Canonical ensemble ground state and correlation entropy of Bose–Einstein condensate

Constraint of a fixed total number of particles yields a correlation between the fluctuation of particles in different states in the canonical ensemble. Here we show that, below the temperature of Bose–Einstein condensation (BEC), the correlation part of the entropy of an ideal Bose gas is cancelled by the ground-state contribution. Thus, in the BEC region, the thermodynamic properties of the gas in the canonical ensemble can be described accurately in a simplified model which excludes the ground state and assumes no correlation between excited levels.

Realization of Bose-Einstein condensation (BEC) of weakly interacting gases has given rise to renewed interest in the canonical ensemble statistics. The trapped atomic clouds are small enough so that finite particle effects are potentially observable. Experimentally, the trapped atoms are isolated so that there is no exchange of particles. Magnetic or optical confinement suggests that the system is also thermally isolated and, hence, microcanonical description is needed. In this microcanonical ensemble the total particle number and the total energy are both exactly conserved. On the other hand, in experiments with two (or many) component BECs, Bose-Fermi mixtures, and additional gas components, e.g., for sympathetic cooling, there is an energy exchange between the components. As a result, each of the components can be described by the canonical ensemble that applies to systems with a conserved particle number while exchanging energy with a heat bath of a given temperature. Such a description is also appropriate for dilute 4 He in a porous medium [1] or for hadronic matter which under certain conditions resembles an ideal Bose gas with its associated Bose condensation [2].
Theoretical treatment of BEC of trapped atoms in the canonical ensemble is challenging since evaluation of the canonical partition function is impeded by the constraint that the total particle number N is fixed. Thermodynamics [3,4] and statistics of such systems [5,6] is an interesting and rich area for scientific analysis. Thermodynamics of BEC clouds was studied experimentally using phase-contrast imaging [7].
Among others, the issue of the fluctuations in the number of condensed atoms is of central importance. It has been shown that the difference between using grand canonical and canonical ensembles is substantial for the condensate statistics. For Bose gas, the fluctuations of the condensate fraction were calculated in a series of papers [5,[8][9][10][11][12][13][14][15][16][17] by a variety of approximate methods for various trapping potentials and in a different number of dimensions. Fairly accurate analytical expression for the condensate statistics in the canonical ensemble was obtained using a non-equilibrium master equation approach based on the quantum theory of laser [18,19]. The number statistics was measured for a degenerate Bose gas confined in an optical trap [20], for a mesoscopic ensemble of cold atoms in a dipole trap [21] and in optical lattices [22,23].
Because of the constraint on the total particle number, exact analytical results in the canonical ensemble are available only for the simplest systems. For an ensemble of bosons trapped in a one-dimensional harmonic potential, an analytical formula for the canonical partition function was found in [24][25][26]. However, there is no known simple analytic expression for the canonical partition function for higher dimensional traps.
Here we study an ideal Bose gas with a fixed number of particles and temperature in a harmonic trap and show that below the temperature of BEC transition the canonical ensemble correlations can be very accurately Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. taken into account analytically in a simple effective model. We consider a system of N particles with energies of the single-particle states e n , n = 0, 1, 2, K. The ground state corresponds to n = 0. In the canonical ensemble the total number of particles N is fixed and the probability to find the system in a microstate The average energy of the system is given by while the Gibbs entropy of the system is Equations (3) and (4) yield For the canonical ensemble, the contributions to the entropy from each state are not independent because of the constraint on the total number of particles. One can write the total entropy as where S ind is the entropy of the system obtained if state contributions are treated as independent and S corr is the remaining part called the correlation entropy. S ind is given by the formula and n p n, is the distribution function for state ν (probability of finding n particles occupying the single-particle state ν).
