PHYSICAL REVIEW E 80, 011126 共2009兲
Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: Exact and interpolation approaches Hideo Hasegawa* Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan 共Received 13 April 2009; revised manuscript received 24 June 2009; published 21 July 2009兲 Generalized Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics have been discussed by the maximum-entropy method 共MEM兲 with the optimum Lagrange multiplier based on the exact integral representation 关A. K. Rajagopal, R. S. Mendes, and E. K. Lenzi, Phys. Rev. Lett. 80, 3907 共1998兲兴. It has been shown that the 共q − 1兲 expansion in the exact approach agrees with the result obtained by the asymptotic approach valid for O共q − 1兲. Model calculations have been made with a uniform density of states for electrons and with the Debye model for phonons. Based on the result of the exact approach, we have proposed the interpolation approximation to the generalized distributions, which yields results in agreement with the exact approach within O共q − 1兲 and in high- and low-temperature limits. By using the four methods of the exact, interpolation, factorization, and superstatistical approaches, we have calculated coefficients in the generalized Sommerfeld expansion and electronic and phonon specific heats at low temperatures. A comparison among the four methods has shown that the interpolation approximation is potentially useful in the nonextensive quantum statistics. Supplementary discussions have been made on the 共q − 1兲 expansion of the generalized distributions based on the exact approach with the use of the un-normalized MEM, whose results also agree with those of the asymptotic approach. DOI: 10.1103/PhysRevE.80.011126
PACS number共s兲: 05.30.⫺d, 05.70.Ce
I. INTRODUCTION
In the last decade, many studies have been made for the nonextensive statistics 关1兴 in which the generalized entropy 共the Tsallis entropy兲 is introduced 共for a recent review, see 关2兴兲. The Tsallis entropy is a one-parameter generalization of the Boltzmann-Gibbs entropy with the entropic index q: the Tsallis entropy in the limit of q = 1.0 reduces to the Boltzmann-Gibbs entropy. The optimum probability distribution or density matrix is obtained with the maximum-entropy method 共MEM兲 for the Tsallis entropy with some constraints. At the moment, there are four possible MEMs: original method 关1兴, un-normalized method 关3兴, normalized method 关4兴, and optimal Lagrange multiplier 共OLM兲 method 关5兴. The four methods are equivalent in the sense that distributions derived in them are easily transformed to each other 关6兴. A comparison among the four MEMs is made in Ref. 关2兴. The nonextensive statistics has been successfully applied to a wide class of subjects in physics, chemistry, information science, biology, and economics 关7兴. One of the alternative approaches to the nonextensive statistics besides the MEM is the superstatistics 关8,9兴 共for a recent review, see 关10兴兲. In the superstatistics, it is assumed that locally the equilibrium state is described by the Boltzmann-Gibbs statistics and that their global properties may be expressed by a superposition over the intensive parameter 共i.e., the inverse temperature兲 关8–10兴. It is, however, not clear how to obtain the mixing probability distribution of fluctuating parameter from first principles. This problem is currently controversial and some attempts to this direction have been proposed 关11–15兴. The concept of the superstatistics has been applied to many kinds of subjects such as hy-
*
[email protected] 1539-3755/2009/80共1兲/011126共19兲
drodynamic turbulence 关16–18兴, cosmic ray 关19兴, and solar flares 关20兴. The nonextensive statistics has been applied to both classical and quantum systems. In this paper, we pay attention to quantum nonextensive systems. The generalized BoseEinstein and Fermi-Dirac distributions in nonextensive systems 共referred to as q-BED and q-FDD hereafter兲 have been discussed by the three methods. 共i兲 The asymptotic approximation 共AA兲 was proposed by Tsallis et al. 关21兴 who derived the expression for the canonical partition function valid for 兩q − 1兩 / kBT → 0. It has been applied to the black-body radiation 关21兴, early universe 关21,22兴, and the Bose-Einstein condensation 关21,23兴. 共ii兲 The factorization approximation 共FA兲 was proposed by Büyükkilic et al. 关24兴 to evaluate the grandcanonical partition function. The FA was criticized in 关25,26兴 but supported in 关27兴, related discussion being given in Sec. IV. The simple expressions for q-BED and q-FDD in the FA have been adopted in many applications such as the blackbody radiation 关23,28–30兴, early universe 关31,32兴, the BoseEinstein condensation 关33–39兴, metals 关40兴, superconductivity 关41,42兴, spin systems 关43–48兴, and metallic ferromagnets 关49兴. 