Interacting quantum gases in confined space: Two- and three-dimensional equations of state

Interacting quantum gases in confined space: Two- and three-dimensionalequations of stateWu-Sheng Dai and Mi Xie Citation: J. Math. Phys. 48, 123302 (2007); doi: 10.1063/1.2821248 View online: http://dx.doi.org/10.1063/1.2821248 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v48/i12 Published by the American Institute of Physics. Related ArticlesCritical rotational speeds for superfluids in homogeneous traps J. Math. Phys. 53, 095203 (2012) Soft matrix and fixed point of Lennard-Jones potentials for different hard-clusters in size at glass transition AIP Advances 2, 022108 (2012) Rank reduction for the local consistency problem J. Math. Phys. 53, 022202 (2012) The propagator of the attractive delta-Bose gas in one dimension J. Math. Phys. 52, 122106 (2011) Symmetry principles in quantum systems theory J. Math. Phys. 52, 113510 (2011) Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

Downloaded 25 May 2012 to 140.254.87.101. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

http://jmp.aip.org/?ver=pdfcov

http://aipadvances.aip.org/resource/1/aaidbi/v2/i1?&section=special-topic-physics-of-cancer&page=1

http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Wu-Sheng Dai&possible1zone=author&alias=&displayid=AIP&ver=pdfcov

http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Mi Xie&possible1zone=author&alias=&displayid=AIP&ver=pdfcov


http://link.aip.org/link/doi/10.1063/1.2821248?ver=pdfcov

http://jmp.aip.org/resource/1/JMAPAQ/v48/i12?ver=pdfcov

http://www.aip.org/?ver=pdfcov







http://jmp.aip.org/about/about_the_journal?ver=pdfcov

http://jmp.aip.org/features/most_downloaded?ver=pdfcov

http://jmp.aip.org/authors?ver=pdfcov

Interacting quantum gases in confined space: Two- andthree-dimensional equations of state

Wu-Sheng Daia� and Mi Xieb�

School of Science, Tianjin University, Tianjin 300072, People’s Republic of China andLiuHui Center for Applied Mathematics, Nankai University and Tianjin University,Tianjin 300072, People’s Republic of China

�Received 26 June 2007; accepted 13 November 2007; published online 18 December 2007�

In this paper, we calculate the equations of state and the thermodynamic quantitiesfor two- and three-dimensional hard-sphere Bose and Fermi gases in finite-sizecontainers. The approach we used to deal with interacting gases is to convert theeffect of interparticle hard-sphere interaction to a kind of boundary effect, and thenthe problem of a confined hard-sphere quantum gas is converted to the problem ofa confined ideal quantum gas with a complex boundary. For this purpose, we firstdevelop an approach for calculating the boundary effect on d-dimensional idealquantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are crossterms involving both the influences of the boundary and of the interparticle inter-action. We compare the effect of the boundary and the effect of the interparticleinteraction. Our result shows that, at low temperatures and high densities, the ratiosof the effect of the boundary to the effect of the interparticle interaction in twodimensions are essentially different to those in three dimensions: in two dimen-sions, the ratios for Bose systems and for Fermi systems are the same and areindependent of temperatures, while in three dimensions, the ratio for Bose systemsdepends on temperatures, but the ratio for Fermi systems is independent of tem-peratures. Moreover, for three-dimensional Fermi cases, compared with the contri-butions from the boundary, the contributions from the interparticle interaction toentropies and specific heats are negligible. © 2007 American Institute of Physics.�DOI: 10.1063/1.2821248�

I. INTRODUCTION

With the decreasing size of physical systems, size, surface, and topology effects are believedto become more and more pronounced.1 Many authors discuss the boundary effects on idealclassical gases2 and ideal quantum gases.3–5 The ideal gas models, however, often suffer fromlimitations when applied to practical systems due to the existence of the interparticle interaction inrealistic gas systems. The theory of interacting gases is an active area of research at all times.6

Once an interacting gas is confined in finite-size space, e.g., an imperfect electron gas in ananosystem7 or Bose-Einstein condensation in a trap,8–10 one needs to take the boundary effectinto account. Nevertheless, the result of the standard statistical mechanics is obtained in thethermodynamic limit, V→�. In doing so, one has lost all information about the system geometry,and the result is valid only when the size of the system is very large in relation to the mean thermalwavelength of particles. In this paper, we will develop an approach to calculate the boundaryeffect on the interacting quantum gas and compare the effect of the boundary and the effect of theinterparticle interaction.

a�Electronic mail: [email protected]�Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS 48, 123302 �2007�

48, 123302-10022-2488/2007/48�12�/123302/20/$23.00 © 2007 American Institute of Physics


http://dx.doi.org/10.1063/1.2821248

http://dx.doi.org/10.1063/1.2821248

http://dx.doi.org/10.1063/1.2821248

In confined space, the boundary will influence the spectrum of the system. For an idealquantum gas, the spectrum of an individual particle would be determined by the precise geometryof the system, and there are some approaches for calculating the boundary effect by directlyperforming the summation over the spectrum.2–4 Nevertheless, a finite-size interacting gas systemis more complex: its spectrum is determined by both interparticle interactions and boundaries;especially, interparticle interactions make “the spectrum of an individual particle” physicallymeaningless. Therefore, it is difficult to calculate the boundary effect on an interacting gas by theapproach developed for ideal gas systems.

To calculate the boundary effect on interacting quantum gases, we first need to develop anapproach to calculate the equation of state and the various thermodynamic quantities for a con-fined interacting quantum gas. In this paper, we take the quantum hard-sphere gas as an examplefor investigating the general properties of interacting gases. Note that often a complete knowledgeof the detailed interaction potential is not necessary for a satisfactory description of the systembecause a particle that is spread out in space sees only an averaged effect of the potential11 or,from the viewpoint of quantum mechanics, for the gases we considered, the collision energy of thecollisions between molecules is low, so one can use the s-wave approximation which is shapeindependent. In Ref. 12, we have developed an approach for calculating the equations of state forBose and Fermi hard-sphere gases with spin j, both for two and three dimensions. Such anapproach is a kind of mean-field treatment. Using this approach, one can calculate the equation ofstate for a two-dimensional quantum hard-sphere gas, which is difficult to calculate by usualmethods. In this approach, the two interplayed effects in an interacting quantum gas, the effect ofthe statistics �the exchange effect� and the effect of the interparticle interaction, are, roughlyspeaking, treated separately by replacing the interacting quantum gas with an ideal quantum gasconfined in a container filled with small excluded spheres �cores� randomly distributed. In otherwords, the interaction among particles is regarded as a kind of boundary effect on an idealquantum gas: the exchange effect on the interacting gas is embodied in the exchange effect on theideal quantum gas and the effect of the interparticle interaction is embodied in the boundary effect.In the following, we will generalize this approach to calculate the boundary effect on a quantumhard-sphere gas. The problem of a confined quantum hard-sphere gas is converted to the problemof a confined ideal quantum gas with two kinds of boundary effects �Fig. 1�: one is the genuineboundary effect from the boundary of the container �the outer boundary� and the other is theeffective boundary effect from the interparticle interaction �the inner boundary�. In a word, insteadof calculating the boundary effect on an interacting quantum gas, we only need to calculate theboundary effect on an ideal quantum gas with complex boundaries, both outer and inner bound-aries.

In this paper, we calculate the equations of state and the thermodynamic quantities for two-and three-dimensional confined Bose and Fermi hard-sphere gases with spin j. Based on theseresults, we compare the contributions from the boundary of the system and from the interparticleinteraction. Our result shows that the effect of the boundary and the effect of the interparticleinteraction in two dimensions and in three dimensions are essentially different. More concretely,we compare the ratios of the effect of the boundary to the effect of the interparticle interaction intwo and three dimensions and in Bose and Fermi systems at low temperatures and high densities.In two dimensions, such ratios in Bose systems and in Fermi systems are the same, and the ratios

FIG. 1. Converting the problem of a hard-sphere gas in a finite-size container to the problem of an ideal gas with two kindsof boundaries.

