arXiv:1710.02819v2 [cond-mat.quant-gas] 18 May 2018 unu a,Itrcigcasclgs neatn quant Interacting gas, classical Interacting gas, quantum Keywords: canonica grand the from b calculated function than int partition rather canonical of polynomial, the coefficients from calculated virial gases is The Gentile gases and pol function. Fermi, Bell Bose, partition the ideal canonical of of terms coefficients in virial quan methods The and expansion Moreo classical cluster interacting quantum gases. and of Gentile functions and partit partition Fermi, canonical t ical Bose, the of ideal of theory calculation including mathematical the gases, the for on method based a paper, suggest this t In of calculation the difficult. Nevertheless, ensemble. q canonical thermodynamic the the speaking, strictly system, finite-size Abstract: h-hnZo n uSegDai Wu-Sheng and Zhou Chi-Chun quantum interacting gases and gases, gases, classical quantum interacting ideal functions: partition Canonical eateto hsc,TajnUiest,Tajn305,P 300350, Tianjin University, Tianjin Physics, of Department ∗ [email protected]. nsaitclmcais o ytmwt xdnme fpa of number fixed with system a for mechanics, statistical In aoia atto ucin ymti ucin elpol Bell function, Symmetric function, partition Canonical ∗ mgs iilcoefficient Virial gas, um ecnnclpriinfnto is function partition canonical he u ae ie yteclassical the by given gases tum sn h xaso fteBell the of expansion the using y att ed ob acltdin calculated be to needs uantity rcigcascladquantum and classical eracting R China .R. potential. l o ucino da quantum ideal of function ion scluae rmteexact the from calculated is e,w xrs h canon- the express we ver, esmercfnto,we function, symmetric he nma nmathematics. in ynomial nma,Ideal ynomial, tce,eg,a e.g., rticles, Contents
1 Introduction 1
2 The integer partition and the symmetric function: a brief review 4
3 The canonical partition function of ideal Bose gases 5
4 The canonical partition function of ideal Fermi gases 7
5 The canonical partition function of ideal Gentile gases 8
6 The canonical partition function of interacting classical gases 10
7 The canonical partition function of interacting quantum gases 10
8 The virial coefficients of ideal quantum gases in the canonical ensemble 11 8.1 The virial coefficients of ideal Bose and Fermi gases in the canonical ensemble 12 8.2 The virial coefficients of Gentile gases in the canonical ensemble 13
9 The virial coefficients of interacting classical and quantum gases in the canonical ensemble 14
10 Conclusions and discussions 15
A Appendix: Details of the calculation for ideal Gentile gases in the canon- ical ensemble 16 A.1 The coefficient QI (q) 16 A.2 The canonical partition function 23 A.3 The virial coefficients 26
1 Introduction
It is often difficult to calculate the canonical partition function. In the canonical ensemble, there is a constraint on the total number of particles. Once there also exist interactions, whether quantum exchange interactions or classical inter-particle interactions, the calcu- lation in canonical ensembles becomes complicated. In ideal quantum gases, there are quantum exchange interactions among gas molecules; in interacting classical gases, there are classical interactions among gas molecules; in interacting quantum gases, there are both quantum exchange interactions and classical inter-particle interactions among gas molecules. In an interacting system there is no single–particle state, and in a canonical ensemble there is a constraint on the total particle number. Therefore, in the calculation of the canonical
– 1 – partition function for such systems, we have to take these two factors into account simul- taneously, and this makes the calculation difficult. It is a common practice to avoid such a difficulty by, instead of canonical ensembles, turning to grand canonical ensembles in which there is no constraint on the total particle number. In grand canonical ensembles, instead of the particle number, one uses the average particle number. In this paper, we suggest a method to calculate canonical partition functions. The method suggested in the present paper is based on the mathematical theory of the symmetric function and the Bell polynomial. The symmetric function was first studied by P. Hall in the 1950s [1, 2]. Now it is studied as the ring of symmetric functions in algebraic combinatorics. The symmetric function is closely related to the integer partition in number theory and plays an important role in the theory of group representations [3]. The Bell polynomial is a special function in combinatorial mathematics [4, 5]. It is useful in the study of set partitions and is related to the Stirling and Bell number. For more details of the Bell polynomial, one can refer to Refs. [4–8]. First, using the method, we will calculate the exact canonical partition function for ideal quantum gases, including ideal Bose gases, ideal Fermi gases, and ideal Gentile gases. In ideal quantum gases, there exist quantum exchange interactions, as will be shown later, the canonical partition functions are symmetric functions. Based on the theory of the symmetric function, the canonical partition function is represented as a linear combination of S-functions. The S-function is an important class of symmetric functions, and they are closely related to integer partitions [3] and permutation groups [1, 2]. Second, we show that the canonical partition functions of interacting classical gases and interacting quantum gases given by the classical and quantum cluster expansions are the Bell polynomial. In interacting classical gases, there exist classical inter-particle interactions and in interacting quantum gases, there exist both quantum exchange interactions and classical inter-particle interactions. After showing that the canonical partition function of interacting gases is the Bell polynomial, we can calculate, e.g., the virial coefficient by the property of the Bell polynomial. We will calculate the virial coefficients of ideal Bose gases, ideal Fermi gases, and ideal Gentile gases from their exact canonical partition functions. Moreover, based on the property of the Bell polynomial, we directly calculate the virial coefficients of interacting classical and quantum gases from the canonical partition function directly, rather than, as in the cluster expansion method, from the grand canonical potential. We also compare the virial coefficients calculated in the canonical ensemble with those calculated in the grand canonical ensemble. From the results one can see that the virial coefficients are different at small N and they are in consistency with each other as N goes to infinity. There are many studies on canonical partition functions. Some canonical partition functions of certain statistical models are calculated, for examples, the canonical partition function for a two-dimensional binary lattice [9], the canonical partition function for quon statistics [10], a general formula for the canonical partition function expressed as sums of the S-function for a parastatistical system [11], the canonical partition function of freely jointed chain model [12], an exact canonical partition function for ideal Bose gases calculated
– 2 – by the recursive method [13], and the canonical partition function of fluids calculated by simulations [14]. Some methods for the calculation of the canonical partition function are developed, for examples, the numerical method [13, 15], the recursion relation of the canonical partition function [15–17]. The behavior of the canonical partition function is also discussed, for example, the zeroes of the canonical partition function [18, 19] and the classical limit of the quantum-mechanical canonical partition function [20].
There are also many studies on the symmetric function and the Bell polynomial. In mathematics, some studies devote to the application of the S-function, also known as the Schur function, for example, the application of S-function in the symmetric function space [21], the application of the S-function in probability and statistics [22], and the supersymmetric S-function [23]. In physics, there are also applications of the S-function, for example, the canonical partition function of a parastatistical system is expressed as a sum of S-functions [11], the factorial S-function, a generalizations of the S-function, times a deformation of the Weyl denominator may be expressed as the partition function of a particular statistical-mechanical system [24] and the application to statistical mechanics and representation theory [25]. The Bell polynomial can be used in solving the water wave equation [26], in seventh-order Sawada–Kotera–Ito equation [27], in combinatorial Hopf algebras [28]. Moreover, various other applications can be found in [29–32].
There are studies devoted to the application of the Bell polynomial in statistical me- chanics. The canonical partition function of an interacting system is expressed in terms of the Bell polynomial by expanding the grand canonical partition function in power series of the fugacity z [33]. As further applications of the bell polynomial, a microscopic interpre- tation of the grand potential is given in Ref. [34]. The partition function of the zero-field Ising model on the infinite square lattice is given by resorting to the Bell polynomial [35]. Ref. [36] proves an assertion suggested in Ref. [34], which provides a formula for the canon- ical partition function using Bell polynomials and shows that the Bell polynomial is very useful in statistical mechanics. In this paper, we will compare our method in obtaining the canonical partition function of interacting gases with that in these literature.