共iii兲 The exact approach 共EA兲 was developed by Rajagopal and co-workers 关50,51兴 who derived the formally exact integral representation for the grand-canonical partition function of nonextensive systems which is expressed in terms of the Boltzmann-Gibbs counterpart. The integral representation approach was originated from the Hilhorst formula 关52兴. Because an actual evaluation of a given integral is generally difficult, it may be performed in an approximate way 关50,51兴 or in the limited cases 关53兴. The validity of the EA is discussed in 关54,55兴. The EA has been applied to nonextensive quantum systems such as black-body radiation 关56,57兴 and the Bose-Einstein condensation 关50,51兴. We believe that it is important and valuable to pursue the EA despite its difficulty. It is the purpose of the present study
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©2009 The American Physical Society
PHYSICAL REVIEW E 80, 011126 共2009兲
HIDEO HASEGAWA
to apply the EA 关50,51兴 to calculations of the generalized distributions of q-BED and q-FDD. The grand-canonical partition function of the nonextensive systems is derived with the use of the OLM scheme in the MEM 关5兴. Self-consistent equations for averages of the number of particles and energy and the grand-canonical partition function are exactly expressed by the integral representation 关50,51兴. The integral representation for q ⬎ 1.0 in the EA is expressed as an integral along the real axis, while that for q ⬍ 1.0 is expressed as the contour integral in the complex plane 关50,51,53兴. We have shown that the 共q − 1兲 expansion by the EA agrees with the result derived by the AA. For q ⱖ 1.0, the self-consistent equations have been numerically solved with the band model for electrons and the Debye model for phonon. It is rather difficult and tedious to obtain the generalized distributions in the EA because they need the self-consistent calculation of averages of number of particles and energy. Based on the exact result obtained, we have proposed the interpolation approximation 共IA兲 to q-BED and q-FDD, which do not need the self-consistently determined quantities and whose results are in agreement with those of the EA within O共q − 1兲 and in high- and low-temperature limits. We may obtain the simple analytic expressions of the q-BED and q-FDD. The paper is organized as follows. In Sec. II, the exact integral representation is derived with the OLM-MEM after Ref. 关50,51,53兴. We have discussed the 共q − 1兲 expansion of physical quantities using the EA and AA. Numerical calculations are performed for electron and phonon models for which we present the q-BED and q-FDD with the temperature-dependent energy. In Sec. III, we propose the IA in which analytical expressions for q-BED and q-FDD are obtained. In Sec. IV, a comparison is made between the generalized distributions calculated by the four methods of the EA, IA, FA 关24兴, and superstatistical approximation 共SA兲. A controversy on the validity of the FA 关24兴 is discussed. With the use of the four methods, the generalized Sommerfeld expansion and low-temperature electronic and phonon specific heats are calculated. Section V is devoted to our conclusion. In Appendix A, we present a study of the EA and AA with the un-normalized MEM 关3,21兴, calculating the 共q − 1兲 expansion of the q-BED and q-FDD. Supplementary discussions on the IA are presented in Appendix B.
Tr ˆ q = 1, Tr兵ˆ qqN其 = cqNq , Tr兵ˆ qqH其 = cqEq , cq = Tr ˆ qq , where Tr stands for the trace, kB is the Boltzmann constant, and Eq and Nq denote the expectation values of the Hamilˆ and the number operator N ˆ , respectively. The OLMtonian H MEM yields 关5,6兴 1 ˆ − Nˆ − E + N 兲兴1/共1−q兲 , 关1 + 共q − 1兲共H q q Xq
ˆ q =
ˆ − Nˆ − E + N 兲兴1/共1−q兲其, 共2兲 Xq = Tr兵关1 + 共q − 1兲共H q q Nq =
1 ˆ − Nˆ − E + N 兲兴q/共1−q兲Nˆ其, Tr兵关1 + 共q − 1兲共H q q Xq 共3兲
Eq =
1 ˆ − Nˆ − E + N 兲兴q/共1−q兲H ˆ 其, Tr兵关1 + 共q − 1兲共H q q Xq 共4兲
where  and denote the Lagrange multipliers. In deriving Eqs. 共1兲–共4兲, we have employed the relation cq = X1−q q . Lagrange multipliers of  and are identified as the inverse physical temperature 共 = 1 / kBT兲 and the chemical potential 共Fermi level兲, respectively 关5,6兴. B. Exact integral representation 1. Case of q ⬎ 1
In the case of q ⬎ 1.0, we adopt the formula for the gamma function ⌫共s兲, x−s =
II. EXACT APPROACH
1 ⌫共s兲
冕
⬁
for Re s ⬎ 0.