123302-2 W.-S. Dai and M. Xie J. Math. Phys. 48, 123302 �2007�


are independent of the temperature of the system. Nevertheless, in three dimensions, such a ratioof a Bose system is in inverse proportion to the temperature, which is different from that of aFermi system: in a Fermi system, such a ratio is independent of the temperature. Moreover, forthree-dimensional Fermi gases, the contributions from the interparticle interaction for differentthermodynamic quantities are very different: the contributions of the interparticle interaction toentropies and specific heats are negligible, but the boundary effects are important. In higher-ordercontributions, there exist cross terms of boundary effects and interparticle interaction effects. Wecalculate some thermodynamic quantities for two- and three-dimensional interacting quantumgases at zero temperature.

In Sec. II, we first generalize the mathematical result given by Kac.13 Based on this result, weprovide an approach for calculating the boundary effects on ideal quantum gases with complexboundaries in various dimensions. In Secs. III and IV, we calculate the equations of state for two-and three-dimensional hard-sphere Bose and Fermi gases that are confined in finite-size contain-ers, respectively. In Sec. V, we discuss the low-temperature and high-density properties of two-and three-dimensional confined hard-sphere Fermi gases. In Sec. VI, we compare the contributionsfrom the boundary and from the interparticle interaction in two and three dimensions, respectively.The conclusions are summarized in Sec. VII. The thermodynamic quantities for two- and three-dimensional quantum hard-sphere gases are given in Appendix A and the zero-temperature limit ofthe thermodynamic quantities for two- and three-dimensional confined hard-sphere Fermi gasesare given in Appendix B.

II. THE EQUATION OF STATE FOR d-DIMENSIONAL IDEAL QUANTUM GASES WITHARBITRARY BOUNDARIES

The approach to calculate the equation of state for confined interacting quantum gases in thispaper is to convert the problem of an interacting quantum gas in a finite-size container to theproblem of a confined ideal quantum gas with two kinds of boundaries �Fig. 1�: one is theboundary of the system and the other is the effective boundary �randomly distributed cores�reflecting the effect of the interparticle interaction. Therefore, we need an approach to calculate theequation of state for a confined ideal quantum gas with both outer and inner boundaries. In thissection, we give a general discussion on confined ideal quantum gases with arbitrary boundaries ind dimensions and calculate the equation of state.

In two-dimensional cases, for the eigenvalue problem

12�2U + �U = 0 in � with U = 0 on � , �1�

where � is the region bounded by curve � and the number of the eigenstates whose eigenvaluesare not larger than 1 / t, Ns, is13

Ns � �n=1

�

e−�nt →�

2�t−

1

4

L�2�t

+�

6�t → 0� . �2�

Here, ��n is the spectrum of eigenvalues of the system, � is the area, and L is the perimeter of thecontainer. � is the Euler-Poincaré characteristic number, in two-dimensional cases �=1−r, wherer is the number of holes in the two-dimensional container. Equation �2� is an important result inmathematics. Weyl proves the first term,14 Pleijel proves the second term,15 and Kac proves thethird term, the topological term.13 Note that from the viewpoint of statistical mechanics, the firstterm Weyl proved is that when the system size is large enough in relation to the mean thermalwavelength of particles �the short-wavelength limit�, one can use the thermodynamic limit ap-proximation, V→�, which ignores the geometry properties of the system. However, Eq. �2� isonly a two-dimensional result and, in principle, is only valid for the convex polygon. For calcu-lating the d-dimensional, especially the three-dimensional, result, we need to first extend themathematical result �Eq. �2�� to more general cases.

For d-dimensional cases, the heat kernel, K�t�, can be expressed as16

123302-3 Interacting quantum gases in confined space J. Math. Phys. 48, 123302 �2007�


K�t� = �n=1

�

e−�nt →�d

o − �di

��2�t�d−

1

2

�d−1o + �d−1

i

��2�t�d−1+

1

4

�d−2o − �d−2

i

��2�t�d−2+ ¯

+ −1

2�d−��d−�

o − �− 1��d−�i

��2�t�d−�+ ¯ + −

1

2�d

��0o − �− 1�d�0

i � , �3�

where �ko and �k

i , k=0,1 ,2 , . . . ,d, are the d+1 valuations �invariant measures� of the containerwith �k

o for the outer boundary and �ki for the inner boundary. The relation between the number of

the eigenstates, Ns, and the heat kernel, K�t�, is given in Ref. 17. Especially, when the density ofthe state is a constant, e.g., in the case of two-dimensional free gases, we have Ns�1 / t�=K�t� in thelimit t→0, the case of Eq. �2�. The mathematical definition of valuations can be found in Ref. 18.A detailed discussion on this mathematical result �Eq. �3�� will be given elsewhere.16 The two- andthree-dimensional cases of Eq. �3� have been used to calculate the equations of state for two- andthree-dimensional hard-sphere gases in Ref. 12, and the three-dimensional result coincides withthat obtained by other methods, e.g., Lee and Yang’s binary collision method.19

Moreover, we can see that the result given by Eq. �3� is also the partition function for ad-dimensional ideal classical gas, which is defined by Z=�se

−s.Based on Eq. �3�, we can calculate the grand potential for a confined ideal quantum gas using

the approach in Ref. 4. The grand potential of an ideal quantum gas is

ln � = � �s

ln�1 � ze− s� . �4�

In this equation and the following, the upper sign stands for bosons and the lower sign forfermions. In principle, if the spectrum of the system is given, one can obtain the exact grandpotential ln � by performing the summation in Eq. �4� exactly. However, when the geometry ofthe system is irregular, first, it is always difficult to obtain the exact spectrum, and second, it isalmost impossible to perform the summation exactly. Equation �3�, in fact, gives an approximateresult of the summation of the spectrum.

Expanding the grand potential �Eq. �4�� gives

ln � = �s

�n

�±1�n+11

n�ze− s�n = �

n

�±1�n+1zn

n �s

e−n s� . �5�

Notice that the energy � s is the spectrum of the Schrödinger equation under the Dirichlet bound-ary condition. By the use of Eq. �3�, the summation over s can be performed,

ln � = �n

�±1�n+1zn

n� 1

nd/2�d

o − �di

�d −1

2

1

n�d−1�/2�d−1

o + �d−1i

�d−1 +1

4

1

n�d−2�/2�d−2

o − �d−2i

�d−2

+ ¯ + −1

2�d−� 1

n�d−��/2�d−�

o − �− 1��d−�i

�d−� + ¯ + −1

2�d

��0o − �− 1�d�0

i � , �6�

where �=h /�2�mkT is the mean thermal wavelength. The summation over n gives the Bose-Einstein and Fermi-Dirac integrals,

�n

�±1�n+1 zn

n� =1

��0

� x�−1

z−1ex � 1dx � h��z� = �g��z� the Bose-Einstein integral

f��z� the Fermi-Dirac integral. �7�

In principle, expanding Eq. �4� requires ze− s �1, and this requires z�1 due to 0� s��. Inother words, the summation over s only converges when z�1, but the range of value of z in theFermi case is 0�z��. The Fermi-Dirac integral is, in fact, an analytic continuation of thefunction given by the series with z�1. Though the sum is divergent in the range z�1, our resultis valid in 0�z��. This fact can be easily understood after an analytic continuation: the radiusof convergence of the series is 1 since all points in the range z�−1 are singular; however, the



function obtained by the analytic continuation is analytic in the range −1�z�� of the real axis.Therefore, the equation of state for a confined ideal quantum gas reads as

PV

kT= ln � = g��d

o − �di

�d h�d+2�/2�z� −1

2

�d−1o + �d−1

i

�d−1 h�d+1�/2�z� +1

4

�d−2o − �d−2

i

�d−2 hd/2�z�

+ ¯ + −1

2�d−��d−�

o − �− 1��d−�i

�d−� h�d−�+2�/2�z� + ¯ + −1

2�d

��0o − �− 1�d�0

i �h1�z� �8�

and

N = g��do − �d

i

�d hd/2�z� −1

2

�d−1o + �d−1

i

�d−1 h�d−1�/2�z� +1

4

�d−2o − �d−2

i

�d−2 h�d−2�/2�z�

+ ¯ + −1

2�d−��d−�

o − �− 1��d−�i

�d−� h�d−��/2�z� + ¯ + −1

2�d

��0o − �− 1�d�0

i �h0�z� ,

�9�

where we have added the factor g for denoting the number of internal degrees of freedom and forspins g=2s+1. The information of the geometry of the system is reflected in the valuations.Notice that the results in Ref. 12 and in Refs. 3 and 4 are some special cases of this result.