This paper is organized as follow. In Sec. 2, we give a brief review for the integer partition and the symmetric function. In Secs. 3 and 4, the canonical partition functions of ideal Bose and Fermi gases are given. In Sec. 5, the canonical partition function of ideal Gentile gases is given. In Secs. 6 and 7, we show that the canonical partition functions of interacting classical and quantum gases are indeed Bell polynomials. In Sec. 8, we calculate the virial coefficients of ideal Bose, Fermi, and Gentile gases in the canonical ensemble and compare them with those calculated in the grand canonical ensemble. In Sec. 9, we calculate the virial coefficients of interacting classical and quantum gases in the canonical ensemble and compare them with those calculated in the grand canonical ensemble. The conclusions are summarized in Sec. 10. In appendix A, the details of the calculation about Gentile gases is given.
– 3 – 2 The integer partition and the symmetric function: a brief review
The main result of the present paper is to represent the canonical partition function as a linear combination of symmetric functions. The symmetric function is closely related to the problem of integer partitions. In this section, we give a brief review of symmetric functions and integer partitions. For more detail of this part, one can refer to Refs. [1–3]. Integer partitions. The integer partitions of N are representations of N in terms of other positive integers which sum up to N. The integer partition is denoted by (λ) = (λ1, λ2, ...), where λ1, λ2, ... are called elements, and they are arranged in descending order. For 2 2 example, the integer partition of 4 are (λ) = (4), (λ)′ = (3, 1), (λ)′′ = 2 , (λ)′′′ = 2, 1 , 4 2 and (λ)′′′′ = 1 , where, e.g., the superscript in 1 means 1 appearing twice, the superscript in 22 means 2 appearing twice, and so on. Arranging integer partitions in a prescribed order. An integer N has many integer partitions and the unrestricted partition function P (N) counts the number of integer par- titions [3]. For a given N, one arranges the integer partition in the following order: (λ),
(λ)′, when λ1 > λ1′ ; (λ), (λ)′, when λ1 = λ1′ but λ2 > λ2′ ; and so on. One keeps compar- ing λi and λi′ until all the integer partitions of N are arranged in a prescribed order. We denote (λ)J the Jth integer partition of N. For example, the integer partitions arranged 2 2 in a prescribed order of 4 are (λ)1 = (4), (λ)2 = (3, 1), (λ)3 = 2 , (λ)4 = 2, 1 , and (λ) = 14 . For any integer N, the first integer partition is always (λ) = (N) and the 5 J=1 last integer partition is always (λ) = 1N . We denote λ the ith element in the J=P (N) J,i integer partition (λ) . For example, for the second integer partition of 4, i.e., (λ) = (3, 1), J 2 the elements of (λ)2 are λ2,1 = 3 and λ2,2 = 1, respectively. For any integer N, we always have λ1,1 = N, and λP (N),1 = λP (N),2 = ... = λP (N),N = 1.
The symmetric function. A function f (x1,x2, ..., xn) is called symmetric functions if it is invariant under the action of the permutation group S ; that is, for σ S , n ∈ n
σf (x1,x2, .., xn)= f xσ(1),xσ(2), .., xσ(n) = f (x1,x2, .., xn) , (2.1) where x1, x2,..., xn are n independent variables of f (x1,x2, ..., xn).
The symmetric function m(λ) (x1,x2, ..., xl). The symmetric function m(λ) (x1,x2, ..., xl) is an important kind of symmetric functions. One of its definition is [2]
m (x ,x , ..., x )= xλI1 xλI2 ....xλIN , (2.2) (λ)I 1 2 l i1 i2 iN perm X where perm indicates that the summation runs over all possible monotonically increasing permutations of x , (λ) is the Ith integer partition of N, and λ is the jth element in P i I Ij (λ)I . Here, the number of variables is l, and l should be larger than N as required in the definition. l, the number of xi, could be infinite and in the problem considered in this paper, l is always infinite. Each integer partition (λ)I of N corresponds to a symmetric function m (x ,x , ...) and vise versa. (λ)I 1 2
– 4 – The S-function (λ) (x1,x2, ...). The S-function (λ) (x1,x2, ...), another important kind of symmetric functions, among their many definitions, can be defined as [1, 2]
P (N) k aJ,m g (λ) (x ,x , ...)= I χI xm , (2.3) I 1 2 N! J i m=1 ! JX=1 Y Xi where (λ)I is the Ith integer partition of N, aJ,m counts the times of the number m I appeared in (λ)J , and χJ is the simple characteristic of the permutation group of order 1 N − N. g is defined as g = N! jaI,j a ! . Here we only consider the case that the I I I,j jY=1 number of variables is always infinite. Each integer partition (λ)I of N corresponds to an S-function (λ)I (x1,x2, ...) and vice versa. The relation between the S-function (λ) (x1,x2, ...) and the symmetric function m(λ) (x1,x2, ...). There is a relation between the S-function (λ) (x1,x2, ...) and the symmetric function m(λ) (x1,x2, ...): the S-function (λ) (x1,x2, ...) can be expressed as a linear combination of the symmetric function m(λ) (x1,x2, ...),
P (N) (λ) (x ,x , ...)= kI m (x ,x , ...) , (2.4) K 1 2 K (λ)I 1 2 XI=1 I where the coefficient kK is the Kostka number [2].
3 The canonical partition function of ideal Bose gases
In this section, we present an exact expression of the canonical partition function of ideal Bose gases.
Theorem 1 For ideal Bose gases, the canonical partition function is
βε1 βε2 ZBE (β, N) = (N) e− , e− , ... , (3.1) where (N) (x1,x2, ...) is the S-function corresponding to the integer partition (N) defined by Eq. (2.3) and εi is the single-particle eigenvalue.
Proof. By definition, the canonical partition function of an N-body system is
βEs Z (β, N)= e− , (3.2) s X where Es denotes the eigenvalue of the system with the subscript s labeling the states, β = 1/ (kT ), k is the Boltzmann constant, and T is the temperature. Rewrite Eq. (3.2) in terms of the occupation number. For an ideal quantum gas, particles are randomly distributed on the single-particle states. Collect the single-particle states occupied by particles by weakly increasing order of energy, i.e., ε ε ε , i1 ≤ i2 ≤ i3 ≤··· and denote the number of particle occupying the jth state by aj. For Bose gases, there
– 5 – is no restriction on aj, since the maximum occupation number is infinite. Note that here a 1 rather than a 0. This is because here only the occupied states are reckoned in. i ≥ i ≥ Eq. (3.2) can be re-expressed as [37]
βEs βεi a1 βεi a2..... ZBE (β, N)= e− = e− 1 − 2 , (3.3)
s ai perm X {X} X where the sum perm runs over all possible mononical increasing permutation of εi and the sum ai runs over all the possible occupation numbers restricted by the constraint { } P P a = N, a 1. (3.4) i i ≥ The constraint (3.4) ensures that theX total number of particles in the system is a constant. Equalling the occupation number ai and the element λi, equalling the variable xi and βεi e− , and comparing the canonical partition function, Eq. (3.3), with the definition of the symmetric function m(λ) (x1,x2, ...), Eq. (2.2), give
P (N) Z (β, N)= m e βε1 , e βε2 , ... . (3.5) BE (λ)I − − XI=1 I Substituting Eq. (2.4) into Eq. (3.5) and setting all the coefficients kK to 1 give Eq. P (N) (3.1), where the relation m e βε1 , e βε2 , ... = (λ) e βε1 , e βε2 , ... [2] and I=1 (λ)I − − 1 − − (λ) = (N) are used. 1 P In a word, the canonical partition function of ideal Bose gases is a S-function. Moreover, by one kind of expression of the S-function given in Ref. [1, 2], we can obtain the canonical partition function of ideal Bose gases given in Ref. [13]. In Ref. [13], the author obtained a recurrence relation of the canonical partition function: ZBE (β, N)= 1 N e βkεn Z (β, N k), based on a result given by Mastsubara [38] and Feyn- N k=1 n − BE − man [39]. Starting from this recurrence relation, the author obtains the canonical partition P P function of ideal Bose gases in matrix form by introducing an triangularized matrix ∞×∞ and computing the inverse of a matrix. In Refs. [1, 2], the S-function is represented as the determinant of a certain matrix: x 1 0 ... 0 i i − x2 x 2 ... 0 i i i i 1 P 3 2 − (N) (x1,x2, ...)= det i xi i xi i xi ...... (3.6) N! P P ...... (N 1) P P P − − xN xN 1 xN 2 ... x i i i i − i i − i i The canonical partition function (3.1),P by Eq.P (3.6), canP be equivalentlyP expressed as
βε1 βε2 ZBE (β, N) = (N) e− , e− , ... Z (β) 1 0 ... 0 − Z (2β) Z (β) 2 ... 0 1 − = det Z (3β) Z (2β) Z (β) ...... , (3.7) N! ...... (N 1) − − Z (Nβ) Z (Nβ β) Z (Nβ 2β) ... Z (β) − −
– 6 – where βεi Z (β)= e− (3.8) Xi is the single-particle partition function of ideal classical gases. For a free ideal classical gas, the single-particle partition function is V Z (β)= (3.9) λ3
β with V the volume and λ = h 2πm the thermal wave length [40]. It is worthy to note thatq different expressions of the S-function will give different expression of canonical partition functions.