us−1e−xudu
共5兲
0
With s = 1 / 共q − 1兲 关or s = q / 共q − 1兲兴 and x = 1 + 共q − 1兲共H − N兲 in Eq. 共5兲, we may express Eqs. 共1兲–共4兲 by 关50,51兴
A. MEM by OLM
We will study nonextensive quantum systems described ˆ . We have obtained the optimum denby the Hamiltonian H sity matrix of ˆ , applying the OLM-MEM to the Tsallis entropy given by 关5,6兴
Nq =
1 Xq
冕 冉 ⬁
G u;
0
冊
q ,1 e共q−1兲u共Eq−Nq兲 q−1
⫻⌶1关共q − 1兲u兴N1关共q − 1兲u兴du,
kB 关1 − Tr ˆ qq兴, Sq = q−1
Eq =
1 Xq
冕 冉 ⬁
G u;
0
with 011126-2
共6兲
冊
q ,1 e共q−1兲u共Eq−Nq兲⌶1关共q − 1兲u兴 q−1
⫻E1关共q − 1兲u兴du, with the constraints
共1兲
共7兲
PHYSICAL REVIEW E 80, 011126 共2009兲
BOSE-EINSTEIN AND FERMI-DIRAC DISTRIBUTIONS…
冕 冉 ⬁
Xq =
G u;
0
冊
1 ,1 e共q−1兲u共Eq−Nq兲⌶1关共q − 1兲u兴du, q−1
具u2典u − 具u典2u =
共8兲 where ˆ
ˆ
⌶1共u兲 = e−u⍀1共u兲 = Tr兵e−u共H−N兲其 = 兿 关1 ⫿ e−u共⑀k−兲兴⫿1 ,
共9兲
k
⍀1共u兲 = ⫾
1 兺 ln关1 ⫿ e−u共⑀k−兲兴, u k
Nq =
1 Xq
k
Eq =
共12兲
k
f 1共⑀,u兲 =
1 eu共⑀−兲 ⫿ 1
共13兲
,
0
1 Xq
冕 冉
with
冕 冉 ⬁
b a−1 −bu G共u;a,b兲 = u e . ⌫共a兲
Xq =
共14兲
The upper 共lower兲 sign in Eqs. 共9兲, 共10兲, and 共13兲 denotes boson 共fermion兲 case, and ⌶1共u兲, ⍀1共u兲, N1共u兲, E1共u兲, and f 1共⑀ , u兲 express the physical quantities for q = 1.0. Equations 共6兲–共8兲 show that physical quantities in nonextensive systems are expressed as a superposition of those for q = 1.0. Although Eqs. 共6兲–共8兲 are formally exact expressions, they have a problem when we perform numerical calculations. The gamma distribution of G关u ; 1 / 共q − 1兲 + ᐉ , 1兴 共ᐉ = 0 , 1兲 in Eqs. 共6兲–共8兲 has the maximum at umax and it has average and variance given by 1 umax = + ᐉ − 1, 共q − 1兲
冉
G u;
1 1 + 1, q−1 共q − 1兲
冊
⬁
G u;
0
1 1 + 1, q−1 共q − 1兲
共18兲
冊
⫻eu共Eq−Nq兲⌶1共u兲E1共u兲du,
a
具u典u =
冕 冉 ⬁
⫻eu共Eq−Nq兲⌶1共u兲N1共u兲du,
共11兲
E1共u兲 = 兺 ⑀k f 1共⑀k,u兲,
共17兲
Equation 共15兲 shows that the gamma distribution in Eqs. 共6兲–共8兲 has the maximum at umax = 1 / 共q − 1兲 → ⬁, while the contribution from ⌶1关共q − 1兲t兴 is dominant at t ⬃ 0 because its argument becomes 共q − 1兲t → 0. Then numerical calculations using Eqs. 共6兲–共8兲 are very difficult. In order to overcome this difficulty, we have adopted a change in variable 共q − 1兲u → u in Eq. 共6兲–共8兲 to obtain alternative expressions given by
共10兲
N1共u兲 = 兺 f 1共⑀k,u兲,
1 + ᐉ. 共q − 1兲
共15兲
1 + ᐉ, 共q − 1兲
共16兲
冊
1 1 + ᐉ, G u; = q−1 共q − 1兲
冦
1
冑2共q − 1兲2 e
0
冊
1 1 eu共Eq−Nq兲⌶1共u兲du. 共20兲 , q − 1 共q − 1兲
1 1 The gamma distribution of G共u ; 共q−1兲 + ᐉ , 共q−1兲  兲 for ᐉ = 0 , 1 in Eqs. 共18兲–共20兲 has the maximum at umax and it has average, mean square, and variance given by
umax = 关1 + 共q − 1兲共ᐉ − 1兲兴 ,
共21兲
具u典u = 关1 + 共q − 1兲ᐉ兴 ,
共22兲
具u2典u = 关1 + 共q − 1兲ᐉ兴关1 + 共q − 1兲共ᐉ + 1兲兴2 ,
共23兲
具u2典u − 具u典2u = 共q − 1兲关1 + 共q − 1兲ᐉ兴2 .
