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Condensate fluctuations of a trapped, ideal Bose gas

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 calculations for 10^2 to 10^6 particles.


I. INTRODUCTION
Large fluctuations are a salient feature of the thermal behavior of systems of bosons.
For example, if n is the mean number of non-interacting particles occupying a particular one-particle state, then the mean-square occupation fluctuation is n(n + 1). This is easily derived in the grand canonical picture by considering diffusive equilibrium with a particle reservoir characterized by a chemical potential [1]. If, however, the system has a fixed total number of particles, N, confined in space by a trapping potential or container, then at low enough temperature T or fixed total energy E when a significant fraction of N are in the ground state, such large fluctuations are impossible. No matter how large N, this aspect of the grand canonical description cannot be even approximately true. This paper addresses what does happen for fixed large N as N → ∞.
A decades-old answer to this question is that any interaction between the particles would eliminate such large fluctuations, even in the presence of a chemical potential. With a weak inter-particle interaction and a chemical potential, fluctuations in the occupations of various states are only weakly correlated. Therefore, the fluctuation in the total number of particles not in the ground state is microscopic. Hence, a macroscopic condensate fluctuation would mean a macroscopic density fluctuation. Even if the particles interact weakly, this would mean a macroscopic energy fluctuation. The consequent macroscopic rise in free energy would suppress the fluctuation. (See Appendix B for a more formal sketch of this argument.) Thus, with interactions producing a finite compressibility, the equivalence of the three standard statistical ensembles is assured in the thermodynamic limit, and the computationally convenient chemical potential can still be used for isolated, large systems [2]. In the context of Bose liquids, the ideal gas is a theoretical curiosity. Large condensate fluctuation is only one of several features for which ignoring interactions gives qualitatively incorrect results [3].
This argument does not address the question of what does happen to condensate fluctuations of the ideal Bose gas. Furthermore, this is not a totally idle or purely theoretical question. In current experimental work on the trapping and cooling of bosonic atoms, there is typically no diffusive particle or thermal energy reservoir [4][5][6]. While the atoms most certainly interact, N = ∞. Hence, one can ask about the system as a whole rather than only describing densities (intensive quantities), which are really just sub-volumes in diffusive and thermal equilibrium with their (much larger) surroundings. For sub-volumes of an infinite system, µ and T give an appropriate description. However, for a finite, isolated system taken as a whole, which has a greater impact on the condensate fluctuations, the particle interactions or the constraint of fixed total N? The answer depends on the density realized in the particular situation. A practical distinction of a gas from a liquid is that the density can be easily varied over many orders of magnitude. In the first successful experiments [4], there are noticeable effects of interparticle repulsion; and many of the more detailed observations currently underway require a mean field (albeit weak) description of the interparticle scattering length to reconcile theory with observations. Nevertheless, it is possible to imagine approaching Bose condensation with a box or trap so large and density so low that the effects of a given inter-atomic interaction, characterized by a fixed scattering length, are negligible, even for density fluctuations of order the equilibrium density. (An estimate of the requisite relation of the scattering length, trap parameters and density is given in Appendix B.) Even though the Bose-Einstein transition temperature decreases with decreasing density, the total energy shift due to a weak fixed-strength inter-particle interaction decreases faster. Also, the actual inter-atomic interactions may not serve to stabilize anything. Rather, the gaseous state may itself only be metastable [5]. In such situations, the equilibrium statistics of the ideal gas are certainly a better starting approximation than the equilibrium statistics of the interacting system.
After a summary of a variety of potentially confusing issues (sec. II), a thoroughly elementary analysis of the problem (sec. III) suggests that the condensate fractional fluctuations vanish with increasing N, but all other significant grand canonical predictions have vanishing corrections. This is also sufficient to establish the equivalence of using either fixed T or fixed total E to characterize the system for large N. The proposed picture provides an explicit prediction (sec. IV) for the condensate fluctuation as well as the values of observables, e.g. two-level correlations, that are identically zero with a chemical potential but are induced by fixing N. (With a natural normalization, such functions are vanishingly small as N → ∞.) The results of a numerical evaluation of the canonical partition function and related functions for N from 10 2 to 10 6 (sec. V) confirm these predictions. Some obvious conclusions are offered (sec. VI), while comments on details of the numerical work are left to Appendix A. Appendix B outlines the simple estimate of the condensate fluctuation damping due to repulsive interactions, which allows a comparison with the effect due to fixing N.