4 The canonical partition function of ideal Fermi gases
In this section, we present an exact expression of the canonical partition function of ideal Fermi gases.
Theorem 2 For ideal Fermi gases, the canonical partition function is
N βε1 βε2 ZFD (β, N)= 1 e− , e− , ... , (4.1) N N where 1 (x1,x2, ...) is the S-function corresponding to the integer partition 1 defined by Eq. (2.3). Proof. Rewrite Eq. (3.2) in terms of occupation numbers. The difference between Fermi and Bose systems is that the maximum occupation number should be 1, i.e., ai = aj = ...ak = 1. Eq. (3.2) then can be re-expressed as [37]
βEs βεi βεi ..... βεi ZFD (β, N)= e− = e− 1 − 2 − N , (4.2) s perm X X where the sum perm runs over all possible mononical increasing permutation of εi. The constraint on the total number of particles of the system now becomes a = a = ... = P 1 2 aN = 1. Equalling the occupation number ai and the element λi, equalling the variable xi and βεi e− , and comparing the canonical partition function, Eq. (4.2), with the definition of the symmetric function m(λ) (x1,x2, ...), Eq. (2.2), give
βε1 βε2 βε1 βε2 Z (β, N)= m e− , e− , ... = m N e− , e− , ... , (4.3) FD (λ)P (N) (1 ) N where (λ)P (N) = 1 is used. Then we have N βε1 βε2 ZFD (β, N)= 1 e− , e− , ... , (4.4) βε1 βε2 βε1 βε2 where the relation m(1N ) e− , e− , ... = (λ)P (N) e− , e− , ... is used [2].
– 7 – N βε1 βε2 Similarly, the S-function 1 e− , e− , ... can be represented as the determinant of a certain matrix [1, 2]:
i xi 1 0 ... 0 2 i xi i xi 2 ... 0 1 P 3 2 (N) (x1,x2, ...)= det i xi i xi i xi ...... (4.5) N! P P ...... (N 1) P P (N 1) P(N 2) − xN x − x − ... x i i i i i i i i The canonical partition function (4.1),P by Eq.P (4.5), canP be equivalentlyP expressed as
Z (β) 1 0 ... 0 Z (2β) Z (β) 2 ... 0 N βε1 βε2 1 ZFD (β, N)= 1 e− , e− , ... = det Z (3β) Z (2β) Z (β) ...... N! ...... (N 1) − Z (Nβ) Z (Nβ β) Z (Nβ 2β) ... Z (β) − − (4.6)
5 The canonical partition function of ideal Gentile gases
The Gentile statistic is a generalization of Bose and Fermi statistics [41, 42]. The maximum occupation number of a Fermi system is 1 and of a Bose system is . As a generalization, ∞ the maximum occupation number of a Gentile system is an arbitrary integer q [41–43]. Be- yond commutative and anticommutative quantization, which corresponds to the Bose case and the Fermi case respectively, there are also some other effective quantization schemes [44, 45]. It is shown that the statistical distribution corresponding to various q-deformation schemes are in fact various Gentile distributions with different maximum occupation num- bers q [46]. There are many physical systems obey intermediate statistics, for example, spin waves, or, magnons, which is the elementary excitation of the Heisenberg magnetic system [47], deformed fermion gases [48, 49], and deformed boson gases [50]. Moreover, there are also generalizations for Gentile statistics, in which the maximum occupation numbers of different quantum states take on different values [51]. In this section, we present the canonical partition function of ideal Gentile gases.
Theorem 3 For ideal Gentile gases with a maximum occupation number q, the canonical partition function is
P (N) J βε1 βε2 Zq (β, N)= Q (q) (λ)I e− , e− , ... , (5.1) XI=1 where the coefficient P (N) J I 1 K Q (q)= kK − Γ (q) (5.2) K=1 X
– 8 – with ΓK (q) satisfying
K Γ (q) = 0, when λK,1 > q, ΓK (q) = 1, when λ q, (5.3) K,1 ≤ I 1 and kK − satisfying p(k) I 1 L L kK − kI = δK (5.4) I=1 X L with kI the Kostka number [2].
Proof. Rewrite Eq. (3.2) in terms of the occupation number. Here, the occupation number a is restricted by 0 < a q. Eq. (3.2) can be re-expressed as j j ≤
βEs βεi a1 βεi a2..... Zq (β, N)= e− = e− 1 − 2 , (5.5)
s ai perm X {X}q X where the sum perm runs over all possible mononical increasing permutation of εi, and the sum ai runs over all possible occupation number restricted by the constraints { }q P P a = N, 1 a q. (5.6) i ≤ j ≤ X Equalling the occupation number ai and the element λi, equalling the variable xi and βεi e− , and comparing the canonical partition function, Eq. (5.5), with the definition of the symmetric function m(λ) (x1,x2, ...), Eq. (2.2), give
P (N) Z (β, N)= ΓK (q) m e βε1 , e βε2 , ... . (5.7) q (λ)K − − KX=1 By introducing ΓK (q), the constraint on a , i.e., 1 a q, is automatically taken into j ≤ j ≤ account. I 1 I 1 Introducing kK − , which satisfies Eq. (5.4), multiplying kK − to both sides of Eq. (2.4), and summing over the indice I, we arrive at P (N) 1 m e βε1 , e βε2 , ... = kI − (λ) e βε1 , e βε2 , ... . (5.8) (λ)K − − K I − − I=1 X Substituting Eq. (5.8) into Eq. (5.7) gives
P (N) P (N) 1 K I − βε1 βε2 Zq (β, N)= Γ (q) kK (λ)I e− , e− , ... . (5.9) I=1 K=1 X X Substituting QJ (q) which is defined by Eq. (5.2) into Eq. (5.9) gives Eq. (5.1) directly. In appendix A.2, as examples, we calculate the canonical partition function of a Gentile gas based on Eq. (5.1) for N = 3, 4, 5, and 6.
– 9 – 6 The canonical partition function of interacting classical gases
In this section, we show that the canonical partition function of an interacting classical gas given by the classical cluster expansion is indeed a Bell polynomial [1–3].