In this paper, we only pay attention to the cases of two- and three-dimensional quantum gases.However, our result is valid also for classical gases and for arbitrary dimensions.

III. TWO-DIMENSIONAL QUANTUM HARD-SPHERE GASES IN CONFINED SPACE

In this section, we discuss a quantum hard-sphere gas confined in a finite-size two-dimensional container. We calculate the equation of state. The thermodynamic properties will begiven in Appendix A.

The approach for dealing with hard-sphere gases is to convert a hard-sphere gas to an idealgas with multicore boundaries, which is developed in Ref. 12. The advantage of this approach isthat it allows us to treat the interparticle interaction and the boundary effect from the containersimultaneously.

The main idea of the approach for studying hard-sphere gases given by Ref. 12 is to replacethe effect of hard-sphere interactions with a boundary effect of randomly distributed core bound-aries. The collision between gas molecules is a low-energy scattering, so we can only take thes-wave contribution into account, i.e., we can replace the interparticle interactions between mov-ing gas molecules by fixed cores. Then, a confined hard-sphere quantum gas is converted to anideal quantum gas confined in a container illustrated in Fig. 1.

The boundary illustrated in Fig. 1 is complex. It contains outer and inner parts and its shapeis irregular. In the above section, we have calculated the grand potential for ideal Bose and Fermigases in such kinds of boundaries. By Eq. �8�, the grand potential for such an ideal quantum gassystem can be expressed as

ln � = g

S − N2�

2s + 1�a2

�2 h2�z� − g1

2

1

2L + N2

�

2s + 1

1

22�a

�h3/2�z� , �10a�

=gS

�2h2 −1

4g

L

�h3/2�z� − Ng

a

�

�

22

�

2s + 1h3/2�z� − Ng a

��2

�2�

2s + 1h2�z� , �10b�

where S is the area, L is the perimeter of the container, a is the diameter of the core, N is the totalnumber of particles in the system, g=2s+1 for spin s, and �=s+1 for bosons and �=s for



fermions. Two parts are in Eq. �10a�: the first part is proportional to the first valuation and thesecond is proportional to the second valuation. Each part consists of two terms: the second termsin these two parts reflect the interparticle interaction, the inner boundary; the first term in thesecond part describes the genuine boundary effect of the container, the outer boundary. In prin-ciple, one can obtain the expression of N from Eq. �10� by the relation N=z� ln � /�z. However,N appears also in the expression of the grand potential, ln �. This means that by N=z� ln � /�z,one obtains a differential equation in N. Such a differential equation for N is exactly solvable, butthe solution is in a complicated form. Thus, we turn to seek a series solution for N. Obviously, thesolution of N must contain the information of the geometry of the container, which is embeddedto some extent in the relation of the area S and the perimeter L and the interparticle interaction,which is reflected by a. Therefore, N can be in general expressed as

N = gS

�2��

A�z� +a

��

B�z� + a

��2

��

C�z�� + gL

��

A�� z� +a

��

B�� z� + a

��2

��

C�� z�� ,

�11�

where A�, B�, C�, A�� , B�� , and C�� are constants to be determined. In the following, we considerBose gases; the result of Fermi gases can be obtained by a similar procedure.

Substituting Eq. �11� into Eq. �10� gives

ln � = gS

�2�l=1

�zl

l2 −1

4g

L

��l=1

�zl

l3/2 − ga

�

�

22

�

2s + 1�l,��g

S

�2A�

1

l3/2 +a

�B�

1

l3/2� + gL

�A��

1

l3/2

+a

�B��

1

l3/2��zl+� − g a

��2

�2�

2s + 1�l,�

gS

�2A�

1

l2 + gL

�A��

1

l2�zl+�, �12�

where the Bose-Einstein integral g� �z� has been expanded as

g��z� = �l=1

�zl

l� . �13�

From Eq. �12�, by the use of N=z� ln � /�z, we have

N = gS

�2��=1

�z�

�−

a

��g

�

2s + 1 ��=2

�

�k=1

�−1

�Ak

�� − k�3/2z� − a

��2

�g�

2s + 1 ��=2

�

�k=1

�−1

�Bk

�� − k�3/2z�

− a

��2

2�g�

2s + 1 ��=2

�

�k=1

�−1

kAk1

�� − k�2z�� + gL

��−

1

4 ��=1

�z�

�1/2

− �ga

�

�

2s + 1 ��=2

�

�k=1

�−1

�Ak�

�� − k�3/2z� − �g a

��2 �

2s + 1 ��=2

�

�k=1

�−1

�Bk�

�� − k�3/2z�

− 2�g a

��2 �

2s + 1 ��=2

�

�k=1

�−1

kAk�1

�� − k�2z�� . �14�

Comparing with Eq. �11�, we can obtain the coefficients

A� =1

�, A�� = −

1

4

1

�1/2 ,

B� = − �g�

2s + 1 �k=1

�−1

�1

k

1

�� − k�3/2 ,



B�� =1

4�g

�

2s + 1 �k=1

�−1

�1

k1/21

�� − k�3/2 , �15�

C� = − �g�

2s + 1 �k=1

�−1

�Bk

�� − k�3/2 − 2�g�

2s + 1 �k=1

�−1

kAk1

�� − k�2 ,

C�� = − �g�

2s + 1 �k=1

�−1

�Bk�

�� − k�3/2 − 2�g�

2s + 1 �k=1

�−1

kAk�1

�� − k�2 .

Substituting these coefficients into Eq. �12� and performing the summations, we obtain the equa-tion of state,

PS

kT= ln � = g

S

�2h2 −1

4g

L

�h3/2 − �g

a

�g

�

2s + 1 S

�2h1h3/2 −1

4

L

�h1/2h3/2�

+ �2g a

��2g

�

2s + 1�2� S

�2 �h1/2h1h3/2 + h0h3/22 � −

1

4

L

��h1/2

2 h3/2 + h−1/2h3/22 ��

− 2�g a

��2

g�

2s + 1 S

�2h1h2 −1

4

L

�h1/2h2� �16�

and

N = gS

�2h1 −1

4g

L

�h1/2 − �g

a

�g

�

2s + 1� S

�2 �h0h3/2 + h1/2h1� −1

4

L

��h−1/2h3/2 + h1/2

2 ��+ �2g a

��2g

�

2s + 1�2� S

�2 �h−1/2h1h3/2 + 3h0h1/2h3/2 + h1/22 h1 + h−1h3/2

2 � −1

4

L

��4h−1/2h1/2h3/2

+ h1/23 + h−3/2h3/2

2 �� − 2�g a

��2

g�

2s + 1� S

�2 �h0h2 + h12� −

1

4

L

��h−1/2h2 + h1/2h1�� . �17�

For including Fermi cases, we have replaced the Bose-Einstein integral g� �z� by h� �z� in theabove result.

