Finite-size effects on the Bose–Einstein condensation critical temperature in a harmonic trap

Abstract

We obtain second and higher order corrections to the shift of the Bose-Einstein critical temperature due to finite-size effects. The confinement is that of a harmonic trap with general anisotropy. Numerical work shows the high accuracy of our expressions. We draw attention to a subtlety involved in the consideration of experimental values of the critical temperature in connection with analytical expressions for the finite-size corrections.

Bose-Einstein statistics for a finite number of particles

“…The thermodynamical quantities that identify the BE condensation were calculated here. The fraction of particles in the ground state is calculated exactly, for arbitrary η, in terms of ξ(β, N ) in (26). Supplemented by the numerical inversion, graphs of this quantity for a gas trapped in a regular box (η = 1 /2) and in a harmonic potential (η = 2) are presented in Fig.…”

“…Figure1: Fraction of particles in the ground state(26) for η = 1 /2 and η = 2. On both cases it can be seen how the fraction of particle in the ground state, for finite N , approach, smoothly, the non-analytical curve for the thermodynamic limit.…”

Representations for the derivative at zero and finite parts of the Barnes zeta function

“…Thus, the total contribution of this term to the sum is d−r−s i=0 (−1) d+r+i d−r−s i u(p) = 0. Hence, terms of the first type totally cancel out in (13).…”

“…Only terms of the form u(p), with p being as above, contribute to the sum (13). We will consider one at a time the contribution of the respective four types of terms to this sum.…”

“…[7,8,9,10,11]). Physical applications that involve the finite parts at the poles include Bose-Einstein condensation [12,13] and functional determinants in quantum field theory [7].…”