Theorem 4 The canonical partition function of an interacting classical gas with N parti- cles is 1 Z (β, N)= B (Γ , Γ , Γ , ..., Γ ) , (6.1) N! N 1 2 3 N where BN (x1,x2, ..., xN ) is the Bell polynomial [4, 5] and Γl is defined as l!V Γ = b (6.2) l λ3 l with bl the expansion coefficient in the classical cluster expansion [40].
Proof. The canonical partition function given by the classical cluster expansion is [40]
N ml 1 l!blV 1 Z (β, N)= 3 . (6.3) " l! λ ml!# ml l=1 {X} Y Comparing Eq. (6.3) with expression of the Bell polynomial [4, 5]
N 1 ml 1 BN (x1,x2, ..., xN )= N! xl , (6.4) " l! ml!# ml l=1 {X} Y we immediately arrive at Eq. (6.1).
Here, for completeness, we list some Γl in the classical cluster expansion. For an 2 interacting classical gas with the Hamiltonian H = N Pl + U , Γ is given as N − l=1 2m j V Γ = , 1 λ3 1 3 3 βU12 Γ = d q d q e− 1 , . (6.5) 2 λ6 1 2 − · · · Z Z 7 The canonical partition function of interacting quantum gases The interacting quantum gas is always an important issue in statistical mechanics. In a quantum hard-sphere gas, there are two interplayed effects: quantum exchange interactions and classical inter-particle interactions [53–55]. In this section, we show that the canonical partition function can be represented by the Bell polynomial. Theorem 5 The canonical partition function of an interacting quantum gas with N parti- cles is 1 Z (β, N)= B (Γ , Γ , Γ , ..., Γ ) , (7.1) N! N 1 2 3 N – 10 – where BN (x1,x2, ..., xN ) is the Bell polynomial [4, 5] and Γl is defined as l!V Γ = b (7.2) l λ3 l with bl the expansion coefficient in the quantum cluster expansion [40, 52]. Here, for completeness, we list some first expansion coefficients in the quantum cluster ~2 expansion. For an interacting quantum gas with the Hamiltonian H = i 2 + i − 2m l=1 ∇l U , the quantum cluster expansion coefficient is given by [40, 52] j N ml 1 l!blV 1 Z (β, N)= d . (7.4) " l! λ ml!# ml l=1 {X} Y Comparing Eq. (7.4) with the expression of the Bell polynomial defined by Eq. (6.4) give Eq. (7.1) directly. Comparing Eqs. (6.1) and (7.1), we can see that the only difference between the canonical partition functions of an interacting classical gas and of an interacting quantum gas is the different between the expansion coefficients. The above result agrees with the canonical partition function given in Ref. [33]. In Ref. [33], the canonical partition function is obtained by starting with an expansion of the grand ∞ m canonical partition function Ξ (z, β) = exp ( βΦ (z, β)) = exp β z φ (β) = − − m! m m=1 ! zN ∂lΦ X 1+ N∞=1 BN (w1, ..., wN ) with wl = βφl (β) = β l and Φ (z, β) the grand N! − − ∂z z=0 potential,P and then comparing with the relation between the grand canonical partition N function and the canonical partition function, Ξ (z, β) = N∞=0 Z (β, N) z [40]. In Ref. [33], the canonical partition function is obtained from a grand partition function. In the P present paper, the canonical partition function is calculated in the canonical ensemble. Con- cretely, in this paper, the canonical partition function for interacting classical and quantum gases is obtained by comparing the canonical partition function of interacting classical and quantum gases given by the cluster expansion method [40] and the definition of the Bell polynomial [4]. This allows us to give the parameter Γl in the canonical partition function explicitly. 8 The virial coefficients of ideal quantum gases in the canonical ensemble Based on the canonical partition function of ideal quantum gases given in sections 3, 4, and 5, we calculate the virial coefficient for ideal quantum gases and then compare them with those calculated in the grand canonical ensemble. – 11 – 8.1 The virial coefficients of ideal Bose and Fermi gases in the canonical en- semble In this section we calculate the virial coefficients of ideal Bose and Fermi gases directly from the canonical partition functions. In terms of the S-function, the canonical partition functions of ideal Bose and Fermi gases can be expressed by the partition function of a classical free particle. Substituting the partition function of a classical free particle (3.9) into Eqs. (3.7) and (4.6) and re-expressing them in terms of the V/λ3 give the canonical partition function of ideal Bose gases N N 1 1 V 1 1 V − ZBE (β, N)= + + (8.1) N! λ3 23/2 2 (N 2)! λ3 · · · − and the canonical partition function of ideal Fermi gases N N 1 1 V 1 1 V − ZFD (β, N)= + . (8.2) N! λ3 − 23/2 2 (N 2)! λ3 · · · − In order to calculate the virial coefficients, we first calculate the equation of state from the canonical partition function. The equation of state is [40] 1 ∂ ln Z (N, β) P = . (8.3) β ∂V The equation of state (8.3) can be expressed by the virial expansion [40]: n 1 PV Nλ3 − = a (8.4) NkT n V n X with al the virial coefficient [40]. Substituting Eqs. (8.1) and (8.2) into Eq. (8.3), the coefficients of Nλ3/V 2 give the second virial coefficients of ideal Bose and Fermi gases, respectively: 1 1 a2 = 1 , (8.5) ±25/2 − N where "+" stands for Bose gases and " " stands for Fermi gases. − Comparing the second virial coefficient obtained in the canonical ensemble, Eq. (8.5) with the second virial coefficient obtained in the grand canonical ensemble [40], 1 agrand = , (8.6) 2 ±25/2 we can see that the second virial coefficient in the canonical partition function depends on the total particle number N. As N tends to infinity, a2 recovers the second virial coefficient grand obtained in the grand canonical ensemble a2 . – 12 – Table 1. The virial coefficients of ideal Gentile gases: q =2 N = 3 N = 4 N = 5 N = 6 grand canonical a1 1111 1 a 0.11785 0.13258 0.14142 0.14731 0.17677 2 − − − − − a3 0.09869 0.15482 0.19317 0.22068 0.386 a 0.04497 0.11330 0.17447 0.22482 0.6124 4 − − − − − Table 2. The virial coefficients of ideal Gentile gases: q =3 N = 3 N = 4 N = 5 N = 6 grand canonical a1 1111 1 a 0.11785 0.13258 0.14142 