Fermi-Dirac statistics and the number theory

Europhysics Letters

PREPRINT

Fermi-Dirac statistics and the number theory Anna Kubasiak 1 , Jaroslaw K. Korbicz 2 , Jakub Zakrzewski 1 , and Maciej Lewenstein 2,3,∗

arXiv:quant-ph/0507201v2 27 Jul 2005

1

Instytut Fizyki im. M. Smoluchowskiego, Uniwersytet Jagiello´ nski, PL-30059 Krak´ ow, Poland 2 Institut f¨ ur Theoretische Physik, Universit¨ at Hannover, D-30167 Hannover, Germany 3 ICFO-Institut de Ci`encies Fot` oniques, E-08034 Barcelona, Spain

PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. PACS. 02.30.Mv – Approximations and expansions.

Abstract. – We relate the Fermi-Dirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic expressions for cumulants of the probability distribution of the number of different partitions.

Introduction. – Standard textbooks use grand canonical ensemble to describe ideal quantum gases [1]. It has been pointed out by Grossmann and Holthaus [2], and Wilkens et al. [3] that such an approach leads to pathological results for fluctuations of particle numbers in Bose-Einstein condensates (BEC). This has stimulated intensive studies of BEC fluctations using canonical and microcanical ensembles for ideal [4, 5] and weakly interacting [6] gases. Grossmann and Holthaus [2] related the problem for an ideal Bose gas in an harmonic trap to the number theoretical studies of partitions of an integer E into M integers [7], and to the famous Hardy-Ramanujan formula [8]. In the series of beautiful papers, they have been able to apply methods of quantum statistics to derive non-trivial number theoretical results (see [9] and references therein). In this Letter we apply the approach of Ref. [9] to Fermi gases, and relate the problem of an ideal Fermi gas in an harmonic trap to studies of partitions of E into M distinct integers. We show that the probability distribution of the number of different partitions is asymptotically Gaussian and we calculate analytically its cumulants. Our results are complementary to those of Tran [10], who has calculate particle number fluctuations in the ground state (Fermi sea) for the fixed energy and number of particles. Ideal Fermi gas. – We consider a microcanonical ensemble of N thermally isolated, spinless, non-interacting fermions, trapped in a harmonic potential with frequency ω0 . The discrete one-particle states have energies ǫν = ~ω0 (ν + 1/2), ν = 0, 1, . . . . A microstate of the system is described by the sequence of the occupation numbers {nν }, where nν = 0, 1 and P∞ n = N . The total energy of a microstate {nν } can be written as: ν ν=0 E({nν }) = N ǫ0 + ~ω0 E ,

c EDP Sciences

(1)

2

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where integer: E=

∞ X

nν ν

(2)

ν=1

determines the contribution to E from one-particle excited states. Note, that E is a sum of M distinct integers, where M = N − 1 if n0 = 1, or M = N otherwise. Hence, in order to construct the microcanonical partition function Γ(E, N ), one has to calculate the number of partitions of the given integer E into M distinct parts. This, in turn, is a standard e problem in combinatorics [11]. If we call the number of distinct partitions Φ(E, M ), then e Γ(E, N ) = Φ(E, M ) and the physical problem of finding Γ(E, N ) is mapped to a combinatorial one. Introduce the total number of distinct decompositions: e Ω(E) =

M max X M=1

e Φ(E, M ),

(3)

and consider the probability (relative frequency): pmc (E, M ) =

e Φ(E, M) , e Ω(E)

(4)

that exactly M distinct terms occur in the decomposition of E. This is the central object of our study. In the case of bosons pmc (E, M ) possesses clear physical interpretation, as it was explained in Ref. [9]. This is the probability of finding M excited particles, when the total excitation energy is ~ω0 E. For fermions, however, the ground state of the whole system corresponds to the filled Fermi sea. Since the number of partitions M differs at most by 1 from the number of particles N , we may regard (4) as a probability density in the fictitious Maxwell demon ensemble [5] with vanishing chemical potential. The distribution (4) describes at the same time an interesting mathematical problem, which, up to our knowledge, has never been treated using physical methods. e Using standard results from combinatorics [11], the exact expression for Φ(E, M ) can be written down:     M e Φ(E, M) = P E − ,M , (5) 2 in terms of a recursively defined function P :   0 for k = 0 or k > n P (n − 1, k − 1) + P (n − k, k) . P (n, k) =  1 for k = n

(6)

The above recursion enables straightforward, although tedious, numerical studies of the distribution (4) (see Fig. 1).