共24兲
Equation 共21兲 shows that the gamma distribution has the maximum at umax =  in the limit of q → 1.0, and an integration over u in Eqs. 共18兲–共20兲 may be easily performed. Indeed, in the case of q ⲏ 1.0 discussed above, the gamma distribution in Eqs. 共18兲–共20兲 becomes
−兵1/关2共q−1兲2兴其共u − 兲2
␦共u − 兲
Although expressions given by Eqs. 共6兲–共8兲 are mathematically equivalent to those given by Eqs. 共18兲–共20兲, the latter expressions are more suitable than the former ones for numerical calculations.
G u;
共19兲
for 共q − 1兲2 Ⰶ 1,
共25兲
for 共q − 1兲2 → 0.
共26兲
2. Case of q ⬍ 1
In the case of q ⬍ 1.0, we adopt the formula that is given by
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HIDEO HASEGAWA
冕
for Re s ⬎ 0, 共27兲
N1共u兲 =
A⌫共r + 1兲共r + 1兲 , ur+1
where a contour integral is performed over the Hankel path C in the complex plane. With s = 1 / 共1 − q兲 关or s = q / 共1 − q兲兴 and x = 1 + 共q − 1兲共H − N兲 in Eq. 共27兲, we obtain 关50,51兴
E1共u兲 =
A⌫共r + 2兲共r + 2兲 , ur+2
xs =
i ⌫共s + 1兲 2
i Nq = 2Xq
共− t兲−s−1e−xtdt
C
冕 冉
冊
q ,1 e−共1−q兲t共Eq−Nq兲 H t; 1−q C
⫻⌶1关− 共1 − q兲t兴N1关− 共1 − q兲t兴dt,
Eq =
i 2Xq
冕 冉
H t;
C
共28兲
冊
q ,1 e−共1−q兲t共Eq−Nq兲 1−q
⫻⌶1关− 共1 − q兲t兴E1关− 共1 − q兲t兴dt,
where r = 1 / 2 for an ideal Bose gas, r = 2 for a harmonic oscillator, A denotes a relevant factor, and 共z兲 stands for the Riemann zeta function. With a repeated use of Eq. 共27兲, Nq, Eq, and Xq may be expressed as sums of gamma functions 关52,56,57兴. Unfortunately, such a sophisticated method cannot be necessarily applied to any models such as Fermi gas. With a change in variable of 共1 − q兲共−t兲 → 共−t兲 in Eqs. 共28兲–共30兲 after the case of q ⬎ 1, they are given by
共29兲
Nq =
冕 冉
H t;
C
1 1 − 1, 1−q 共1 − q兲
冊
⫻e−t共Eq−Nq兲⌶1共− t兲N1共− t兲dt,
with
Xq =
i 2Xq
i 2
冕 冉
H t;
C
冊
1 ,1 e−共1−q兲t共Eq−Nq兲⌶1关− 共1 − q兲t兴dt, 1−q
Eq =
共30兲 H共t;a,b兲 = ⌫共a + 1兲b−a共− t兲−a−1e−bt ,
共31兲
where ⌶1共u兲, N1共u兲, E1共u兲, and f 1共⑀ , u兲 are given by Eqs. 共9兲–共12兲 with complex u. In the case of q ⬍ 1.0, Nq, Eq, and Xq given by Eqs. 共28兲–共30兲 are expressed by an integral along the Hankel contour path C in the complex plane. The Hankel path may be modified to the Bromwich contour which is parallel to the imaginary axis from c − i⬁ to c + i⬁ 共c ⬎ 0兲 关56,57兴. The Bromwich contour is usually understood as counting the contributions from the residues of all poles located in the left side of Re z ⬍ c of the complex plane z when the integrand is expressed by simple analytic functions. If the integrand is not expressed by simple analytic functions, we have to evaluate it by numerical methods. Unfortunately, we have not succeeded in evaluating Eqs. 共28兲–共30兲 with the sufficient accuracy. It is not easy to numerically evaluate the integral along the Hankel or Bromwich contour, which is required to be appropriately deformed for actual numerical calculations 关58,59兴. This subject has a long history and it is still active in the field of the numerical methods for the inverse Laplace transformation 关58兴 and for the gamma functions 关59兴. It is worthwhile to remark that for a Bose gas model with the density of states of 共⑀兲 = A⑀r, we obtain 共with = 0兲 关52,56,57兴