0.14731 0.17677 2 − − − − − a 0.01316 0.01048 0.00842 0.00685 0.0033 3 − a4 0.00039 0.03667 0.07390 0.10616 0.374889 Table 3. The virial coefficients of ideal Gentile gases: q =4 N = 4 N = 5 N = 6 grand canonical a1 1 1 1 1 a 0.13258 0.14142 0.14731 0.17677 2 − − − − a 0.01048 0.00842 0.00685 0.0033 3 − a 0.00152 0.00190 0.00199 0.00011 4 − Table 4. The virial coefficients of ideal Gentile gases: q =5 N = 5 N = 6 grand canonical a1 1 1 1 a 0.14142 0.14731 0.17677 2 − − − a 0.00842 0.00685 0.0033 3 − a 0.00190 0.00199 0.00011 4 − 8.2 The virial coefficients of Gentile gases in the canonical ensemble For Gentile gases, the maximum occupation number is an given integer q. In this section, we list some virial coefficients of Gentle gases. The details of the calculation will be given in the Appendix A.1. In the following, we compare the results obtained in canonical ensembles and in grand Table 5. The virial coefficients of ideal Gentile gases: q =6 N = 6 grand canonical a1 1 1 a 0.14731 0.17677 2 − − a 0.00685 0.0033 3 − a 0.0020 0.00011 4 − – 13 – canonical ensembles in Tables (1)-(5), respectively. The virial coefficient of Gentile gases in the grand canonical ensemble can be found in Ref. [56]. From Tables (1)-(5), one can see that the virial coefficients calculated from in the canonical ensemble are different from those calculated in the grand canonical ensemble. However, as N increases, the result obtained in the canonical ensemble tends to the result obtained in the grand canonical ensemble. 9 The virial coefficients of interacting classical and quantum gases in the canonical ensemble In this section, based on the expansion of the Bell polynomial, we calculate the second virial coefficient of interacting classical and quantum gases in canonical ensemble and then compare the results with those calculated in the grand canonical ensemble. The Bell polynomial can be expended as N N! N 2 B [Γ , Γ , ..., Γ ]=Γ + Γ − Γ + ... +Γ . (9.1) N 1 2 N 1 2 (N 2)! 1 2 N − Then the canonical partition function (6.1) becomes 1 N 1 N 2 1 Z (β, N)= Γ + Γ − Γ + ... + Γ . (9.2) N! 1 2 (N 2)! 1 2 N! N − The second virial coefficients of interacting classical and quantum gases can be directly V obtained from the canonical partition function. By Eq. (6.2), we have Γ1 = λ3 and 2V Γ2 = λ3 b2. Then Eq. (9.2) becomes N N 1 1 V 1 V − Z (β, N)= + b + . (9.3) N! λ3 (N 2)! λ3 2 · · · − Substituting Eq. (9.3) into Eq. (8.3) and selecting the coefficients of Nλ3/V 2 give the second virial coefficients: 1 a = b 1 . (9.4) 2 2 − N One can see that the second virial coefficients are functions of the total particle number N. Comparing the second virial coefficient obtained in the canonical ensemble, Eq. (9.4), with the second virial coefficient obtained in the grand canonical ensemble [40], grand a2 = b2, (9.5) we can see that the result obtained in the canonical partition function depends on the total particle number N and recovers the result obtained in the grand canonical ensemble when N tends to infinite. – 14 – 10 Conclusions and discussions Solving an N-body system is an important problem in physics. There are many mechanical and statistical-mechanical methods developed to this problem. The mechanical treatment for an N-body system is to find the eigenvalues and eigen- states of the N-body Hamiltonian of the system. In the mechanical treatment, both in- formations of eigenvalues and eigenfunctions are taken into account. However, as is well known, solving an N-body system in mechanics is very difficult. In an interacting quan- tum system, two factors, classical inter-particle interactions and the quantum exchange interactions, intertwined together. The statistical-mechanical treatment for an N-body system, essentially, is to use av- erage values instead of exact values. Only the information of eigenvalues is taken into account. All thermodynamic quantities can be obtained from, for example, the canonical βEs partition function Z (β, N) = s e− which is determined only by the eigenvalues Es of the N-body system. At the expense of losing the information of eigenfunctions, the P statistical-mechanical treatment is much easier compared with the mechanical treatment. In a word, the statistical-mechanical treatment can be regarded as an approximate method for N-body problems. A rigorous description of an N-body system in statistical mechanics should be made in a canonical ensemble with fixed particle number N. The core task in a canonical ensemble is to seek the canonical partition function Z (β, N). The calculation of the canonical partition function is in fact a sum over all states with a constraint of the total particle number of the system. Nevertheless, it is very difficult to calculate the canonical partition function when one has to deal with the difficulties from the classical inter-particle interactions and quantum exchange interactions and take the constraint of the total particle number of the system into consideration simultaneously. In order to avoid the difficulty caused by the constraint of the total particle number, conventionally, one introduces the grand canonical ensemble which has no constraint on the particle number, and, instead of canonical partition function, one turns to calculate the grand canonical partition function. The advantage is that the constraint of particle numbers on the sum is now removed; the price is that the particle number now is not a constant. In the grand canonical ensemble, the total particle number is not rigorously equal to an exact number N, but is equal to the average particle number N . That is to say, h i an N-body problem is approximately converted to an N -body problem, with a deviation h i of 1 . Unless in the thermodynamic limit, i.e., N , the result obtained in the grand √N →∞ canonical ensemble is not the same as that obtained in the canonical ensemble. Thus, for an N-body problem, the grand canonical