Moreover, there is also an alternative way to calculate the equation of state. The abovetreatment is to expand N as a series of the fugacity z; alternatively, N can also be expanded as aseries of h� �z�,

N = gS

�2��

A�h� + ��

A�� h�h� + ��

A�� h�h�h� + ¯ � +a

�� B�h� + ��

B�� h�h�

+ ��

B�� h�h�h� + ¯ � + a

��2�

�

C�h� + ��

C�� h�h� + ��

C�� h�h�h� + ¯ ��+ g

L

��

D�h� + ��

D�� h�h� + ��

D�� h�h�h� + ¯ +a

�� E�h� + ��

E�� h�h�

+ ��

E�� h�h�h� + ¯ � + a

��2�

�

F�h� + ��

F�� h�h� + ��

F�� h�h�h� + ¯ �� ,

�18�

where A�, B�, C�. . ., are constants to be determined. A similar procedure gives



A1 = 1, B0,3/2� = − g��

2s + 1, B1/2,1� = − g�

�

2s + 1,

C0,2� = C1,1� = − 2�g�

2s + 1, C−1,3/2,3/2� = C−1/2,1,3/2� = C1/2,1/2,1� = g�

�

2s + 1�2

,

C0,1/2,3/2� = 3g��

2s + 1�2

, D1/2 = −1

4, E−1/2,3/2� =

1

4g�

�

2s + 1, �19�

E1/2,1/2� =1

4g�

�

2s + 1, F−1/2,2� = F1/2,1� =

�

2g

�

2s + 1,

F−3/2,3/2,3/2� = F1/2,1/2,1/2� = −1

4g�

�

2s + 1�2

, F−1/2,1/2,3/2� = − g��

2s + 1�2

,

and the other coefficients equal 0. We again obtain the equation of state, �16� and �17�.Comparing the equation of state for a two-dimensional hard-sphere gas in confined space with

the equation of state for a two-dimensional hard-sphere gas in free space given in Ref. 12 and theequation of state for a two-dimensional ideal gas in confined space given in Refs. 3 and 4, we seethat the first-order contributions from the effect of the boundary and from the effect of theinterparticle interaction agree with these two effects calculated separately in Refs. 12, 3, and 4.Nevertheless, cross terms appear in the second-order contribution. This means that if one wants touse the result of the boundary effect on ideal gases or the result of hard-sphere gases in free spaceto approximate the result of hard-sphere gases in confined space, only the first-order approxima-tion is reliable. The reason of the existence of the cross terms is that the boundary will influencethe spectrum of the system.

IV. THREE-DIMENSIONAL QUANTUM HARD-SPHERE GASES IN CONFINED SPACE

In this section, we discuss a quantum hard-sphere gas confined in a finite-size three-dimensional container. We calculate the equation of state. The thermodynamic quantities will begiven in Appendix A.

Similar to the two-dimensional case, we can express the grand potential of a confined hard-sphere gas as the grand potential of a confined ideal gas with multicore boundaries,

ln � = gV

�3h5/2 −1

2g

1

2S + N2

�

2s + 1

1

24�a2

�2 h2 +1

4g

L − N2�

2s + 14a

�h3/2, �20�

where V is the volume of the system, S is the area of the surface, and L is the third valuation ofthe system, which is related to the linear size of the system. Three parts are in Eq. �20�: the firstpart only consists of one term, which is proportional to the volume V, since the contribution fromthe interparticle interaction, which is only of the order �a /��3, is ignored; the second and the thirdparts consist of both contributions from the boundary and from the interparticle interaction. Simi-lar to the two-dimensional case, for seeking a series solution, we express N as



N =V

�3g��

A�z� +a

��

B�z� + a

��2

��

C�z�� +S

�2g��

A�� z� +a

��

B�� z� + a

��2

��

C�� z��+

L

�g��

�

A�� z� +a

��

B�� z� + a

��2

��

C�� z�� , �21�

where A�, B�, C� , . . . are constants to be determined. Substituting Eq. �21� into Eq. �20� and usingthe relation N=z� ln � /�z and Eq. �13� give an expression of N. Comparing with Eq. �21� givesthe coefficients

A� =1

�3/2 , A�� = −1

4

1

�, A�� =

1

4

1

�1/2 ,

B� = − 2g�

2s + 1 �k=1

�−1

�Ak

�� − k�3/2 ,

B�� = − 2g�

2s + 1 �k=1

�−1

�Ak�

�� − k�3/2 ,

B�� = − 2g�

2s + 1 �k=1

�−1

�Ak�

�� − k�3/2 , �22�

C� = − 2�g�

2s + 1 �k=1

�−1

�Ak

�� − k�2 − 2g�

2s + 1 �k=1

�−1

�Bk

�� − k�3/2 ,

C�� = − 2�g�

2s + 1 �k=1

�−1

�Ak�

�� − k�2 − 2g�

2s + 1 �k=1

�−1

�Bk�

�� − k�3/2 ,

C�� = − 2�g�

2s + 1 �k=1

�−1

�Ak�

�� − k�2 − 2g�

2s + 1 �k=1

�−1

�Bk�

�� − k�3/2 .

Substituting these coefficients into Eqs. �20� and �21� gives the equation of state,

PV

kT= ln � = g

V

�3h5/2 −1

4g

S

�2h2 +1

4g

L

�h3/2 − 2g

a

�g

�

2s + 1 V

�3h3/22 −

1

4

S

�2h1h3/2 +1

4

L

�h1/2h3/2�

− 2�g a

��2

g�

2s + 1 V

�3h3/2h2 −1

4

S

�2h1h2 +1

4

L

�h1/2h2� + 4g a

��2g

�

2s + 1�2� V

�32h1/2h3/22

−1

4

S

�2 �h1/2h1h3/2 + h0h3/22 � +

1

4

L

��h1/2

2 h3/2 + h−1/2h3/22 �� 23�

and



N = gV

�3h3/2 −1

4g

S

�2h1 +1

4g

L

�h1/2 − 2g

a

�g

�

2s + 1� V

�32h3/2h1/2 −1

4

S

�2 �h0h3/2 + h1/2h1�

+1

4

L

��h−1/2h3/2 + h1/2

2 �� − 2�g a

��2

g�

2s + 1� V

�3 �h1/2h2 + h1h3/2� −1

4

S

�2 �h0h2 + h12�

+1

4

L

��h−1/2h2 + h1/2h1�� + 4g a

��2g

�

2s + 1�2� V

�32�h−1/2h3/22 + 2h1/2

2 h3/2� −1

4

S

�2 �h−1/2h1h3/2

+ 3h0h1/2h3/2 + h1/22 h1 + h−1h3/2

2 � +1

4

L

��4h−1/2h1/2h3/2 + h1/2

3 + h−3/2h3/22 �� , �24�

where g� �z� has been replaced by h� �z� for including the Fermi case.Like that in the two-dimensional case, we can also calculate the equation of state by expand-

ing N as a series of h�.

V. TWO- AND THREE-DIMENSIONAL CONFINED FERMI HARD-SPHERE GASES ATLOW TEMPERATURES AND HIGH DENSITIES

In this section, we discuss the low-temperature properties of confined Fermi hard-spheregases. In this paper, we will not consider the low-temperature properties of Bose gases since atlow temperatures, there may display Bose-Einstein condensation, which is a special domain. Thediscussion of the boundary influence on the Bose-Einstein condensation can be found in Ref. 8 andthe Bose-Einstein condensation of interacting Bose gases can be found in Ref. 20.

In the following, according to the expressions of the thermodynamic quantities listed inAppendix A, we compare the contributions from the boundary and from the interparticle interac-tion. The zero-temperature limit will be given in Appendix B.