Canonical distribution. – To proceed further with the analytical study of pmc (E, M ) we introduce the corresponding canonical description. Instead of fixing the energy E of the each member of the ensemble, we consider an ensemble of systems in a thermal equilibrium with a heat bath of temperature T . The canonical partition function Z(β, N ), β = 1/kT , generating the statistical description in this case, can be written using (1) as: Z(β, N ) =

X E

e−βE Γ(E, N ) = e−βN ǫ0

∞ X

E=0

e e−~ω0 βE Φ(E, M) ,

(7)

Anna Kubasiak, Jaroslaw K. Korbicz, Jakub Zakrzewski, and Maciej Lewenstein: Fermi-Dirac statistics and the number

pmc(ε,M)

0.1

0

20

40

60

M

Fig. 1 – The distribution pmc (E , M ) for E = 1000, 2000, 3000, and 4500.

where E = 0 term vanishes. Introducing b = ~ω0 β we can define a combinatorial partition function: ∞ X e Zc (b, M ) = e−bE Φ(E, M) , (8) E=1

e which can be viewed as the generating function of the microcanonical quantities Φ(E, M ). The canonical probability, corresponding to (4), is then given by: Zc (b, M ) pcn (b, M ) = P∞ , −bE Ω(E) e E=1 e

(9)

A convenient and physically relevant way of an analytical description of the distribution (9) is provided by the cumulants. For a probability distribution {p(M ), M ≥ 0} of an integer-valued random variable, kth cumulant κ(k) is defined as [12]: k  d (k) ln pˆ(z) , (10) κ = z dz z=1 P where pˆ(z) = M e−Mz p(M ) is the (real) characteristic function of the distribution. The relation of the first few culumants to the central moments µ(k) is the following: ¯ κ(1) = µ(1) = M κ(2) = µ(2) κ(3) = µ(3) κ(4) = µ(4) − 3(µ(2) )2 , (11) P P ∞ M ¯ = where M M M p(M ) is the mean value. Denote as Ξc (b, z) = M=0 z Zc (b, M ) = Q ∞ −bν ) the grand partition function, corresponding to the canonical partition funcν=0 (1 + ze tion (8) with z being an analog of fugacity. We are interested in excited part so we take not Ξc (b, z) but rather Ξex (b, z). Ξex (b, z) =

∞ Y

ν=1


1 + ze−bν .

(12)

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We substitute (9) in to the definition (10) and obtain:  k ∂ (k) κcn (b) = z ln Ξex (b, z) . ∂z z=1

(13)

Now, we can apply the procedure of Ref. [9] in order to obtain a compact expression for (k) the cumulants κcn (b) as well as their asymptotic behavior for b ≪ 1. First, we note that: ln Ξex (b, z) =

∞ X

ν=1


ln 1 + ze−bν .

(14)

and we can always find such a neighborhood of z = 1 that ze−b < 1. Since, according to (13), we are only interested in the behavior of Ξex in the vicinity of z = 1, the logarithms in the second term can be Taylor expanded; using then the Mellin transformation of the Euler gamma function Γ(t) [13], we obtain the following integral representation of ln Ξex : Z τ +i∞ 1 ln Ξex (b, z) = dt b−t ζ(t)Γ(t)ft+1 (z) , (15) 2πi τ −i∞ where ζ(t) is the Riemann zeta function [12], and fα (z) =

∞ X (−1)n z n . nα n=0

(16)

is the fermionic function [1]. The parameter τ > 0 was chosen in such a way, that the integration line in (15) is to the right from the last pole of the integrand. Finally, we substitute Eq. (15) into (13) and with the help of the following properties of the fermionic function fα : z dfα (z)/dz = fα−1 (z), fα (1) = (1 − 21−α )ζ(α), we find that: (k)

(k)

k κ(k) cn (b) = κcn,boson (b) − 2 κcn,boson (2b) ,

where (k) κcn,boson (b)

1 = 2πi

Z

(17)

τ +i∞

τ −i∞

dt b−t Γ(t)ζ(t)ζ(t + 1 − k)

(18)

are the canonical cumulants of the bosonic counterpart of the distribution (9). Equation (17) is of central importance for this letter. It links the fermionic cumulants to that corresponding for bosons. The latter has been studied in detail in Ref. [9]. In particular [9] gives the asymptotic expressions of the first few bosonic cumulants for b ≪ 1. Using those together with (17-18) allow us to find the asymptotic expressions for the cumulants of the (0) (1) fermionic canonical distribution (9) : κcn (b) = π 2 /12b + ln 2/2 + b/24 + O(b15.5 ), κcn (b) = (3) (2) ln 2/b − 1/4 + b/48 + O(b2.5 ), κcn (b) = 1/2b − 1/8 + O(b14.5 ), κcn (b) = 1/4b − b/96 + O(b2.5 ), (4) κcn (b) = −1/16 + O(b11.5 ), etc. The dependence on b in the fermionic case is of lower order (4) than the correspponding one in the bosonic case. For example κcn is constant up to the terms 11.5 of the order b . Microcanonical distribution. – We come back to the study of the distribution (4). Our (k) aim now is to find the cumulants κmc (E) of this distribution. Applying the definition (10) to (4) we find that:  k ∂ κ(k) (E) = z ln Υ(E, z) , (19) mc ∂z z=1