ensemble is an approximate method even in statistical-mechanical treatment. The method suggested in the paper is based on the mathematical theory of the sym- metric function and the Bell polynomial. In this paper, for ideal quantum gases, including Bose, Fermi, and Gentile gases, we suggest a method to calculate exact canonical partition functions, and show that the canonical partition functions of ideal Bose, Fermi, and Gentile gases can be represented as linear combinations of the S-function. For interacting classical – 15 – and quantum gases, we point out that the canonical partition functions given by the clus- ter expansion method are indeed the Bell polynomial. Starting from the exact canonical partition functions of ideal quantum gases, we calculate the virial coefficients in canonical ensemble. For interacting gases, we calculate the virial coefficients in canonical ensembles instead of in grand canonical ensembles as that in literature. A Appendix: Details of the calculation for ideal Gentile gases in the canonical ensemble In this appendix, we give some details for the calculation of the ideal Gentile gases in the canonical ensemble. A.1 The coefficient QI (q) The canonical partition function of ideal Gentile gases, Eq. (5.1), can be represented as a βε1 βε2 linear combination of the S-function (λ)I e− , e− , ... and the corresponding coefficient QI (q) is defined by Eq. (5.2). That is to say, the canonical partition function of ideal Gentile gases will be obtained once the coefficient QI (q) is known. In this Appendix, we show how to calculate the coefficient QI (q) for some given N. J (3) 1 (3) 1 N = 3. The Kostka number kK for N = 3 reads k(3) = k1 = 1, k(2,1) = k2 = 0, 3 (3) 1 (2,1) 2 (2,1) 2 (2,1) 2 (1 ) 3 k(13) = k3 = 0, k(3) = k1 = 1, k(2,1) = k2 = 1, k(13) = k3 = 0, k(3) = k1 = 1, 3 3 (1 ) 3 (1 ) 3 k(2,1) = k2 = 2, and k(13) = k3 = 1 [1, 2]. For clarity , we can rewrite the Kostka number in a matrix form: 1 0 0 k = 1 1 0 , (A.1) 1 2 1 J where we take the upper index of kK as a row index and the lower index as a column index. K For q = 3, by the definition of Γ (q), Eq. (5.3), we have Γ(3) (3) = Γ1 (3) = 1, K Γ(2,1) (3) = Γ2 (3) = 1, and Γ(13) (3) = Γ3 (3) = 1. For clarity, we express Γ (q) as 1 Γ (3) = 1 , (A.2) 1 where we take the upper index of ΓK (q) as a row index. Again we express Eq. (5.2) in the matrix form 1 Q (q)= k− Γ (q) , (A.3) where Q (q) denotes the vector form of QI (q) by taking the upper index as a row index, 1 1 J and k− is the matrix form of k− K . Then substituting Eqs. (A.1) and (A.2) into Eq. (A.3) gives 1 1 0 0 − 1 1 Q (3) = 1 1 0 1 = 0 . (A.4) 1 2 1 1 0 – 16 – 3 That is to say Q(3) (3) = Q1 (3) = 1, Q(2,1) (3) = Q2 (3) = 0, Q(1 ) (3) = Q3 (3) = 0. Therefore, when N = 3 and q = 3, Eq. (5.1) can be expressed as βε1 βε2 βε1 βε2 3 βε1 βε2 Z (β, 3) = 1 (3) e− , e− , ... + 0 (2, 1) e− , e− , ... + 0 1 e− , e− , ... 3 × × × βε 1 βε2 = (3) e− , e− , ... , (A.5) βε1 βε2 βε1 βε2 βε1 βε2 βε1 βε2 where (λ)1 e− , e− , ... = (3) e− , e− , ... , (λ)2 e− , e− , ... = (2, 1) e− , e− , ... , and (λ) e βε1 , e βε2 , ... = 13 e βε1 , e βε2 , ... . Eq. (A.5) is just the canonical partition 3 − − − − function for Gentile gases with the particle number N = 3 and the maximum occupation number q = 3. For q = 2, by Eq. (5.3), we have 0 Γ (2) = 1 . (A.6) 1 Substituting Eqs. (A.6) and (A.1) into Eq. (A.3) gives 1 1 0 0 − 0 0 Q (2) = 1 1 0 1 = 1 . (A.7) 1 2 1 1 1 − Substituting the coefficient Q (2), Eq. (A.7), into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 3 βε1 βε2 Z (β, 3) = 0 (3) e− , e− , ... + 1 (2, 1) e− , e− , ... 1 1 e− , e− , ... 2 × × − × βε1 βε2 3 βε1 βε2 = (2, 1) e− , e− , ... 1 e− , e− , ... . (A.8) − For q = 1, by Eq. (5.3), we have 0 Γ (1) = 0 . (A.9) 1 Substituting Eqs. (A.9) and (A.1) into Eq. (A.3) gives 1 1 0 0 − 0 0 Q (1) = 1 1 0 0 = 0 . (A.10) 1 2 1 1 1 Substituting the coefficient Q (1) given by Eq. (A.10) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 3 βε1 βε2 Z (β, 3) = 0 (3) e− , e− , ... + 0 (2, 1) e− , e− , ... + 1 1 e− , e− , ... 1 × × × 3 βε1 βε2 = 1 e− , e− , ... . (A.11) – 17 – N = 4. The Kostka number for N = 4 is [1, 2] 10000 11000 k = 11100 . (A.12) 12110 13231 For q = 4, by Eq. (5.3), we have 1 1 Γ (4) = 1 . (A.13) 1 1 Substituting Eqs. (A.12) and (A.13) into Eq. (A.3) gives 1 10000 − 1 1 11000 1 0 Q (4) = 11100 1 = 0 . (A.14) 12110 1 0 13231 1 0 Substituting the coefficient Q (4) given by Eq. (A.14) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 2 βε1 βε2 Z (4, β) = 1 (4) e− , e− , ... + 0 (3, 1) e− , e− , ... + 0 2 e− , e− , ... 4 × × × 2 βε1 βε2 5 βε1 βε2 + 0 2, 1 e− , e− , ... + 0 1 e− , e− , ... × × βε1 βε2 = (4) e− , e− , ... . (A.15) For q = 3, by Eq. (5.3), we have 0 1 Γ (3) = 1 . (A.16) 1 1 Substituting Eqs. (A.12) and (A.16) into Eq. (A.3) gives 1 10000 − 0 0 11000 1 1 Q (3) = 11100 1 = 0 . (A.17) 12110 1 1 − 13231 1 1 – 18 – Substituting the coefficient Q (3) given by (A.17) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 2 βε1 βε2 Z (β, 4) = 0 (4) e− , e− , ... + 1 (3, 1) e− , e− , ... + 0 2 e− , e− , ... 3 × × × 2 βε1 βε2 5 βε1 βε2 1 2, 1 e− , e− , ... + 1 1 e− , e− , ... − × × βε1 βε2 2 βε1 βε2 5 βε1 βε2 = (3, 1) e− , e− , ... 2, 1 e − , e− , ... + 1 e− , e− , ... . − (A.18) For q = 2, by Eq. (5.3), we have 0 0 Γ (2) = 1 . (A.19) 1 1 Substituting Eqs. (A.12) and (A.19) into Eq. (A.3) gives 1 10000 − 0 0 11000 0 0 Q (2) = 11100 1 = 1 . (A.20) 12110 1 0 13231 1 1 − Substituting the coefficient Q (2) given by Eq. (A.20) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 2 βε1 βε2 Z (β, 4) = 0 (4) e− , e− , ... + 0 (3, 1) e− , e− , ... + 1 2 e− , e− , ... 2 × × × 2 βε1 βε2 5 βε1 βε2 + 0 2, 1 e− , e− , ... 1 1 e− , e− , ... × − × 2 βε1 βε2 5 βε1 βε2 = 2 e− , e− , ... 