For a non-self-interacting Bose gas with a fixed, large number of particles confined to a trap, as the ground-state occupation becomes macroscopic, the condensate number fluctuations remain microscopic. However, this is the only significant aspect in which the grand canonical description differs from canonical or microcanonical in the thermodynamic limit. General arguments and estimates including some of the vanishingly small quantities are compared to explicit, fixed-number calculations for
102--
106 particles. \textcopyright{} 1996 The American Physical Society.
For ensemble of bosons trapped in a one-dimensional harmonic potential well we have found an analytical formula for the canonical partition function and shown that, for 100 trapped atoms, the discrepancy between the grand canonical and the canonical predictions for the condensate fraction reaches 10% in the vicinity of the Bose-Einstein threshold. This discrepancy decreases only logarithmically as the number of atoms increases. Furthermore, we investigate numerically the case of a three-dimensional ``cigar-shape'' trap in the range of parameters corresponding to current Bose-Einstein condensation experiments.
For ideal Bose atoms in an isotropic harmonic trap, we consider thermodynamic variables obtained from microcanonical, canonical, and grand canonical ensembles, each with certain variables specified and other variables fluctuating. For the first two of these ensembles, we derive recursion relations that link partition functions for different dimensions. We discuss fluctuations in general, and obtain expressions for variances of the atom number N, the chemical potential \ensuremath{\mu}, and the temperature T for small, Gaussian fluctuations in the grand canonical ensemble. Then from our recursion relations and others given elsewhere, we obtain probability distributions for N, for ground-state occupation number
n0, for \ensuremath{\mu}, and for T. Below the critical temperature, the shape of the distributions for N,
n0, and \ensuremath{\mu} are definitely not Gaussian for the grand canonical ensemble. For given temperature and small N, we find that the chemical potential values pertaining to the three ensembles differ. We compare the specific-heat function
CN(T) for the three ensembles and propose to use the minimum of
dCN/dT to define the critical temperature to facilitate comparisons with similar configurations of interacting atoms.
We analyse occupation number fluctuations of an ideal Bose gas in a trap which is isolated from the environment with respect to particle exchange (canonical ensemble). We show that in contrast to the predictions of the grandcanonical ensemble, the counting statistics of particles in the trap ground state changes from monotonously decreasing above the condensation temperature to single-peaked below that temperature. For the exactly solvable case of a harmonic oscillator trapping potential in one spatial dimension we extract a Landau–Ginzburg functional which–despite the non-interacting nature of the system–displays the characteristic behaviour of a weakly interacting Bose gas. We also compare our findings with the usual treatment which is based on the grand-canonical ensemble. We show that for an ideal Bose gas neither the grand-canonical and canonical ensemble thermodynamically equivalent, nor the grand-canonical ensemble can be viewed as a small system in diffusive contact with a particle reservoir.
We discuss the exact particle number counting statistics of degenerate ideal Bose gases in the microcanonical, canonical, and grand-canonical ensemble, respectively, for various trapping potentials. We then invoke the Maxwell's Demon ensemble [Navez et el., Phys. Rev. Lett. (1997)] and show that for large total number of particles the root-mean-square fluctuation of the condensate occupation scales n0 / [T=Tc] r N s with scaling exponents r = 3=2, s = 1=2 for the3D harmonic oscillator trapping potential, and r = 1,s= 2=3 for the 3D box. We derive an explicit expression for r and s in terms of spatial dimension D and spectral index sigma of the single-particle energy spectrum. Our predictions also apply to systems where Bose-Einstein condensation does not occur. We point out that the condensate fluctuations in the microcanonical and canonical ensemble respect the principle of thermodynamic equivalence.
We develop a reliable procedure for calculating the microcanonical fluctuations of the ground state occupation number for harmonically trapped ideal Bose gases, and show that these fluctuations vanish uniformly when the temperature approaches zero. The key point is the precise determination of the number of microstates from the canonical partition sum, thus avoiding a failure of the usual saddle point method. We also demonstrate why the magnitude of the condensate fluctuations does not depend on the total particle number.