Anna Kubasiak, Jaroslaw K. Korbicz, Jakub Zakrzewski, and Maciej Lewenstein: Fermi-Dirac statistics and the number

where: ln Υ(E, z) =

∞ X

M=0

e z M Φ(E, M)

(20)

e is another generating function of Φ(E, M ), complementary to Zc (b, M ). In order to calculate Υ(E, z), we first note, that: Ξex (b, z) =

∞ X

zM

M=0

∞ X

E=1

e e−bE Φ(E, M) =

∞ X

e−bE Υ(E, z) .

e ex (x, z) we obtain: Introducing a new variable x = e−b and witting Ξex (− ln x, z) = Ξ e ex (x, z) = Ξ

∞ X

(21)

E=1

xE Υ(E, z) ,

(22)

E=1

e ex (x, z) w.r.t. x. We that is, Υ(E, z) are the coefficients in the power series expansion of Ξ treat x as a complex variable and use an integral identity: I

e ex (x, z) Ξ , xE+1

(23)

∂ ln Ξex (b0 (z), z) , ∂b

(24)

1 Υ(E, z) = 2πi

dx

where the integration contour is any closed loop surrounding the origin, oriented anti-clockwise. The integral in (23) can be approximately evaluated using the saddle point method similar to that used in Refs. [1, 9]. The implicit equation for the saddle point x0 (z) is: E +1=−

with b0 (z) = − ln x0 (z). From Eq. (24) we obtain in the Gaussian approximation: ln Υ(E, z) ≈ ln Ξex (b0 (z), z) + Eb0 (z) −

1 1 ln 2π − ln 2 2

2   ∂ ln Ξex (b0 (z), z) . − ∂b

(25)

The above solution for Υ(E, z) is of course similar to the one obtained in a bosonic case in Ref. [9], the difference being just in the form of the grand canonical partition function Ξex . The microcanonical cumulants are then calculated from (25) according to the equation (19): k  ∂ db0 (z) ∂ (k) κmc (E) = z +z ln Υ(E, z) , (26) ∂z dz ∂b0 z=1 (1) (0) where from Eq. (24) db0 (z)/dz = −∂b κcn (b0 (1))/∂b2 κcn (b0 (1)), where in the last expresz=1

sion we used the definitions of the canonical cumulants (13). Generally, from (13),(25) and (k) (13) it follows that the microcanonical cumulants κmc (E) can be expressed in terms of the (k) canonical cumulants κcn (b) and their derivatives w.r.t. the temperature parameter b. For (1) example for the first cumulant κmc one obtains: ! (1) 2 (1) 3 (0) 1 ∂ κ (b (1)) ∂ κ (b (1)) 1 ∂ κ (b (1)) cn cn cn 0 0 b 0 (1) b b κ(1) 1+ . (27) + mc (E) = κcn (b0 (1)) − (0) 2 ∂ 2 κ(0) 2 ∂ 2 κ(0) ∂ 2 κcn (b0 (1)) cn (b0 (1)) cn (b0 (1)) b

b

b

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For b ≪ 1 we can use calculate the first few microcanonical cumulants explicitly: √ 1 2 3 ln 2 √ 1 3 ln 2 κ(1) (E) = E − + 2 + O(E − 2 ) (28) mc π 4 π √ √ 36(ln 2)2 − 23 π 2 1 1 3 2 2 − π + 12(ln 2) − + O(E − 2 ) κ(2) (E) = − (29) E − mc 3 4 π 8 √π √ 3 2 4 3 E − 36π ln 2 + π + 432(ln 2) κ(3) mc (E) = π5 3 4 π − 54π 2 + 864(ln 2)3 1 +4 (30) + O(E − 2 ) 6 π √ √ 3 3 4 4π ln 2 + 2160(ln 2)4 − 216π 2 (ln 2)2 + 3π 4 E κ(4) mc (E) = − 7 π i 1 1 h1 8 π + 31104(ln 2)4 + 27π 4 − 2592π 2(ln 2)2 + 36π 4 ln 2 + O(E − 2 ) .(31) + 8 π 16 Using (28-31) we obtain analytic expressions for the skewness and excess [12] of the distri3 (2) (4) (2) (3) bution (4) γ1 (E) = κmc (E)/(κmc (E)) 2 , γ2 (E) = κmc (E)/(κmc (E))2 ; these parameters measure the deviation of the distribution (4) from the Gaussian one. We obtain: 3 0.12894 √ + O(E − 4 ) 4 E 1.2001 + O(E −1 ) , γ2 (E) = − √ E