A. Two-dimensional cases

At low temperatures and high densities, the fact that the fugacity z�1 allows us to take someapproximations.21 First, by the use of the results in Appendix A, from the asymptotic expansion ofEq. �A2�, we obtain the relation between the chemical potential and the temperature approxi-mately, taking �=s,

� = F0��1 + �1 + �2� − �2

24�1 −

�2

40�2� kT

F0 �2� . �25�

Here,

�1 =1

2

L�S� g

�N�26�

reflects the contribution from the container boundary,

�2 =10s

3a��N

gS�27�

reflects the contribution from the interparticle interaction, and

F0 =

�2

2m

4�

g

N

S

is the Fermi energy of the ideal gas in free space. From Eq. �25�, we can calculate the Fermienergy of a hard-sphere Fermi gas in confined space,



F = F0�1 + �1 + �2� . �28�

Comparing with the Fermi energy of the ideal Fermi gas in free space, F0 , we can see that there are

two additional terms: one comes from the effect of the container boundary and the other comesfrom the interparticle interaction.

From Eq. �25� and the result given in Appendix A, we can obtain the low-temperature andhigh-density expansions of the thermodynamic quantities.

Pressure,

P =1

2

N

S F

0�1 +2

3�1 +

6

5�2� +

�2

31 −

3

4�1 −

3

20�2� kT

F0 �2� . �29�

Internal energy,

U =1

2N F

0�1 +4

3�1 +

4

5�2� +

�2

31 −

1

2�1 −

3

10�2� kT

F0 �2� . �30�

Free energy,

F =1

2N F

0�1 +4

3�1 +

4

5�2� −

�2

31 −

1

2�1 −

3

10�2� kT

F0 �2� . �31�

Entropy,

S =�2

3Nk1 −

1

2�1 −

3

10�2� kT

F0 � . �32�

Specific heat,

CV =�2

3Nk1 −

1

2�1 −

3

10�2� kT

F0 � . �33�

In the above results, for brevity, only the leading contributions of the boundary and theinterparticle interaction are taken into account. The contribution from the container boundary isproportional to the factor L /�S which reflects the geometry property of the container to someextent; the contribution from the interparticle interaction is proportional to a which is the diameterof the hard core.

From these results, the ratios between the contributions from the boundary and from theinterparticle interaction to various thermodynamic quantities can be calculated directly,

�U2D = �F

2D = �S2D = �CV

2D = 3aP2D =

5

3� F

2D =5

3

�1

�2=

1

4�

g

s

L

aN. �34�

B. Three-dimensional cases

Similar to the two-dimensional case, we calculate the expressions of the thermodynamicquantities for three-dimensional confined hard-sphere Fermi gases at low temperatures and highdensities.

From Eq. �A9� in Appendix A, the chemical potential can be calculated approximately,

� = F0��1 + �1 + �2� −

�2

12 kT

F0 �2� , �35�

where



�1 =1

4

S

V�

6

gV

N�1/3

�36�

reflects the contribution from the container boundary and

�2 =8s

3a 6N

�gV�1/3

�37�

reflects the contribution from the interparticle interaction. Here,

F0 =

�2

2m6�2N

gV�2/3

is the Fermi energy of the ideal Fermi gas in free space. The Fermi energy of a confined hard-sphere Fermi gas is the chemical potential at zero temperature,

F = ��T→0 = F0�1 + �1 + �2� . �38�

Based on this result, we can calculate the asymptotic expansions of the thermodynamic quan-tities.

Pressure,

P =2

5

N

V F

0�1 +5

8�1 +

5

4�2� +

5�2

121 −

3

4�1� kT

F0 �2� . �39�

Internal energy,

U =3

5N F

0�1 +5

4�1 +

5

6�2� +

5�2

121 −

1

2�1� kT

F0 �2� . �40�

Free energy,

F =3

5N F

0�1 +5

4�1 +

5

6�2� −

5�2

121 −

1

2�1� kT

F0 �2� . �41�

Entropy,

S = Nk��2

21 −

1

2�1� kT

F0 �� . �42�

Specific heat,

CV = Nk��2

21 −

1

2�1� kT

F0 �� . �43�

Like that in two dimensions, the ratios between the contribution from the boundary and thecontribution from the interparticle interaction can be calculated directly,

�U3D = �F

3D = 3�P3D =

3

2� F

3D =3

2

�1

�2=

9

64s

S

aV�gV

6N�2/3

. �44�

These results show that, different from the two-dimensional case, the behaviors of different three-dimensional thermodynamic quantities are different: at low temperatures and high densities, forthe internal energy, the pressure, and the free energy, the effect of the boundary and the effect ofinterparticle interaction both play important roles, but for the entropy and the specific heat, onlythe effect of boundary is important, i.e., in the leading contribution there are no interparticleinteractions.



VI. BOUNDARY VERSUS INTERPARTICLE INTERACTION

The above results provide the equations of state for two- and three-dimensional quantumhard-sphere gases confined in finite-size containers. Based on these results, we can compare theeffect of the boundary and the effect of the interparticle interaction for confined hard-sphere gases.

A. Two-dimensional cases

From Eq. �16�, we can compare the contributions from the boundary and from the interparticleinteraction in two dimensions. The leading contributions of the boundary and of the interparticleinteraction to the grand potential of two-dimensional gases are

ln �boundary2D = −

1

4g

L

�h3/2, �45�

ln �interaction2D = − �g

a

�g

�

2s + 1

S

�2h1h3/2. �46�

The ratio is

�2D =ln �boundary

2D

ln �interaction2D =

1

4�

L�/Sa/�

1

�

1

h1�z�. �47�

For estimating the magnitude of �2D, we approximately use

N = gS

�2h1�z� ,

a result of an ideal gas, to determine the relation of the fugacity z and the temperature T. Note thatthe result obtained here is valid for both Bose and Fermi cases since in the derivation, we only usethe function h1�z�, which includes both the Bose-Einstein and the Fermi-Dirac integrals. Then, wehave

�2D =1

4�

g

�

L

aN. �48�

From this result, one can see that for a confined two-dimensional hard-sphere gas, whether thesystem is bosonic or fermionic, the ratio between the leading contribution of boundaries and theleading contribution of interparticle interactions is independent of the temperature T.

Besides the ratio �2D which is the ratio between the boundary and the interparticle interactioncontributions to the grand potential, the ratios of other thermodynamic quantities for a two-dimensional Fermi gas have been given in Sec. V A, and the result shows that in such a system theratios are of the same order no matter what thermodynamic quantities are.

Take 23Na as an example. The scattering length a=2.75 nm and the spin s=0. Supposing thenumber density n=1014 m−2 and the container is a square box of sides d, from Eq. �48�, we have

�2D =1.16 � 10−6

d. �49�

We can see that when the linear size of the system is about 1 �m, the contributions from theboundary and from the interparticle interaction are of the same order. When the size is smallerthan 1 �m, the boundary effect dominates. Note that this result is valid only when the temperatureis higher than the critical temperature.

In summary, in two dimensions, the ratios of the effect of boundary to the effect of theinterparticle interaction in Bose systems and in Fermi systems are the same and the ratios areindependent of the temperature.