冋
⌶1共u兲 = exp
i 2Xq
冕 冉
H t;
C
1 1 − 1, 1−q 共1 − q兲
共32兲
冊
⫻e−t共Eq−Nq兲⌶1共− t兲E1共− t兲dt,
共33兲
with
Xq =
i 2
冕 冉
H t;
C
冊
1 1 , e−t共Eq−Nq兲⌶1共− t兲dt. 1 − q 共1 − q兲 共34兲
1 1 − ᐉ , 共1−q兲 Average and mean square over H共t , 1−q  兲 for ᐉ = 0 , 1 are given by
具共− t兲典t = 关1 − 共1 − q兲ᐉ兴 ,
共35兲
具共− t兲2典t = 关1 − 共1 − q兲ᐉ兴关q − 共1 − q兲ᐉ兴2 .
共36兲
Equations 共32兲–共34兲 are useful in making the 共q − 1兲 expansion, as will be discussed in the following. C. (q − 1) expansion 1. Exact approach
We will consider the 共q − 1兲 expansion of the expectation ˆ in the EA. By using Eqs. 共18兲 and value of an operator O 共32兲, we obtain
册
A⌫共r + 1兲共r + 2兲 , ur+1
ˆ 其, ˆ 典 = 1 Tr兵关1 − 共1 − q兲Kˆ兴q/共1−q兲O 具O q Xq 011126-4
共37兲
PHYSICAL REVIEW E 80, 011126 共2009兲
BOSE-EINSTEIN AND FERMI-DIRAC DISTRIBUTIONS…
ˆ典 = = 具O q
冦
1 Xq
冕 冉 冕 冉 ⬁
G u;
0
i 2Xq
H t;
C
冊
1 q , Y 1共u兲O1共u兲du q − 1 共q − 1兲
for q ⬎ 1,
共38兲
1 q Y 1共− t兲O1共− t兲dt for q ⬍ 1, , 1 − q 共1 − q兲
共39兲
冊
with
ˆ
ˆ其 Tr兵e−uKO , O1共u兲 = Y 1共u兲
共40兲
ˆ
Y 1共u兲 = Tr兵e−uK其 = eu共Eq−Nq兲⌶1共u兲,
共41兲
ˆ − N ˆ − E + N , Kˆ = H q q
共42兲
where Xq is given by Eq. 共20兲 for q ⬎ 1 and by Eq. 共34兲 for q ⬍ 1. It is noted that Y 1共u兲 includes the self-consistently calculated Nq and Eq. We first consider the case of q ⲏ 1 for which the integral including an arbitrary function W共u兲 is assumed to be given by
冕 冉 ⬁
J=
G u;
0
冊
1 1 + ᐉ, W共u兲du q−1 共q − 1兲
expression for J as Eq. 共46兲, which is then valid both for q ⱗ 1.0 and q ⲏ 1.0. For W共u兲 = Y 1共u兲 and W共u兲 = Y 1共u兲O1共u兲 in Eq. 共46兲, we obtain 1 2Y 1 Xq = Y 1 + 共q − 1兲2 2 + ¯ , 2  Oq =
冋
共Y 1O1兲 1 Y 1O1 + 共q − 1兲  Xq
Y1 = − 具Kˆ典1Y 1 , 
共50兲
2Y 1 = 具Kˆ2典1Y 1 , 2
共51兲
O1 ˆ 典 − 具KˆO ˆ典 , = 具Kˆ典1具O 1 1 
共52兲
1 1 + ᐉ , 共q−1兲 Since G共u ; q−1  兲 has the maximum around u =  as mentioned before 关Eq. 共21兲兴, W共u兲 may be expanded as
W 1 2W + 共u − 兲2 2 + ¯ .  2  共44兲
2O 1 ˆ 典 − 具Kˆ2典 具O ˆ 典 + 2关具Kˆ典 具KˆO ˆ 典 − 具Kˆ典2具O ˆ典 兴 = 具Kˆ2O 1 1 1 1 1 1 1 2
Substituting Eq. 共44兲 to Eq. 共43兲 and using the relations given by Eqs. 共22兲 and 共23兲, we obtain J in a series of 共q − 1兲 as J = W共兲 + 具共u − 兲典u
W 1 W + 具共u − 兲2典u 2 + ¯ ,  2 
共53兲 to Eqs. 共48兲 and 共49兲, we finally obtain the O共q − 1兲 expansion of Oq given by
共
共45兲
冋
J = W共兲 + 共q − 1兲 ᐉ
册
W 1 2 2W +  + ¯  2 2
Next we consider the case of q ⱗ 1 for which a similar integral along the Hankel path C is given by i 2
冕 冉
H t;
C
冊
1 q − ᐉ, W共− t兲dt 1−q 共1 − q兲
ˆ 典 + 1 2关具K ˆ 2典 具O ˆ 典 − 具Kˆ2O ˆ典 兴 Oq ⯝ O1 + 共1 − q兲 具KˆO 1 1 1 1 2
兲 共54兲
+ ¯.