1 e− , e− , ... . (A.21) − For q = 1, by Eq. (5.3), we have 0 0 Γ (1) = 0 . (A.22) 0 1 Substituting Eqs. (A.12) and (A.22) into Eq. (A.3) gives 1 10000 − 0 0 11000 0 0 Q (1) = 11100 0 = 0 . (A.23) 12110 0 0 13231 1 1 – 19 – Substituting the coefficient Q (2) given by Eq. (A.20) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 2 βε1 βε2 Z (β, 4) = 0 (4) e− , e− , ... + 0 (3, 1) e− , e− , ... + 0 2 e− , e− , ... 1 × × × 2 βε1 βε2 5 βε1 βε2 + 0 2, 1 e− , e− , ... + 1 1 e− , e− , ... × × 5 βε1 βε2 = 1 e− , e− , ... . (A.24) N = 5. The Kostka number for N = 5 is [1, 2] 1000000 1100000 1110000 k = 1211000 . (A.25) 1221100 1333210 1456541 For q = 5, by Eq. (5.3), we have 1 1 1 Γ (5) = 1 . (A.26) 1 1 1 Substituting Eqs. (A.25) and (A.26) into Eq. (A.3) gives 1 1000000 − 1 1 1100000 1 0 1110000 1 0 Q (5) = 1211000 1 = 0 . (A.27) 1221100 1 0 1333210 1 0 1456541 1 0 Substituting the coefficients Q (5) given by Eq. (A.27) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 βε1 βε2 Z (β, 5) = 1 (5) e− , e− , ... + 0 (4, 1) e− , e− , ... + 0 (3, 2) e− , e− , ... 5 × × × 2 βε1 βε2 2 βε1 βε2 3 βε1 βε2 + 0 3, 1 e− , e− , ... + 0 2 , 1 e− , e− , ... + 0 2, 1 e− , e− , ... × × × 5 βε1 βε2 + 0 1 e− , e− , ... × βε1 βε2 = (5) e− , e− , ... . (A.28) – 20 – For q = 4, by Eq. (5.3), we have 0 1 1 Γ (4) = 1 . (A.29) 1 1 1 Substituting Eqs. (A.25) and (A.29) into Eq. (A.3) gives 1 1000000 − 0 0 1100000 1 1 1110000 1 0 Q (4) = 1211000 1 = 1 . (A.30) − 1221100 1 0 1333210 1 1 1456541 1 1 − Substituting the coefficient Q (4) given by Eq. (A.30) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 βε1 βε2 Z (β, 5) = 0 (5) e− , e− , ... + 1 (4, 1) e− , e− , ... + 0 (3, 2) e− , e− , ... 4 × × × 2 βε1 βε2 2 βε1 βε2 3 βε1 βε2 1 3, 1 e− , e− , ... + 0 2 , 1 e− , e− , ... + 1 2, 1 e− , e− , ... − × × × 5 βε1 βε2 1 1 e− , e− , ... − × βε1 βε2 2 βε1 βε2 = (4, 1) e− , e− , ... 3, 1 e− , e− , ... − 3 βε1 βε2 5 βε1 βε2 + 2, 1 e− , e− , ... 1 e− , e− , ... . (A.31) − For q = 3, by Eq. (5.3), we have 0 0 1 Γ (3) = 1 . (A.32) 1 1 1 – 21 – Substituting Eqs. (A.25) and (A.32) into Eq. (A.3) gives 1 1000000 − 0 0 1100000 0 0 1110000 1 1 Q (3) = 1211000 1 = 0 . (A.33) 1221100 1 1 − 1333210 1 0 1456541 1 1 Substituting the coefficients Q (3) given by Eq. (A.33) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 βε1 βε2 Z (β, 5) = 0 (5) e− , e− , ... + 0 (4, 1) e− , e− , ... + 1 (3, 2) e− , e− , ... 3 × × × 2 βε1 βε2 2 βε1 βε2 3 βε1 βε2 + 0 3, 1 e− , e− , ... 1 2 , 1 e− , e− , ... + 0 2, 1 e− , e− , ... × − × × 5 βε1 βε2 + 1 1 e− , e− , ... × βε1 βε2 2 βε1 βε2 3 βε1 βε2 = (3, 2) e− , e− , ... 2 , 1 e− , e− , ... + 2, 1 e− , e− , ... − 5 βε1 βε2 + 1 e− , e− , ... . (A.34) For q = 2, by Eq. (5.3), we have 0 0 0 Γ (2) = 0 . (A.35) 1 1 1 Substituting Eqs. (A.25) and (A.35) into Eq. (A.3) gives 1 1000000 − 0 0 1100000 0 0 1110000 0 0 Q (2) = 1211000 0 = 0 . (A.36) 1221100 1 1 1333210 1 1 − 1456541 1 0 Substituting the coefficients Q (2) given by Eq. (A.36) into Eq. (5.1) gives the canonical – 22 – partition function βε1 βε2 βε1 βε2 βε1 βε2 Z (β, 5) = 0 (5) e− , e− , ... + 0 (4, 1) e− , e− , ... + 0 (3, 2) e− , e− , ... 2 × × × 2 βε1 βε2 2 βε1 βε2 3 βε1 βε2 + 0 3, 1 e− , e− , ... + 1 2 , 1 e− , e− , ... 1 2, 1 e− , e− , ... × × − × 5 βε1 βε2 + 0 1 e− , e− , ... × 2 βε1 βε2 3 βε1 βε2 = 2 , 1 e− , e− , ... 1 2, 1 e− , e− , ... . (A.37) − × For q = 1, by Eq. (5.3), we have 0 0 0 Γ (1) = 0 . (A.38) 0 0 1 Substituting Eqs. (A.25) and (A.38) into Eq. (A.3) gives 1 1000000 − 0 0 1100000 0 0 1110000 0 0 Q (1) = 1211000 0 = 0 . (A.39) 1221100 0 0 1333210 0 0 1456541 1 1 Substituting the coefficients Q(2) given by Eq. (A.36 ) into Eq. (5.1) gives the canonical partition function βε1 βε2 βε1 βε2 βε1 βε2 Z (β, 5) = 0 (5) e− , e− , ... + 0 (4, 1) e− , e− , ... + 0 (3, 2) e− , e− , ... 1 × × × 2 βε1 βε2 2 βε1 βε2 3 βε1 βε2 + 0 3, 1 e− , e− , ... + 0 2 , 1 e− , e− , ... + 0 2, 1 e− , e− , ... × × × 5 βε1 βε2 + 1 1 e− , e− , ... × 5 βε1 βε2 = 1 e− , e− , ... . (A.40) A.2 The canonical partition function In Appendix A.1, we show how to calculate the coefficients QI (q) from Eq. (5.2). The canonical partition function of ideal Gentile gases can be represented as a linear combination of the S-functions provided the coefficients QI (q) are given. In this section, we calculate the canonical partition function and express the canonical partition function in terms of the single-particle partition function defined by Eq. (3.8), Z (β). – 23 – K N = 3. The simple characteristics χI of the permutation group S3 for N = 3 is [1] 1 1 1 χ = 2 0 1 , (A.41) − 1 1 1 − K where χI is expressed in the matrix form by taking the upper index as the row index K and the lower index as the column index. Substituting the simple characteristics χI , Eq. (A.41), into the definition of the S-function, Eq. (2.3), gives the expression of S- βε1 βε2 functions, (λ)I e− , e− , ... , in terms of the single-particle partition functions Z (β). Then substituting the S-function (λ) e βε1 , e βε2 , ... into the canonical partition func- I − − tion, Eq. (A.5), gives the expression of the canonical partition function in terms of the single-particle partition function Z (β): βε1 βε2 3 βε1 βε2 Z (β, 3) = (2, 1) e− , e− , ... 1 e− , e− , ... 2 − 1 1 2 = Z (β)3 + Z (β) Z (2β) Z (3β) . (A.42) 3! 2 − 3 K N = 4. The simple characteristic χI of the permutation group S4 for N = 4 is [1] 11 1 1 1 3 0 1 1 1 − − χ = 2 12 0 0 . (A.43) − 3 0 1 1 1 − − 1 1 1 1 1 − − Substituting the simple characteristics (A.43) into the S-function in Eqs. (A.21) and (A.18), respectively, gives 2 βε1 βε2 4 βε1 βε2 Z (β, 4) = 2 e− , e− , ... 1 e− , e− , ... 2 − 1 1 1 2 1 = Z(β)4 + Z (β)2 Z (2β )+ Z (2β)2 Z (3β) Z (β)+ Z (4β) (A.44) 4! 4 8 − 3 4 and βε1 βε2 2 βε1 βε2 4 βε1 βε2 Z (β, 4) = (3, 1) e− , e− , ... 2, 1 e− , e− , ... + 1 e− , e− , ... 3 − 1 1 1 1 3 = Z (β)4 + Z (β)2 Z (2β)+ Z (2β)2 + Z (3β) Z (β ) Z (4β) . (A.45) 4! 4 8 3 − 4 K N = 5. The simple characteristic χI of the permutation group S5 for N = 5 is [1] 1111111 1 0 11 0 14 − − 0 1 1 1 1 1 5 − − − χ = 1 0 0 0 2 0 6 . (A.46) − 0 1 1 1 1 1 5 − − − 10 1 1 0 14 − 1 1 11 1 11 − − – 24 – Substituting the simple characteristics (A.46) into the S-functions in Eqs. (A.37), (A.34), and (A.34), respectively, gives 1 Z (β, 5) = Z (β)5 2 5! 