B. Three-dimensional cases

For three-dimensional cases, from Eq. �23�, the leading contributions of the boundary and ofthe interparticle interaction are

ln �boundary3D = −

1

4g

S

�2h2, �50�

ln �interaction3D = − 2g

a

�g

�

2s + 1

V

�3h3/22 . �51�

The ratio is

�3D =ln �boundary

3D

ln �interaction3D =

1

8

S�/Va/�

1

�

h2

h3/22 . �52�

For three-dimensional Bose gases, there exists a critical temperature above which the gasesare nondegenerate. At temperatures above degeneracy, both Bose and Fermi gases behave almostas a classical gas. At low temperatures, for three-dimensional Fermi gases, we have calculatedsuch ratios for various thermodynamic quantities in Sec. V B, see Eq. �44�. For the grand poten-tial, the ratio is

�Fermi3D =

3

64s

S

aV�gV

6N�2/3

. �53�

The ratio is independent of the temperature.Different from two-dimensional cases, in three dimensions, the ratios between the effect of the

boundary and the effect of the interparticle interaction for Bose cases and for Fermi cases aredifferent. For Bose gases, the ratio depends on the temperature, but for Fermi gases, the ratio isindependent of the temperature. Moreover, for the Fermi case, even for different thermodynamicquantities, the ratios are different: for entropies and specific heats, such ratios are nearly infinite,i.e., the effect of interparticle interaction on entropies and specific heats is negligible at lowtemperatures and high densities, though such an effect is important for other thermodynamicquantities. Such a difference arises from the difference between the chemical potentials in two andthree dimensions. From Eqs. �25� and �35�, we can see that the second-order contribution, whichis proportional to �kT / F

0�2, in two dimensions includes both the boundary and the interparticleinteraction effects, but in three dimensions, the second-order term is independent of these twofactors. �Notice that the second-order contribution in the chemical potential will contribute to theleading order of the effects of the boundary and of the interparticle interaction in entropies andspecific heats.� In fact, even in the case of ideal Fermi gases in free space, there exists a similardifference between the chemical potentials in two and three dimensions.22 Concretely, the chemi-cal potential for a two-dimensional ideal Fermi gas is

�free2D = F

0�1 + O kT

F0 �4�� 54�

and for a three-dimensional ideal Fermi gas is

�free3D = F

0�1 −�2

12 kT

F0 �2

+ O kT

F0 �4�� . �55�

There are no contributions of order �kT / F�2 in two-dimensional cases.



VII. CONCLUSIONS AND DISCUSSIONS

In this paper, we calculate the equations of state and the thermodynamic quantities for two-and three-dimensional Bose and Fermi hard-sphere gases in finite-size containers. The approachwe used is to replace the interparticle hard-sphere interaction with a kind of boundary, and then theeffect of the hard-sphere interaction is converted to an effect of boundaries. Consequently, theproblem of a hard-sphere gas in a finite-size container is converted to the problem of an ideal gaswith two kinds of boundaries: one is the boundary of the container, which leads to boundaryeffects, and the other reflects, in fact, the interparticle interaction. For calculating the boundaryeffect on a hard-sphere gas, we first develop an approach for calculating the boundary effect onideal quantum gases in arbitrary dimensions. Such an approach can be, of course, used to calculatethe boundary effects on other gas systems. It should be emphasized that though Kac’s result �ourresult is a generalization of Kac’s result� is obtained under the assumption that the wavelength ofthe particle is short, the result is valid also for the case of low temperatures. Such validity has beendemonstrated by the validity of the thermodynamic limit approximation which assumes that thewavelength tends to infinity. The result based on the thermodynamic limit approximation is thezero-order approximation of Kac’s result. In other words, compared to the thermodynamic limitapproximation, Kac’s result provides a more complete and precise approximation.

Our results, the equations of state and the thermodynamic quantities, include both influencesof boundaries and interparticle interactions. There exist cross terms in the higher-order contribu-tions of boundaries and interparticle interactions in the equation of state and in the thermodynamicquantities. Based on these results, we compare the influences of boundaries and interparticleinteractions. The ratio of the boundary effect to the interparticle interaction effect shows that �1� intwo dimensions, the ratio is independent of the temperature, and the ratios for Bose systems andfor Fermi systems are the same. �2� In three dimensions, the properties of Bose systems and ofFermi systems are different: in Bose cases, the ratio depends on the temperature, but in Fermicases, the ratio is independent of the temperature. �3� Different from two-dimensional cases, forthree-dimensional Fermi systems, the ratios for different thermodynamic quantities are very dif-ferent: for entropies and specific heats, the ratios are nearly infinite. Such a difference also appearsin two- and three-dimensional ideal gases.

In this paper, we only pay attention to low-temperature and high-density cases since theboundary effect becomes important often at low temperatures. In principle, one can also considerthe boundary effect at high temperatures by virial expansion. Nevertheless, it should be empha-sized that the results for interacting quantum gases in free space given in Ref. 12 are moreeffective for low-temperature and high-density cases rather than high-temperature and low-densityones. Concretely, in this approach, the interparticle interactions between moving gas molecules arereplaced by fixed cores, i.e., only the s-wave contribution is taken into account. Therefore, such anapproach is valid only when the collision between molecules is low-energy scattering. At hightemperatures, the collision energy is relatively high and the p-wave modification is needed to beconsidered. The s-wave scattering cross section is �s=4�a2 �for simplicity, here, we do notconsider the indistinguishability of identical particles�, while the cross section including thep-wave modification is �p=4�a2�1− �1 /3��ka�2�, where k=2� /�dB and �dB is the de Brogliewavelength. Replacing the de Broglie wavelength by the mean thermal wavelength �, we have�p�4�a2�1− �4�2 /3��a /��2�. Clearly, if one wants to take the second-order contribution �a /��2

into account, he must consider the p-wave modification. Consequently, for the case of high tem-peratures and low densities, the virial expansion coefficients compare well with, for example,Huang-Yang-Luttinger’s result11 only to the order of a /�. However, at low temperatures and highdensities, the s-wave approximation is valid, and our result compares well almost to the order of�a /��2.12

The advantage of the approach of replacing the interparticle interactions by randomly distrib-uted fixed cores is that one can easily deal with the interacting quantum gases in any dimensionsand take the boundary modification into account directly.



ACKNOWLEDGMENTS

We are very indebted to Dr. G. Zeitrauman for his encouragement. This work is supported inpart by NSF of China, under Project Nos. 10605013 and 10675088.

APPENDIX A: THE THERMODYNAMIC QUANTITIES

1. The thermodynamic quantities for two-dimensional quantum hard-sphere gases

From the equation of state �Eqs. �16� and �17��, we can calculate the thermodynamic quanti-ties directly,

ln � = gS

�2h2 −1

4g

L

�h3/2 − ��g

S

�2

a

�h1h3/2 +

1

4��g

L

�

a

�h1/2h3/2 − 2��g

S

�2 a

��2

h1h2

+ �2�2gS

�2 a

��2

�h1/2h1h3/2 + h0h3/22 � , �A1�

N = gS

�2h1 −1

4g

L

�h1/2 − ��g

S

�2

a

��h0h3/2 + h1/2h1� +

1

4��g

L

�

a

��h−1/2h3/2 + h1/2

2 �

− 2��gS

�2 a

��2

�h0h2 + h12� + �2�2g

S

�2 a

��2

�h−1h3/22 + h−1/2h1h3/2 + 3h0h1/2h3/2 + h1/2

2 h1� .

