hiroshi Tamura, Valentin Anatol'evich Zagrebnov

Published May 1, 2008• 0 citations
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Abstract

日 概要We study a mean fleld model of boson gases trapped in a harmonic potential.The behaviors of the position distribution in the weak potential limit are classified into two types.In the high temperature region and in the weak potential limit, the position distributions converge to that of the hee boson gas.In the low temperature region, the position distributions is not uniform (diverge) because of the Bose condensation.

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日 概要We study a mean fleld model of boson gases trapped in a harmonic potential.The behaviors of the position distribution in the weak potential limit are classified into two types.In the high temperature region and in the weak potential limit, the position distributions converge to that of the hee boson gas.In the low temperature region, the position distributions is not uniform (diverge) because of the Bose condensation.
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\kappaκ -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\kappaκ\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
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.
Abstract The critical properties of the phase transition from a normal gas to a Bose–Einstein condensate (superfluid) of a harmonically confined Bose gas are addressed with the knowledge of an equation of state of the underlying homogeneous Bose fluid. It is shown that while the presence of the confinement trap arrests the usual divergences of the isothermal compressibility and heat capacities, the critical behavior manifests itself now in the divergence of derivatives of the mentioned susceptibilities. This result is illustrated with a mean-field like model of an equation of state for the homogeneous particle density as a function of the chemical potential and temperature of the gas. The model assumes the form of an ideal Bose gas in the normal fluid while in the superfluid state a function is proposed such that, both, asymptotically reaches the Thomas–Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the global thermodynamics of the harmonically confined gas, from which we analyze its critical properties. We discuss how these properties can be experimentally tested.
For some time, the theoretical result for the transition temperature of a dilute three-dimensional Bose gas in an arbitrarily wide harmonic trap has been known to first order in the interaction strength. We extend that result to second order. The first-order result for a gas trapped in a harmonic potential can be computed in mean-field theory (in contrast to the first-order result for a uniform gas, which cannot). We show that, at second order, perturbation theory suffices for relating the transition temperature to the chemical potential at the transition, but the chemical potential is nonperturbative at the desired order. The necessary information about the chemical potential can be extracted, however, from recent lattice simulations of uniform Bose gases.
We use a diffusion Monte Carlo method to calculate the lowest-energy state of a uniform gas of bosons interacting through different model potentials, both strictly repulsive and with an attractive well. We explicitly verify that at low density the energy per particle follows a universal behavior fixed by the gas parameter na3.{\mathrm{na}}^{3}.na3. In the regime of densities typical for experiments in trapped Bose-condensed gases, the corrections to the mean-field energy greatly exceed the differences due to the details of the potential.