II. POTENTIAL ISSUES
It is only the non-interacting particles in the ground state of a trap or confining potential that do not satisfy the hypotheses of the standard demonstration [2] of the equivalence of the grand canonical and canonical ensembles in the thermodynamic limit. Hence, the questions raised here only arise if the ground state occupation is macroscopic. At ultra-low T when almost all particles are in the ground state, the condensate serves as a particle reservoir for all the excited states, and so some form of the grand canonical description for excited states T . And were this the case, the equivalence of fixing E and fixing T might be lost in the thermodynamic limit.
Chemical potential is not just a calculational convenience. There is really no practical alternative for analytic calculations because not much is known directly about the large but fixed N asymptotics of the canonical or microcanonical partition functions, even for systems as simple as the ideal Bose gas. If this analytic tool were lost, theory would be reduced almost entirely to numerical techniques.

III. FIXED-N STATISTICS
The resolution of these conundrums lies in the observation that the grand canonical excited state occupations in the thermodynamic limit are independent of not only the condensate fluctuations but the condensate occupation itself. Hence, if the behavior of the excited state occupancies can be reliably estimated using the concept of a chemical potential, one can deduce the behavior of the condensate from the constraint of fixed N. This argument is really just a minor extension of the traditional one used to compute the condensate fraction [7,1]. In particular, it goes as follows.
Let i label the one-particle (or trap) states and ε i be their energies. Take i = 0 to be the lowest energy level, and take ε i = 0. In the presence of a chemical potential µ, the mean occupation numbers N i for non-interacting bosons are With the chosen zero of energy, (defining the fugacity λ, to be used later). Once N 0 ≫ 1 (which may still be for N 0 ≪ N), The expression for the expected total number of particles with i > 0, N e , and how it depends on µ is determined by the density of states. For an isotropic harmonic oscillator potential in three dimensions with level spacing ǫ, as long as N e < N and T /ǫ ≫ 1 [8]. Under the latter condition, the asymptotic behavior of the sum over states is given by an integral. (ζ(3) ≈ 1.202 is the Riemann Zeta function.) Under these circumstances, the fixed-T corrections to eq. (3) are O(1/N 0 ). The root-meansquare fluctuation of any occupation number is precisely For the isotropic oscillator, this implies . The corrections to eq. (5) for µ not exactly zero are again The success of using a µ to characterize a system with a large but fixed total number of particles N relies on the fact that each individual energy level is a system in diffusive equilibrium with the much larger remainder of the total system. This remainder acts as the single level's particle reservoir. Once N 0 is not much less than N, the utility of µ is no longer clear. Certainly there exists no yet-much-larger particle reservoir for the ground state.
Referring back to eq. (1), once N 0 is large, the only role of the particular value of µ is to determine N 0 . The N i>0 are insensitive to µ or N 0 . So, if we consider each individual excited level with i > 0 as a system in contact with the reservoir of all the other levels, we need not know exactly what the chemical potential actually is, only that it is nearly zero.
In fact, there need not be any precise meaning to µ, only that it is nearly zero. It may be impossible to disentangle the effects of "µ = 0" from other 1/N consequences of fixing the total N. From this perspective, N 0 is determined not by a µ but by N and N e : However, this is precisely the same value of N 0 that is deduced from eq. (1) when N is interpreted as an expectation in the presence of an external µ.
At the level of occupation expectations, the assignments given by eq. (1) for i > 0 minimize the total free energy (energy minus T ×entropy) irrespective of the actual value of N 0 or N as long as N e is fixed. This is because adding or removing particles from the i = 0 condensate changes neither the energy nor the entropy of the entire system. Hence, for large N 0 , the occupation numbers for i > 0 are unchanged from their grand canonical values if, instead of being determined by a diffusive equilibrium, N is fixed at some value and N 0 is large. Once there is a condensate, the only thing that can change as particles are added at The total expected energy E at fixed T depends only on the i > 0 occupations. Thus canonical and grand canonical evaluations of the total energy must agree as N → ∞. For the isotropic harmonic trap Since it is a canonical ensemble identity that the root-mean-square total energy fluctuation satisfies the equivalence of the canonical and microcanonical ensembles is assured as long as (This is true for any trapping potential, not just the explicit example given.)