�A2�

Internal energy,

U

kT= g

S

�2h2 −1

8g

L

�h3/2 −

3

2��g

S

�2

a

�h1h3/2 +

1

4��g

L

�

a

�h1/2h3/2 − 4��g

S

�2 a

��2

h1h2

+ 2�2�2gS

�2 a

��2

�h1/2h1h3/2 + h0h3/22 � . �A3�

Free energy,

F

kT= N ln z − �g

S

�2h2 −1

4g

L

�h3/2 − ��g

S

�2

a

�h1h3/2 +

1

4��g

L

�

a

�h1/2h3/2 − 2��g

S

�2 a

��2

h1h2

+ �2�2gS

�2 a

��2

�h1/2h1h3/2 + h0h3/22 �� . �A4�

Entropy,

S

k= 2g

S

�2h2 −3

8g

L

�h3/2 −

5

2��g

S

�2

a

�h1h3/2 +

1

2��g

L

�

a

�h1/2h3/2 − 6��g

S

�2 a

��2

h1h2

+ 3�2�2gS

�2 a

��2

�h1/2h1h3/2 + h0h3/22 � − N ln z . �A5�

Specific heat,



CV

k= 2g

S

�2h2 −3

16g

L

�h3/2 −

15

4��g

S

�2

a

�h1h3/2 +

1

2��g

L

�

a

�h1/2h3/2 − 12��g

S

�2 a

��2

h1h2

+ 6�2�2gS

�2 a

��2

�h1/2h1h3/2 + h0h3/22 � + �g

S

�2h1 −1

8g

L

�h1/2 −

3

2��g

S

�2

a

��h0h3/2 + h1/2h1�

+1

4��g

L

�

a

��h−1/2h3/2 + h1/2

2 � − 4��gS

�2 a

��2

�h0h2 + h12� + 2�2�2g

S

�2 a

��2

�h−1h3/22

+ h−1/2h1h3/2 + 3h0h1/2h3/2 + h1/22 h1��T

z

�z

�T, �A6�

where we have used the relation between the fugacity z and the temperature T,

T

z

�z

�T= − � S

�2h1 −1

8

L

�h1/2 −

3

2��

S

�2

a

��h0h3/2 + h1/2h1� +

1

4��

L

�

a

��h−1/2h3/2 + h1/2

2 �

− 4��S

�2 a

��2

�h0h2 + h12� + 2�2�2 S

�2 a

��2

�h−1h3/22 + h−1/2h1h3/2 + 3h0h1/2h3/2 + h1/2

2 h1�� S

�2h0 −1

4

L

�h−1/2 − ��

S

�2

a

��h−1h3/2 + h−1/2h1 + 2h0h1/2� +

1

4��

L

�

a

��h−3/2h3/2

+ 3h−1/2h1/2� − 2��S

�2 a

��2

�h−1h2 + 3h0h1� + �2�2 S

�2 a

��2

�h−2h3/22 + h−3/2h1h3/2

+ 5h−1h1/2h3/2 + 4h−1/2h0h3/2 + 3h−1/2h1/2h1 + 4h0h1/22 ��−1

, �A7�

obtained by Eq. �A2�.

2. The thermodynamic quantities for three-dimensional quantum hard-sphere gases

From the equation of state �Eqs. �23� and �24��, we can calculate the thermodynamic quanti-ties directly,

ln � = gV

�3h5/2 −1

4g

S

�2h2 − 2�gV

�3

a

�h3/2

2 +1

4g

L

�h3/2 +

1

2�g

S

�2

a

�h1h3/2 − 2��g

V

�3 a

��2

h3/2h2

+ 8�2gV

�3 a

��2

h1/2h3/22 , �A8�

N = gV

�3h3/2 −1

4g

S

�2h1 − 4�gV

�3

a

�h1/2h3/2 +

1

4g

L

�h1/2 +

1

2�g

S

�2

a

��h0h3/2 + h1/2h1�

− 2��gV

�3 a

��2

�h1/2h2 + h1h3/2� + 8�2gV

�3 a

��2

�h−1/2h3/22 + 2h1/2

2 h3/2� . �A9�

Internal energy,

U

kT=

3

2g

V

�3h5/2 −1

4g

S

�2h2 − 4�gV

�3

a

�h3/2

2 +1

8g

L

�h3/2 +

3

4�g

S

�2

a

�h1h3/2 − 5��g

V

�3 a

��2

h3/2h2

+ 20�2gV

�3 a

��2

h1/2h3/22 . �A10�

Free energy,



F

kT= N ln z − �g

V

�3h5/2 −1

4g

S

�2h2 − 2�gV

�3

a

�h3/2

2 +1

4g

L

�h3/2 +

1

2�g

S

�2

a

�h1h3/2

− 2��gV

�3 a

��2

h3/2h2 + 8�2gV

�3 a

��2

h1/2h3/22 � . �A11�

Entropy,

S

k=

5

2g

V

�3h5/2 −1

2g

S

�2h2 − 6�gV

�3

a

�h3/2

2 +3

8g

L

�h3/2 +

5

4�g

S

�2

a

�h1h3/2 − 7��g

V

�3 a

��2

h3/2h2

+ 28�2gV

�3 a

��2

h1/2h3/22 − N ln z . �A12�

Specific heat,

CV

k=

15

4g

V

�3h5/2 −1

2g

S

�2h2 − 12�gV

�3

a

�h3/2

2 +3

16g

L

�h3/2 +

15

8�g

S

�2

a

�h1h3/2

−35

2��g

V

�3 a

��2

h3/2h2 + 70�2gV

�3 a

��2

h1/2h3/22 + �3

2g

V

�3h3/2 −1

4g

S

�2h1 − 8�gV

�3

a

�h1/2h3/2

+1

8g

L

�h1/2 +

3

4�g

S

�2

a

��h0h3/2 + h1h1/2� − 5��g

V

�3 a

��2

�h1/2h2 + h3/2h1�

+ 20�2gV

�3 a

��2

�h−1/2h3/22 + 2h1/2h1/2h3/2��T

z

�z

�T, �A13�

where

T

z

�z

�T= −

3

2� V

�3h3/2 −1

6

S

�2h1 −16

3�

V

�3

a

�h1/2h3/2 +

1

12

L

�h1/2 +

1

2�

S

�2

a

��h0h3/2 + h1/2h1�

−10

3��

V

�3 a

��2

�h1/2h2 + h1h3/2� +40

3�2 V

�3 a

��2

�h−1/2h3/22 + 2h1/2

2 h3/2�� V

�3h1/2 −1

4

S

�2h0 − 4�V

�3

a

��h−1/2h3/2 + h1/2

2 � +1

4

L

�h−1/2

+1

2�

S

�2

a

��h−1h3/2 + h−1/2h1 + 2h0h1/2� − 2��

V

�3 a

��2

�h−1/2h2 + h0h3/2 + 2h1/2h1�

+ 8�2 V

�3 a

��2

�h−3/2h3/22 + 6h−1/2h1/2h3/2 + 2h1/2

3 ��−1

, �A14�

which is the relation between the fugacity z and the temperature T obtained by Eq. �A9�.The existence of cross terms is due to the complexity of the two interplayed effects: the effect

of the boundary and the effect of the interparticle interaction in a confined interacting gas.

APPENDIX B: THE ZERO-TEMPERATURE LIMIT FOR CONFINED HARD-SPHERE FERMIGASES

1. Two-dimensional cases

In the following, we calculate the boundary and the interparticle interaction contributions tohigher orders at zero temperature, which contain cross terms of these two effects.

The ground state pressure,



� P

2s + 1�

T→0=

1

8�

2m

�2 �2 −s

6�

�2m�3/2

�3 a�5/2 +1

2�5s2

9−

s

8� �2m�2

�4 a2�3

−L

S� 1

6�

�2m

��3/2 −

s

6�

2m

�2 a�2 +1

�2s2

9−

s

16� �2m�3/2

�3 a2�5/2� . �B1�

The particle number density at zero temperature,

� N

�2s + 1�S�

T→0=

1

4�

2m

�2 � −5s

12�

�2m�3/2

�3 a�3/2 +1

2�5s2

3−

3s

8� �2m�2

�4 a2�2

−L

S� 1

4�

�2m

�� −

s

3�

2m

�2 a� +5

� s2

9−

s

32� �2m�3/2

�3 a2�3/2� . �B2�

The internal energy at zero temperature,

� U

�2s + 1�S�

T→0=

1

8�

2m

�2 �2 −s

4�

�2m�3/2

�3 a�5/2 +1

�5s2

9−

s

8� �2m�2

�4 a2�3

−L

S� 1

12�

�2m

��3/2 −

s

6�

2m

�2 a�2 +1

� s2

3−

3s

32� �2m�3/2

�3 a2�5/2� . �B3�

At zero temperature, the chemical potential �= F is just the Fermi energy. The cross terms inthese results show the complexity of a confined interacting gas.

