Using the Euler-Maclaurin summation we calculate analytically the internal energy for non-interacting bosons confined within a harmonic oscillator potential. The specific heat shows a sharp λ-like peak indicating a condensation into the ground state at a well-defined transition temperature. Full agreement is obtained with direct numerical calculation of the same quantities. When the number of trapped particles is very large and at temperatures near and above the transition temperature, the results also agree with previous approximate calculations. At extremely low temperatures both the specific heat and the number of particles excited from the condensate are exponentially suppressed.Bose-Einstein condensation has now been experimentally demonstrated in magnetic traps of rubidium[1], lithium [2] and most recently sodium[3] gases. To a good approximation one describes the trapping potentials as 3-dimensional anisotropic oscillators which in the rubidium experiments[1] have frequencies typically around ω = 500 − 1000 Hz. For the sake of simplification we will here ignore the anisotropy and the weak interactions between the alkali gas atoms. The energy levels of one particle are then simply given as ε n =hωn where the quantum number n takes the values n = 0, 1, 2 . . . when we drop the zero-point energy. Since each energy level has a degeneracy g n = (n + 1)(n + 2)/2, the total number N of particles in such a trap at temperature T and chemical potential µ is given by the Bose-Einstein distribution aswhere β = 1/k B T . The ground state has quantum number n = 0 and thus contains N 0 = λ/(1 − λ) particles where λ = exp (βµ) is the fugacity. We then havewhere the number of particles in the higher states isg n λe −bn 1 − λe −bn1

We consider systems of a small number of interacting bosons confined to harmonic potentials in one and two dimensions. By exact numerical diagonalization of the many-body Hamiltonian we determine the low lying excitation energies and the ground state energy and density profile. We discuss the dependence of these quantities on both interaction strength g and particle number N . The ground state properties are compared to the predictions of the Gross-Pitaevskii equation, and the agreement is surprisingly good even for relatively low particle numbers. We also calculate the specific heat based on the obtained energy spectra.

The one-loop effective potential for nonrelativistic bosons with a delta function repulsive potential is calculated for a given chemical potential using functional methods. After renormalization and at zero temperature it reproduces the standard ground state energy and pressure as function of the particle density. At finite temperatures it is found necessary to include ring corrections to the one-loop result in order to satisfy the Goldstone theorem. It is natural to introduce an effective chemical potential directly related to the order parameter and which uniformly decreases with increasing temperatures. This is in contrast to the ordinary chemical potential which peaks at the critical temperature. The resulting thermodynamics in the condensed phase at very low temperatures is found to be the same as in the Bogoliubov approximation where the degrees of freedom are given by the Goldstone bosons. At higher temperatures the ring corrections dominate and result in a critical temperature unaffected by the interaction. Academic Press

We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.

A comparative analysis of transaction time-series is made, for light to moderately loaded hosts, motivated by the problem of anomaly detection in computers. Criteria for measuring the statistical state of hosts are examined. Applying a scaling transformation to the measured data, it is found that the distribution of fluctuations about the mean is closely approximated by a steady-state, maximum-entropy distribution, modulated by a periodic variation. The shape of the distribution, under these conditions, depends on the dimensionless ratio of the daily/weekly periodicity and the correlation length of the data. These values are persistent or even invariant. We investigate the limits of these conclusions, and how they might be applied in anomaly detection.