γ1 (E) = −

(32)

and hence in the limit E → ∞ both γ1 and γ2 vanish, implying that for large integers E the distribution (4) approaches Gaussian. This is a different situation from the bosonic case, where, as shown in Ref. [9], both skewness and excess are non-zero in the limit of large integers, so that bosonic microcanonical distribution does not approach Gaussian. In Fig.2 we compare the numerically obtained dependence of the skewness and excess with the analytical predictions of Eq. (32). Summarizing, we have studied relations between the Fermi-Dirac statistics for an ideal, harmonically trapped Fermi gas, and the theory of partitions of integers into distinct parts. Using methods of quantum statistical physics, we have described analytically properties of the probability distribution of the number of different partitions. We anticipate that the results concerning microcanonical ensemble for fermions can be used to characterize degeneracies of fermionic many body states, that play an essential role in the process of laser cooling of small samples of atoms in microtraps in the so called Lamb-Dicke limit [14]. We acknowledge discussions with M. Holzmann, and support from the Deutsche Forschungsgemeinschaft (SFB 407, 432 POL), EU IP ”SCALA” and the Socrates Programme. This work was supported (A.K.) by Polish government scientific funds (2005-2008) as a research grant. Support of Polish Scientific funds PBZ-MIN-008/P03/2003 is also acknowledged (J.Z.). REFERENCES [*] [1] [2] [3]

Also at Instituci´ o Catalana de Recerca i Estudis Avan¸cats. K. Huang, Statistical mechanics, (Wiley and Sons), 1963. S. Grossmann and M. Holthaus, Phys. Rev. E, 54 (3495-3498) , (1996). M. Wilkens and C. Weiss, J. Mod. Opt., 44 (1801-1814) , (1997).

Anna Kubasiak, Jaroslaw K. Korbicz, Jakub Zakrzewski, and Maciej Lewenstein: Fermi-Dirac statistics and the number

-0.01

γ1 -0.02

-0.03

γ2

0

-0.05 -0.1

-0.04

-0.15 -0.2

-0.05 0

1000

0

2000

1000

2000

3000

3000

4000

ε

4000

5000

ε

5000

Fig. 2 – The skewness γ1 obtained from numerically generated cumulants as compared with the asymptotic analytic prediction (dashed line). The insert shows the comparison for the excess γ2 . Observe that the analytical asymptotics (dashed line) practically coincides with the numerical data.

[4] M. Gajda and K. Rza ¸ z˙ ewski, Phys. Rev. Lett., 78 (2686-2689) , (1997); S. Grossman and M. Holthaus, Phys. Rev. Lett., 79 (3557-3560) , (1997); M. Holthaus M, E. Kalinowski E, and K. Kirsten, Ann. Phys. , 270 (198-230) , (1998); M. N. Tran, M. V. N. Murthy, and R. K. Bhaduri, Ann. Phys., 311 (204-219) , (2004). [5] P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rza ¸ z˙ ewski, Phys. Rev. Lett., 79 (1789-1792) , (1997). [6] S. Giorgini S, L.P. Pitaevskii, and S. Stringari, Phys. Rev. Lett., 80 (5040-5043) , (1998); Z. Idziaszek, M. Gajda, P. Navez, M. Wilkens, and K. Rza ¸ z˙ ewski, Phys. Rev. Lett., 82 (4376-4379) , (1999); Z. Idziaszek, Phys. Rev. A, 71 (053604) , (2005); [7] G.E. Andrews, The Theory of Partitions (Encyclopedia of Mathematics and its Applications), Vol. 2, (Addison-Wesley), 1976. [8] G. H. Hardy, S. Ramanujan (Editor), Proc. Lond. Math. Soc., 17 (75) , (1918). [9] M. Holthaus, K.T. Kapale, V.V. Kocharovsky, and MO Scully, Physica A, 300 (433467) , (2001); C. Weiss and M. Holthaus, Europhys. Lett., 59 (486-492) , (2002); C. Weiss, M. Block, M. Holthaus, and G. Schmieder, J. Phys. A, 36 (1827) , (2003). [10] M.N. Tran, J. Phys. A, 36 (961) , (2003). [11] H. Rademacher, Topics in Analytic Number Theory, (Springer, Berlin), 1973. [12] M. Abramowitz and I. Stegun (ed.), Handbook of mathematical function, (Dover Publications, New York), 1972. elyi (ed.), Tables of integral transforms, (McGraw-Hill, New York), 1954. [13] A. Erd´ [14] M. Lewenstein M, J.I. Cirac, and L. Santos, J. Phys. A, 33 (4107-4129) , (2000).