IV. FLUCTUATION ESTIMATES
From the discussion above, it is expected that all occupations approach their grand canonical values as N → ∞, even if either or both N and E are fixed. One can go further and estimate the leading behavior of various quantities that vanish in this limit. As examples I consider the condensate fluctuations and the occupation correlations between levels.
As long as N 0 ≪ N, the root-mean-square fluctuation in the condensate number, ∆N 0 , satisfies eq. (4). Once N 0 ∼ O(N), eq. (6) implies The cross over between these two behaviors is an example of the phenomena that make a direct analysis of the fixed-N partition function difficult. It is appropriate to introduce the "critical" temperature T c , given by the point at which N e reaches N or, rather, at which N 0 goes from macroscopic to microscopic. For the isotropic harmonic potential, eq.(3) implies As N increases, T c remains fixed in absolute, physical units only if the trap size is increased, e.g. ǫ decreased. The transition occurs when the central density in the trap reaches the infinite volume critical value [9]. In terms of the natural temperature variable for the study of Bose-Einstein condensation, T /T c , the transition between eq. (4) and eq. (9) takes place in a vanishingly small interval as N → ∞.
In the thermodynamic limit with N, N 0 , and N e all very large, eqs. (3,5,6,9,10) can be combined to give a simple estimate of the leading behavior: A set of quantities that are of interest in the calculation of the angular dependence of light scattering off cold, trapped atoms [10] are the two-level occupation expectations, n i n j . (I use the notation "n i " for the actual ith level occupation number in a particular configuration of the thermal ensemble.) In the grand canonical analysis of an ideal Bose gas, these are given precisely by N i N j . In particular, there is no correlation between the fluctuations in one level and another. However, with N fixed, this cannot be exactly true. A refinement of the argument of the previous section allows one to estimate the leading behavior of these correlations. As an example consider the two states with the largest fluctuations, i = 0 and j = 1, because their fixed-N induced correlation must, therefore, be the largest: At fixed N, if n 0 fluctuates down, say, then n e must fluctuate up by an equal amount.
The impact on the n i>0 can be estimated by computing the particular expected N i given that N e is larger than its original equilibrium value by the negative of the i = 0 fluctuation.
This implies (writing ∆n i for n i − n i ) The fugacity, λ, is defined by eq.(2). For the isotropic, harmonic trap in the thermodynamic limit, this can be evaluated to give (with the natural normalization factor N 0 N 1 )  It is the next correction, the difference between the two ensembles' predictions at a given N (as illustrated in fig. (2)) that reflects the residual difference in physics between the ensembles. This difference is particularly pronounced as N 0 makes the transition from microto macroscopic just below T c . There, the grand canonical -canonical discrepancy decreases only very slowly with N. The sign and shape of the difference is such that the canonical N 0 does not rise quite as sharply as the grand canonical, but the width of the relevant region of T /T c vanishes with increasing N. Above T c , the distinction between fixing N and fixing µ has rapidly vanishing consequences.
The dashed lines in fig. (1) are the results of a numerical evaluation of the canonical ∆N 0 /[N 0 (N 0 + 1)] 1/2 versus the same T /T c 's for N = 10 2 , 10 3 , and 10 4 . For T T c , this ratio approaches 1, in agreement with the grand canonical eq. (4). However, for T < T c , it goes to zero, more dramatically with increasing N. This same ∆N 0 data is plotted again on a log scale as the solid lines in fig. (3). The dotted lines are plots of eq. (11) for the same N's. As long as neither N 0 nor N e are too small, eq. (11) clearly captures the N and T dependence of ∆N 0 , and the agreement improves with increasing N. In particular, the fractional discrepancy between the canonical and eq. (11) values appears to go roughly like presumably those calculations could be extended to higher N with algorithmic improvements that avoided the simultaneous evaluation of numbers of vastly different magnitudes.

VI. DISCUSSION AND CONCLUSIONS
The general arguments presented here, while heuristic, have an internal consistency. For example, to compute ∆N e , which is used implicitly in eqs. (12,13), one assumes that the correlations between level occupations are negligible. One then deduces non-zero correlations that are induced by particle conservation. However, the induced correlations are, indeed, small enough to be ignored in the calculation of the leading behavior of ∆N e and of the correlations themselves. This is nowhere near to a "theory" of the large N asymptotics of the canonical ideal Bose gas. The leading behavior of some interesting observables were estimated and confirmed numerically. But in these cases, the leading behavior either was simply given by or could be deduced from the grand canonical ensemble. The next level of approximation, e.g. to account for fig. (2), would require a detailed analysis of the canonical or microcanonical partition function and may be very difficult to determine analytically.
Starting with the grand canonical description with µ and T as independent variables, one finds large fluctuations in N below T c . Hence, fixing N may have been expected to be of some consequence. However, the grand canonical total energy fluctuations are always small and vanish relative to the mean total energy in the thermodynamic limit. Nothing special happens in E at T c . So fixing E should have no dramatic consequences. Overall, the switch from T to E should be of even less consequence than the switch from µ to N. A direct numerical evaluation of the microcanonical partition function would be considerably more difficult.
However, from a practical standpoint, the modest results here are useful. The largest consequence of going from a chemical potential to fixed N for an ideal Bose gas is that An overall factor in Z has no effect on physical observables. This can be used to consider-