2. Three-dimensional cases

In the zero-temperature limit, by expanding the Fermi-Dirac integral, we obtain the thermo-dynamic quantities.

The ground state pressure,

� P

2s + 1�

T→0=

1

15�2

�2m�3/2

�3 �5/2 −2s

9�3

�2m�2

�4 a�3 +4s

�42s

9−

�2

96� �2m�5/2

�5 a2�7/2

−S

V� 1

32�

2m

�2 �2 −s

12�2

�2m�3/2

�3 a�5/2 + 5s2

18�3 −s

64�� 2m�2

�4 a2�3�+

L

V� 1

6�

�2m

��3/2 −

s

3�2

2m

�2 a�2 + 8s2

9�3 −s

16�� 2m�3/2

�3 a2�5/2� . �B4�

The particle number density at zero temperature,

� N

�2s + 1�V�

T→0=

1

6�2

�2m�3/2

�3 �3/2 −2s

3�3

�2m�2

�4 a�2 +14s

�4 2s

9−

�2

96� �2m�5/2

�5 a2�5/2

−S

V� 1

16�

2m

�2 � −5s

24�2

�2m�3/2

�3 a�3/2 + 5s2

6�3 −3s

64�� 2m�2

�4 a2�2�+

L

V� 1

4�

�2m

�� −

2s

3�2

2m

�2 a� + 20s2

9�3 −5s

32�� 2m�3/2

�3 a2�3/2� . �B5�

The internal energy at zero temperature,



� U

�2s + 1�V�

T→0=

1

10�2

�2m�3/2

�3 �5/2 −4s

9�3

�2m�2

�4 a�3 +10s

�4 2s

9−

�2

96� �2m�5/2

�5 a2�7/2

−S

V� 1

32�

2m

�2 �2 −s

8�2

�2m�3/2

�3 a�5/2 + 5s2

9�3 −s

32�� 2m�2

�4 a2�3�+

L

V� 1

12�

�2m�1/2

��3/2 −

s

3�2

2m

�2 a�2 + 4s2

3�3 −3s

32�� 2m�3/2

�3 a2�5/2� .

�B6�

The free-space result can be obtained by setting S=0 and L=0, which compares well with Lee andYang’s result based on the binary collision method:19 the difference only appears in the secondterm of the second-order �a /��2 contribution. In the above result, the coefficient of the second-order term includes the factor �2s /9−�2 /96�, but in Lee and Yang’s result, such factor is �2s /9− �11−2 ln 2� /105�. The difference between �2 /96�0.10 and �11−2 ln 2� /105�0.09 is about0.01.

1 H. Potempa and L. Schweitzer, J. Phys.: Condens. Matter 10, L431 �1998�; D. Braun, G. Montambaux, and M. Pascaud,Phys. Rev. Lett. 81, 1062 �1998�; V. E. Kravtsov and V. I. Yudson, ibid. 82, 157 �1999�; M. Jałochowski, E. Bauer, H.Knoppe, and G. Lilienkamp, Phys. Rev. B 45, 13607 �1992�; W. P. Halperin, Rev. Mod. Phys. 58, 533 �1986�.

2 A. Sisman and I. Muller, Phys. Lett. A 320, 360 �2004�; A. Sisman, J. Phys. A 37, 11353 �2004�.3 W.-S. Dai and M. Xie, Phys. Rev. E 70, 016103 �2004�.4 W.-S. Dai and M. Xie, Phys. Lett. A 311, 340 �2003�.5 H. Pang, W.-S. Dai, and M. Xie, J. Phys. A 39, 2563 �2006�.6 F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 �1999�; A. J. Leggett, ibid. 73, 307�2001�; Y. E. Kim and A. L. Zubarev, Phys. Lett. A 327, 397 �2004�; E. Diez, Y. P. Chen, S. Avesque, M. Hilke, E.Peled, D. Shahar, J. M. Cerveró, D. L. Sivco, and A. Y. Cho, Appl. Phys. Lett. 88, 052107 �2006�.

7 Q. Chen, J. Stajic, and K. Levin, Phys. Rev. Lett. 95, 260405 �2005�; S. M. Reimann and M. Manninen, Rev. Mod. Phys.74, 1283 �2002�; J. Adamowski, M. Sobkowicz, B. Szafran, and S. Bednarek, Phys. Rev. B 62, 4234 �2000�; Y.Alhassid, Rev. Mod. Phys. 72, 895 �2000�.

8 R. M. Ziff, G. E. Uhlenbeck, and M. Kac, Phys. Rep. 32, 169 �1977�; D. W. Robinson, Commun. Math. Phys. 50, 53�1976�; L. Vandevenne, A. Verbeure, and V. A. Zagrebnov, J. Math. Phys. 45, 1608 �2004�; L. J. Landau and I. F. Wilde,Commun. Math. Phys. 70, 43 �1979�; J.-B. Bru and V. A. Zagrebnov, J. Phys. A 33, 449 �2000�.

9 V. Bagnato and D. Kleppner, Phys. Rev. A 44, 7439 �1991�; V. Bagnato, D. E. Pritchard, and D. Kleppner, ibid. 35, 4354�1987�.

10 M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 �1995�; C. C.Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 �1995�; K. B. Davis, M.-O. Mewes, M.R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, ibid. 75, 3969 �1995�; D. G. Fried, T. C.Killian, L. Willmann, D. Landhuis, S. C. Moss, D. Kleppner, and T. J. Greytak, ibid. 81, 3811 �1998�.

11 K. Huang and C. N. Yang, Phys. Rev. 105, 767 �1957�; K. Huang, C. N. Yang, and J. M. Luttinger, ibid. 105, 776�1957�.

12 W.-S. Dai and M. Xie, Europhys. Lett. 72, 887 �2005�.13 M. Kac, Am. Math. Monthly 73, 1 �1966�.14 H. Weyl, in Gesammelte Abhandlungen, edited by K. Chandrasekharan �Springer, Berlin, 1968�, Vol. I.15 A. Pleijel, Ark. Mat. 2, 553 �1954�.16 W.-S. Dai and M. Xie �unpublished�.17 W.-S. Dai and M. Xie, preprint, arXiv:math-ph/0703847.18 D. A. Klain and G.-C. Rota, Introduction to Geometric Probability �Cambridge University Press, Cambridge, 1997�.19 T. D. Lee and C. N. Yang, Phys. Rev. 116, 25 �1959�. Before the publication of this paper, the results had been reported

in T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 �1957�.20 M. Holzmann, J.-N. Fuchs, G. A. Baym, J.-P. Blaizot, and F. Laloë, C. R. Phys. 5, 21 �2004�; G. Baym, J.-P. Blaizot, M.

Holzmann, F. Laloë, and D. Vautherin, Eur. Phys. J. B 24, 107 �2001�; S. Ledowski, N. Hasselmann, and P. Kopietz,Phys. Rev. A 69, 061601�R� �2004�; P. Arnold, G. D. Moore, and B. Tomázik, ibid. 65, 013606 �2002�; J. O. Andersen,Rev. Mod. Phys. 76, 599 �2004�.

21 R. K. Pathria, Statistical Mechanics, 2nd ed. �Butterworth-Heinemann, Oxford, 1996�.22 W.-S. Dai and M. Xie, Ann. Phys. �N.Y.� 309, 295 �2004�. In this paper, there is a misprint in Eq. �26�; this equation

should be �= F�1− ��2 /3��1 / �n+1�� /s−1��kT / F�2+ ¯ �.



Documents

Interacting quantum gases in confined space: Two- and three-dimensional equations of state