1 1 + Z (β)3 Z (2β)+ Z (β) Z (2β)2 (A.47) 12 8 1 1 1 1 Z (β)2 Z (3β) Z (2β) Z (3β)+ Z (β) Z (4β)+ Z (5β) , (A.48) − 3 − 3 4 5 1 Z (β, 5) = Z (β)5 3 5! 1 1 + Z (β)3 Z (2β)+ Z (β) Z (2β)2 (A.49) 12 8 1 1 3 1 + Z (β)2 Z (3β)+ Z (2β) Z (3β) Z (β) Z (4β)+ Z (5β) (A.50) 6 6 − 4 5 and 1 1 1 Z (β, 5) = Z (β)5 + Z (β)3 Z (2β)+ Z (β) Z (2β)2 4 5! 12 8 1 1 1 4 + Z (β)2 Z (3β)+ Z (2β) Z (3β)+ Z (β) Z (4β) Z (5β) . 6 6 4 − 5 (A.51) K N = 6. The simple characteristics χI of the permutation group S6 for N = 6 can be found in Ref. [1]. A similar procedure gives 1 Z (β, 6) = Z (β)6 2 6! 1 1 + Z (β)4 Z (2β)+ Z (β)2 Z (2β)2 48 16 1 1 1 + Z (2β)3 Z (β)3 Z (3β) Z (β) Z (2β) Z (3β) 48 − 9 − 3 2 1 1 1 1 + Z (3β)2 + Z (β)2 Z (4β)+ Z (2β) Z (4β)+ Z (5β) Z (β) Z (6β) , 9 8 8 5 − 3 (A.52) 1 Z (β, 6) = Z (β)6 3 6! 1 1 + Z (β)4 Z (2β)+ Z (β)2 Z (2β)2 48 16 1 1 1 + Z (2β)3 + Z (β)3 Z (3β)+ Z (β) Z (2β) Z (3β) 48 18 6 1 3 3 1 1 + Z (3β)2 Z (β)2 Z (4β) Z (2β) Z (4β)+ Z (5β) Z (β)+ Z (6β) , 18 − 8 − 8 5 6 (A.53) – 25 – 1 Z (β, 6) = z (β)6 4 6! 1 1 + z (β)4 z (2β)+ z (β)2 z (2β)2 48 16 1 1 1 + z (2β)3 + z (β)3 z (3β)+ z (β) z (2β) z (3β) 48 18 6 1 1 1 4 1 + z (3β)2 + z (β)2 z (4β)+ z (2β) z (4β) z (5β) z (β)+ z (6β) , (A.54) 18 8 8 − 5 6 and 1 Z (β, 6) = Z (β)6 5 6! 1 1 + Z (β)4 Z (2β)+ Z (β)2 Z (2β)2 48 16 1 1 1 + Z (2β)3 + Z (β)3 Z (3β)+ Z (β) Z (2β) Z (3β) 48 18 6 1 1 1 1 5 + Z (3β)2 + Z (β)2 Z (4β)+ Z (2β) Z (4β)+ Z (5β) Z (β) Z (6β) . 18 8 8 5 − 6 (A.55) K The simple characteristics χI of the permutation group S6 is not listed in the paper, but it can be found in Ref. [1]. The calculation of the canonical partition function for Gentile gases for N = 6 are not listed in this paper, but can be calculated by the method provided in this paper. From the results, Eqs. (A.42), (A.44), (A.45), (A.48), (A.50), (A.51), (A.52), (A.53), (A.54), and (A.55), one can see that the canonical partition functions Zq (β, N) of ideal 1 N Gentile gases can be written in the form of Zq (β, N) = N! Z (β) + corrections, where Z (β)N is the canonical partition function of ideal classical gases with N particles and the correction depends on both N and q. A.3 The virial coefficients In this section, we calculate the virial coefficients from the canonical partition function. The exact canonical partition function of ideal Gentile gases can be obtained by sub- stituting the single-particle partition function Z (β), Eq. (3.9), into the canonical partition functions, Eqs. (A.42), (A.44), (A.45), (A.48), (A.50), (A.51), (A.52), (A.53), (A.54), and (A.55). For q = 2, 1 V 3 1 V 2 2 V Z (β, 3) = + , (A.56) 2 6 λ3 √ λ3 − √ λ3 4 2 9 3 1 V 4 1 V 3 1 V 2 2 V 2 1 V Z (β, 4) = + + + , (A.57) 2 24 λ3 √ λ3 64 λ3 − √ λ3 32 λ3 8 2 9 3 1 V 5 1 V 4 1 V 3 Z (β, 5) = + + 2 120 λ3 √ λ3 64 λ3 24 2 1 V 3 1 V 2 1 V 2 1 V + + , (A.58) − √ λ3 32 λ3 − √ λ3 √ λ3 9 3 18 6 25 5 – 26 – 1 V 6 1 V 5 1 V 4 1 V 4 Z2 (β, 6) = + + 720 λ3 96√2 λ3 128 λ3 − 27√3 λ3 1 V 3 1 V 3 1 V 3 2 V 2 + + 64 λ3 − 768√2 λ3 − 18√6 λ3 243 λ3 1 V 2 1 V 2 1 V 2 + + . (A.59) 128√2 λ3 25√5 λ3 − 18√6 λ3 Substituting Eqs. (A.56-A.59) into the equation of state (8.3) respectively gives the virial coefficients al listed in Table (1). For q = 3, 1 V 3 1 V 2 1 V Z3 (β, 3) = + + , (A.60) 6 λ3 4√2 λ3 9√3 λ3 1 V 4 1 V 3 1 V 2 Z3 (β, 4) = + + 24 λ3 8√2 λ3 64 λ3 1 V 2 3 V + , (A.61) 9√3 λ3 − 32 λ3 1 V 5 1 V 4 1 V 3 Z3 (β, 5) = + + 120 λ3 24√2 λ3 64 λ3 1 V 3 3 V 2 1 V 2 1 V + + + , (A.62) √ λ3 − 32 λ3 √ λ3 √ λ3 18 3 36 6 25 5 1 V 6 1 V 5 1 V 4 1 V 4 Z3 (β, 6) = + + + 720 λ3 96√2 λ3 128 λ3 54√3 λ3 3 V 3 1 V 3 1 V 3 1 V 2 + + + − 64 λ3 √ λ3 √ λ3 486 λ3 768 2 36 6 3 V 2 1 V 2 1 V 2 + + . (A.63) − √ λ3 √ λ3 √ λ3 128 2 25 5 36 6 Substituting Eqs. (A.60)-(A.63) into the equation of state (8.3) respectively gives the virial coefficients al listed in Table (2). For q = 4, 1 V 4 1 V 3 1 V 2 Z (β, 4) = + + 4 24 λ3 √ λ3 64 λ3 8 2 1 V 2 1 V + + , (A.64) √ λ3 32 λ3 9 3 1 V 5 1 V 4 1 V 3 Z (β, 5) = + + 4 120 λ3 √ λ3 64 λ3 24 2 1 V 3 1 V 2 1 V 2 4 V + + + , (A.65) √ λ3 32 λ3 √ λ3 − √ λ3 18 3 36 6 25 5 – 27 – 1 V 6 1 V 5 1 V 4 1 V 4 Z4 (β, 6) = + + + 720 λ3 96√2 λ3 128 λ3 54√3 λ3 1 V 3 1 V 3 1 V 3 1 V 2 + + + + 64 λ3 768√2 λ3 36√6 λ3 486 λ3 1 V 2 4 V 2 1 V 2 + + . (A.66) 128√2 λ3 − 25√5 λ3 36√6 λ3 Substituting Eqs. (A.64-A.66) into the equation of state (8.3) respectively gives the virial coefficients al listed in Table (3). For q = 5, 1 V 5 1 V 4 1 V 3 Z (β, 5) = + + 5 120 λ3 √ λ3 64 λ3 24 2 1 V 3 1 V 2 1 V 2 1 V + + + + , (A.67) √ λ3 32 λ3 √ λ3 √ λ3 18 3 36 6 25 5 1 V 6 1 V 5 1 V 4 1 V 4 Z (β, 6) = + + + 5 720 λ3 √ λ3 128 λ3 √ λ3 96 2 54 3 1 V 3 1 V 3 1 V 3 1 V 2 + + + + 64 λ3 √ λ3 √ λ3 486 λ3 768 2 36 6 1 V 2 1 V 2 5 V 2 + + . (A.68) √ λ3 √ λ3 − √ λ3 128 2 25 5 36 6 Substituting Eqs. (A.67) and (A.68) into Eq. (8.3) respectively gives the virial coefficients al listed in Table (4). For q = 6, 1 V 6 1 V 5 1 V 4 1 V 4 Z6 (β, 6) = + + + 720 λ3 96√2 λ3 128 λ3 54√3 λ3 1 V 3 1 V 3 1 V 3 1 V 2 + + + + 64 λ3 768√2 λ3 36√6 λ3 486 λ3 1 V 2 1 V 2 1 V 2 + + + . (A.69) √ λ3 √ λ3 √ λ3 128 2 25 5 36 6 Substituting Eq (A.69) into the equation of state (8.3) gives the virial coefficients al listed in Table (5). Acknowledgments We are very indebted to Dr G. Zeitrauman for his encouragement. This work is supported in part by NSF of China under Grant No. 11575125 and No. 11675119. – 28 – References [1] D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups: Second Edition. AMS CHELSEA PUB, 2005. [2] I. Macdonald, Symmetric Functions and Hall Polynomials. Oxford classic texts in the physical sciences. Clarendon Press, 1998. [3] G. Andrews, The Theory of Partitions. Cambridge mathematical library. Cambridge University Press, 1998. [4] F. S. 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