Exact ground state number fluctuations of trapped ideal and interacting fermions

We consider a small and fixed number of fermions (bosons) in a trap. The ground state of the system is defined at T=0. For a given excitation energy, there are several ways of exciting the particles from this ground state. We formulate a method for calculating the number fluctuation in the ground state using microcanonical counting, and implement it for small systems of noninteracting fermions as well as bosons in harmonic confinement. This exact calculation for fluctuation, ...

We consider a fixed number of noninteracting bosons in a harmonic trap. The determination of the exact microcanonical ground state number fluctuation is a difficult enterprise. There have been several theoretical attempts to solve this problem approximately, especially in 1D models where analytic results were found using some asymptotic formulae from number theory. Here, we obtain the exact number fluctuation curves, and show that these exact curves are substantially differen...

We question the validity of the grand canonical ensemble for the description of Bose-Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and the specific heat capacity can be well approximated with the help of the conventional grand canonical arguments, the calculation of the fluctuation of the number of particles contained in the condensate requires a microcanonical approach. Resorting to the theory of restricted ...

The statistical properties of non-interacting bosons and fermions confined in trapping potentials are most easily obtained when the system may exchange energy and particles with a large reservoir (grand-canonical ensemble). There are circumstances, however, where the system under consideration may be considered as being isolated (micro-canonical ensemble). This paper first reviews results relating to micro-canonical ensembles. Some of them were obtained a long time ago, parti...

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 micrscopic. 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 vanishingly small quantities are compared to explicit, fixed-number calculat...

We study spin-1/2 fermions, interacting via a two-body contact potential, in a one-dimensional harmonic trap. Applying exact diagonalization, we investigate their behavior at finite interaction strength, and discuss the role of the ground-state degeneracy which occurs for sufficiently strong repulsive interaction. Even low temperature or a completely depolarizing channel may then dramatically influence the system's behavior. We calculate level occupation numbers as signatures...

It is well-known that the number fluctuation in the grand canonical ensemble, which is directly proportional to the compressibility, diverges for an ideal bose gas as T -> 0. We show that this divergence is removed when the atoms interact in one dimension through an inverse square two-body interaction. In two dimensions, similar results are obtained using a self-consistent Thomas-Fermi (TF) model for a repulsive zero-range interaction. Both models may be mapped on to a system...

The mean ground state occupation number and condensate fluctuations of interacting and non-interacting Bose gases confined in a harmonic trap are considered by using a canonical ensemble approach. To obtain the mean ground state occupation number and the condensate fluctuations, an analytical description for the probability distribution function of the condensate is provided directly starting from the analysis of the partition function of the system. For the ideal Bose gas, t...

The thermodynamic properties of bosons moving in a harmonic trap in an arbitrary number of dimensions are investigated in the grand canonical, canonical and microcanonical ensembles by applying combinatorial techniques developed earlier in statistical nuclear fragmentation models. Thermodynamic functions such as the energy and specific heat are computed exactly in these ensembles. The occupation of the ground or condensed state is also obtained exactly, and signals clearly th...

One-particle properties of non-interacting Fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for large Fermion number N are given for the particle density distribution n_0(z,N). For large N and near the classical boundary at the Fermi energy the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of t...