Here we consider an ideal Bose gas of N particles confined in an isotropic harmonic trapping potential. Energies of a particle in the trap are where n, l and m are non-negative integers. We assume that the ground-state energy is equal to zero. The thermodynamic limit formula for the critical temperature of BEC transition is where z ( ) x is the Riemann zeta function. For the canonical ensemble, one can calculate the partition function ( ) Z T N using the following recursion relation [12] If we know Z N (T), then entropy S can be obtained using equation (4). On the other hand, in the canonical ensemble the distribution function for the state ν is given by [12] = - Using this formula and equation (6), one can find S ind and then calculate = -S S S corr ind . In figure 1 we plot S S corr as a function of temperature obtained numerically for N=200 particles in the harmonic trap. The figure shows that S corr is negative and its absolute value becomes comparable with the total entropy S at low T. Thus, correlation between particles gives substantial contribution to the thermodynamic quantities at low temperatures and must be taken into account in the proper description of the system.
Here we show, however, that canonical ensemble correlations can be accurately accounted for by replacing the system of correlated particles with an effective model which has no correlations. One can decompose S ind into the ground and excited state contributions  potential μ. Recall that in the grand canonical ensemble the grand canonical partition function is given by Our findings indicate that below T c the entropy in the canonical ensemble can be very accurately calculated using the grand canonical formula with zero chemical potential in which contribution from the ground state is excluded. Namely, the canonical partition function is approximated as It is interesting to compare equation ( To illustrate the accuracy of equation (14), we calculate entropy for N=1000 particles in the harmonic trap numerically using equations (4), (7) for the canonical ensemble and equation (14). Figure 5 shows entropy S as a function of temperature T obtained in the two approaches. The two curves are essentially identical below T c . In figure 6 we plot the relative difference between the two curves for N=200 and 1000. The figure shows that for N=1000 the relative difference is smaller than -10 20 when < T T 0.77 c . Since other thermodynamic quantities can be expressed in terms of the entropy, they also can be very accurately calculated using the simplified effective model. For example, the mean energy of the system can be  obtained from equation (5), which then can be used to calculate Helmholtz free energy = -F E TSand heat capacity.
The result obtained above by numerical calculations can be also proved analytically. Indeed, canonical partition function (2) involves summation over particle configurations . Separating the sum over n 0 and introducing summation index =n N n 0 we obtain å å ..., 1 2 where Z is the grand canonical partition function with m = 0 and no ground-state contribution given by equation (10). Dividing both sides by Z yields where > P n N is the probability that in the grand canonical ensemble with m = 0 the number of particles in excited states n is greater then N. Below T c this probability is exponentially small. One can estimate > P n N by approximating the distribution function for the particle number in the excited states P n as Gaussian and > P n N is given by we obtain that > P n N is exponentially small, namely which coincides with the photon distribution in a thermal field. Equation (15) is not applicable for the ground state. p n,0 can be obtained as a mirror image of statistics of the number of particles in the excited states: Our findings can be understood from the following arguments. Below T c , the noncondensate particles can be treated as being in contact with a big reservoir of condensate particles. As a consequence, description of the noncondensate particles in the grand canonical picture is accurate below T c . In this picture there is no correlation between excited levels and fluctuations of particles at the excited states occur mainly via their exchange with the large condensate reservoir. However, such exchange yields correlations between the ground and excited states. Since fluctuations of particles in the ground state is the mirror image of the noncondensate particle fluctuations (which are already included in the grand canonical picture) the addition of the ground state to the system should not change the total entropy. Thus, the ground-state contribution must be cancelled by the term describing correlation between ground and excited states.
In summary, we show that in canonical ensemble below the temperature of BEC transition the correlation part of the entropy of an ideal Bose gas is equal to the ground-state contribution with the opposite sign. Thus, in the BEC region, the thermodynamic properties of the gas in the canonical ensemble can be accurately described in a simplified model which excludes the ground state and assumes no correlation between excited levels. Mathematically, the model corresponds to the grand canonical description with zero chemical potential and no ground state. Our findings provide new insight on the canonical ensemble correlations and yield substantial simplification of calculation of various thermodynamic quantities in the canonical ensemble.