Abstract
We study the problem of BoseEinstein condensation in the perfect Bose gas in the canonical
ensemble,
in anisotropically dilated rectangular parallelpipeds (Casimir
boxes). We prove that in the canonical ensemble
for these anisotropic boxes there is the same type of generalized BoseEinstein condensation
as in the grandcanonical ensemble for the equivalent geometry.
However the amount of condensate in the individual states is different in some cases and
so are the fluctuations.
Keywords: Generalized BoseEinstein Condensation, Canonical
Ensemble, Fluctuations
PACS:
05.30.Jp, 03.75.Fi, 67.40.w. AMS:82B10 , 82B23, 81V70
070504
The Canonical Perfect Bose Gas in Casimir Boxes
Joseph V. Pulé^{a}^{a}aResearch Associate, School of Theoretical Physics, Dublin Institute for Advanced Studies. Department of Mathematical Physics University College Dublin
Belfield, Dublin 4, Ireland Email: Joe.P
and
Valentin A. Zagrebnov Université de la Méditerranée and Centre de Physique Théorique CNRSLuminyCase 907 13288 Marseille, Cedex 09, France Email:
1 Introduction
Many calculations in the grandcanonical ensemble (GCE) show a dependence of BoseEinstein condensation (BEC) on the way the infinite volume limit is taken. For example, in [1] and [3] the authors study the the perfect boson gas (PBG) in the GCE in rectangular parallelepipeds whose edges go to infinity at different rates (Casimir boxes, see [7]). They showed that this anisotropic dilation can modify the standard groundstate BEC, converting it into a generalized BEC of type II or III. For a short history of the notion of generalized BEC we refer the reader to [1] and [3].
On the other hand, due to the lack of (strong) equivalence of ensembles, the PBG in the canonical ensemble (CE) and in the GCE gives different expectations and fluctuations for many observables. For example, it was shown in [5] that for the isotropic dilation of the canonical PBG the distribution of groundstate occupation number is different from the one in the GCE. The same is true for the fluctuations of the occupation numbers, which are shape dependent and are not normal or Gaussian. Therefore this lack of equivalence of ensembles does not allow us to deduce the same shape dependence for BEC in the CE as for its grandcanonical counterpart and so far the question of whether it is true in the CE has not been considered except in a special case [6].
The aim of the present paper is to fill this gap. We study the problem of BEC in the PBG in the CE, in anisotropically dilated rectangular boxes. We shall prove that in the CE for these anisotropic boxes there is the same type of generalized BEC as in the GCE for the equivalent geometry. However the amount of condensate in the individual states is different in some cases and so are the fluctuations.
We would like to note that there is a renewed interest in generalized BEC both from the theoretical [14], [12], [18] and experimental [16], [8] point of view. This due to recent experiments which produce “fragmentation” of BEC (see e.g. [9], [13]), that is, the single state condensation can be “smeared out” over two or more quantum states. We return to this point in Section 4.
The structure of the present paper is as follows. In the rest of this section we give the mathematical setting. In Section 2 we collect together the results about PBG in the GCE that we shall need. In Section 3 we study the PBG in the CE for the system of anisotropic parallelpipeds. We start by giving some results which are common to the three cases corresponding to the three characteristic ways of taking the thermodynamic limit. These are determined by how fast the longest edge grows: (a) faster than the square root of the volume, (b) like the square root of the volume and (c) slower than the square root of the volume. In the three subsections of Section 3 we study these cases separately. In Section 4 we discuss the results.
We finish this section by establishing the general setting and notation.
Let be a rectangular parallelepiped of volume V :
(1.1) 
where
(1.2) 
The space of oneparticle wavefunctions is and the oneparticle Hamiltonian is the selfadjoint extension of the operator determined by the Dirichlet boundary conditions on . We denote by the ordered eigenvalues of :
We also introduce the boson Fock space on defined by , where stands for the space of particle symmetric functions. Then denotes the particle free Hamiltonian determined by on , and the corresponding Hamiltonian in the Fock space.
Now the expectations for the PBG in the canonical ensemble at temperature and density are defined by the Gibbs state
(1.3) 
where
(1.4) 
is the particle canonical partition function. As usual we put . The grandcanonical Gibbs state is defined by
(1.5) 
where is the corresponding chemical potential. Here is the particle number operator, that is, where denotes the operator for the number of particles in the th oneparticle state. The grandcanonical partition function at chemical potential is
(1.6) 
Because of their commutative nature it is useful to think of and as random variables rather than operators.
Notice that the oneparticle Hamiltonian spectrum coincides with the set
(1.7) 
described by the multiindex . Then and the groundstate eigenvalue .
Let . For a given we define by
(1.8) 
Note that is a nondecreasing function on with for . is the distribution of the eigenvalues (integrated density of states) of the oneparticle Hamiltonian . One can prove in many ways, for example by using Lemma 3.1 or by taking the Laplace transform, that
(1.9) 
We shall show (see Lemma 3.1) that if for some . This bound and (1.9) imply that the critical density of the PBG:
(1.10) 
is finite for any nonzero temperature.
By (1.5) the mean occupation number of the PBG in the grandcanonical ensemble in the state is given by
(1.11) 
Let be the unique root of the equation
(1.12) 
for a given . Then a standard result [3] shows that the boundedness of the critical density (1.10) implies the existence of generalized BEC with condensate density, , given by:
(1.13) 
Following the van den BergLewisPulé classification
[1] and [3], it is useful to
identify three categories of generalized BEC:
I. The condensation is of type I when a finite number
of singleparticle states are macroscopically occupied.
II. It is of type II when an infinite number of states
are macroscopically occupied.
III. It is of type III when none of the states is
macroscopically occupied.
For a specific geometry we have more detailed information at our
disposal. In the next section we collect the results from
[1] that we shall need later about the GCE in the
case of the anisotropically dilated parallelepipeds
(1.1).
Remark 1.1 Though we have chosen here to work with
Dirichlet boundary conditions, the proofs in this paper can be
adapted without difficulty to periodic or Neumann boundary
conditions.
Remark 1.2 Note that according to the classification
presented above the condensate “fragmentation” is nothing but a
generalized BEC of type I or II.
2 Generalized BoseEinstein Condensation of the Perfect Bose Gas in the Grand Canonical Ensemble
Proposition 2.1
([1],Theorem 1)
Let . Then the behaviour of
is as follows:

For , where is the unique root of the equation
(2.14) 
For , and for ,
(2.15) (2.16) (2.17) where is the unique root of the equation
(2.18)
The next statement by the same authors shows that there are different types of generalized BEC corresponding to different asymptotics (2.15)(2.17).
Proposition 2.2
([1])
For there is no generalized BEC and therefore no BEC of any type.
For there is generalized BEC and all three types of BEC occur:

For only the groundstate is macroscopically occupied (BEC of type I):
(2.19) 
For there is macroscopic occupation of an infinite number of lowlying levels (BEC of type II):
(2.20) 
Finally, for no singleparticle state is macroscopically occupied (BEC of type III):
(2.21) (2.22)
We shall need an easy generalization of the foregoing proposition to obtain the distribution of the random variables through their Laplace transform.
Theorem 2.1
Let . Then:

For ,
(2.23) 
For ,

For ,
(2.25)
We shall require some properties of the Kac distribution , see e.g. [1, 3, 10, 19]. The Kac distribution relates the canonical (1.3) and grandcanonical (1.5) expectations in a finite volume:
(2.28) 
The limiting Kac distribution gives the decomposition of the limiting grandcanonical state into limiting canonical states . In the particular case of the PBG it is more convenient to define the Kac distribution in terms of the mean particle density, rather then the chemical potential. Therefore we define
(2.29) 
so that
(2.30) 
The next proposition proved in [1] gives the limiting Kac density for anisotropically dilated parallelepipeds:
Proposition 2.3
Let
(2.31) 
If , then the PBG limiting Kac distribution has the onepoint support:
(2.32) 
If , then:

For ,
(2.33) 
For ,

For ,
(2.35)
3 Generalized BoseEinstein Condensation and Fluctuations of the Perfect Bose Gas in the Canonical Ensemble
In this section we prove results for the CE analogous to those for the GCE. We are forced to use different methods for the three regimes, so we treat them in separate subsections. But first we give some results which will be useful in all three cases. The basic identity for the canonical expectations at density of the occupation numbers is (see [5] equation (10)):
(3.36) 
The canonical expectations are notoriously difficult to calculate and are only accessible
through the grandcanonical expectations. In the two cases and
we shall exploit the fact that the sum on the righthand side of equation
(3.36) is very similar to the grandcanonical partition function.
The next theorem shows that the canonical expectations are monotonic increasing in the
density. Note that this theorem holds for the PBG with any oneparticle spectrum.
Theorem 3.1
For fixed and fixed , the canonical expectations for the PBG, , are monotonic decreasing functions of the density for while the moments , are monotonic increasing functions of the density.
Proof: From (3.36) we get
(3.37) 
Since
by the inequalities (see [10]):
and by (3) we get the monotonicity:
(3.38) 
By differentiating (3) times with respect to at ,
(3.39) 
which is positive by the same argument.
Remark: In this paper whenever we take the limit
(3.40) 
we shall mean that we take the system with particles in a container of volume and then let , that is,
(3.41) 
The next theorem is valid for containers of any geometry and not just for rectangular boxes.
Theorem 3.2
For the generalized condensate in the CE at density is equal to , that is
(3.42) 
Proof: The statement is true for the imperfect (meanfield) Bose gas in the GCE, see [2]. Since the meanfield term in the CE is irrelevant, the theorem follows from monotonicity and the fact that the Kac density for the imperfect Bose gas has onepoint support.
In the next theorem we shall make certain assumptions that are clearly satisfied for the parallelpipeds we are considering. We believe that in fact they hold much more generally.
Theorem 3.3
Suppose that and are continuous in at and that . Then implies that .
Proof: Using the decomposition (2.30) and monotonicity we get for any :
Since and are continuous in , letting tend to zero, we get
and because does not vanish the result follows.
Remark: Note that this lemma implies that for , there is never BEC in the CE.
Before looking at the three cases , and we first obtain lower and upper bounds on the density of states.
Lemma 3.1
(3.43) 
for some .
Proof:
(3.44) 
that is is the number of points of inside the ellipsoid
(3.45) 
If we associate the point with the volume of the unit cube centered at
(3.46) 
and so
(3.47) 
Let , let and , , and . If the point in the first quadrant satisfies , then it satisfies . That is, each point inside the first quadrant of the ellipsoid lies in a unit cube with the corner (with , and ) inside the ellipsoid . Therefore
(3.48) 
yielding
(3.49) 
3.1 Case .
We study this case first because it is the simplest since the limiting Kac distribution is a delta measure concentrated at and we have strong equivalence of ensembles (see Proposition 2.3). We shall use this fact together with the monotonicity properties of Theorem 3.1 to show that in this case the limiting canonical and grandcanonical expectations are identical.
Lemma 3.2
For and the following inequalities hold
(3.50) 
Proof: We start with the first inequality. Using the decomposition (2.30) we get for any :
In the penultimate inequality equality we have used the monotonicity established in Theorem 3.1 and in the last one we have used (2.32) and (2.35). The last inequality in (3.2) is proved similarly:
Therefore
(3.53) 
The following theorem and corollary give the distribution and the mean of and therefore they give the fluctuations about the mean.
Theorem 3.4
If and then the limiting distribution in the canonical ensemble of has Laplace transform for given by
Proof: This follows from the preceding lemma and Theorem 2.1.
Corollary 3.1
If and then
Proof: Again we have to check that there exists such that for all
(3.56) 
Then the corollary follows from the preceding theorem. We have again for any :
if is large enough. This implies the existence of as above since
converges as .
Corollary 3.2
When , there is type III BEC in the canonical ensemble.
Proof: From the preceding corollary or from Theorem 3.3 we can deduce immediately that
(3.59) 
for any .
3.2 Case .
For this case BEC into the ground state is treated in [6]. Here we extend the result to higher levels.
Because for the spectral series (1.7) corresponding to has the smallest energy spacing , it plays a specific role in calculations of the limiting occupation densities. Let
(3.60) 
for .
In ([6]) the following result was proved:
Let . Then for
(3.61) 
Here we give an extension of (3.61) to other ’s. Note that by Theorem 3.3 and comparison with the GCE, in this case there can only be condensation in states corresponding to . The main tool in the technique developed in [5] and [6] is the following identity:
Let be (nonnormalized) measures whose distributions are the functions
(3.62) 
for and some . Then we can rewrite equation (3.36) as follows
(3.63)  