for q ⯝ 1.0. 共46兲
J=
共49兲
Note that the O关共q − 1兲兴 term in Eq. 共48兲 vanishes because ᐉ = 0 in Eq. 共46兲. Substituting the relations given by
for ᐉ = 0,1.
2
册
1 2共Y 1O1兲 + 共q − 1兲2 +¯ . 2 2
共43兲
W共u兲 = W共兲 + 共u − 兲
共48兲
2. Asymptotic approach
On the other hand, we may adopt the AA 关21兴 to obtain Oq given by Eq. 共37兲 valid for O共q − 1兲. By using the relation exq ⯝ ex关1 − 共1 − q兲x2 / 2 + ¯兴 in Eqs. 共2兲 and 共37兲, we may expand Xq and Oq up to O共q − 1兲 as
关
for ᐉ = 0,1. 共47兲
By expanding W共−t兲 at −t =  and using the relations for averages given by Eqs. 共35兲 and 共36兲, we obtain the same 011126-5
兴
Xq ⯝ X1 1 − 21 共1 − q兲2具Kˆ2典1 + ¯ , Oq =
共55兲
1 ˆ其 Tr兵关1 − 共1 − q兲Kˆ兴−1关1 − 共1 − q兲Kˆ兴1/共1−q兲O Xq 共56兲
PHYSICAL REVIEW E 80, 011126 共2009兲
HIDEO HASEGAWA
⯝
再
册冎
冋
D. Generalized distributions
1 ˆ ˆ ˆ 兴 1 − 1 共1 − q兲2Kˆ2 O Tr e−K关1 + 共1 − q兲K 2 Xq
1. O(q − 1) expansion
Equations for Nq and Eq given by Eqs. 共18兲, 共19兲, 共32兲, and 共33兲 may be expressed as
共57兲
+¯
共
ˆ 典 − 具K ˆ 2O ˆ典 兴 ˆO ˆ 典 + 1 2关具K ˆ 2典 具O ⯝O1 + 共1 − q兲 具K 1 1 1 1 2
兲
N q = 兺 f q共 ⑀ k,  兲 =
冕
f q共⑀, 兲共⑀兲d⑀ ,
共59兲
E q = 兺 f q共 ⑀ k,  兲 ⑀ k =
冕
f q共⑀, 兲⑀共⑀兲d⑀ ,
共60兲
k
共58兲
+ ¯.
Equation 共58兲 agrees with Eq. 共54兲 obtained by the EA within O共q − 1兲. In Appendix A, we have shown that the same equivalence holds between the AA and EA with the un-normalized MEM 关3,21兴.
f q共 ⑀ ,  兲 =
冦
1 Xq
冕 冉 冕 冉 ⬁
G u;
0
i 2Xq
where f q共⑀ , 兲 关⬅f q共⑀兲兴 signifies the generalized distributions, q-BED and q-FDD, given by
冊
1 q , Y 1共u兲f 1共⑀,u兲du q − 1 共q − 1兲
H t;
C
k
for q ⬎ 1,
共61兲
1 q Y 1共− t兲f 1共⑀,− t兲dt for q ⬍ 1, , 1 − q 共1 − q兲
共62兲
冊
with the density of states 共⑀兲 given by
f q共⑀,T = 0兲 = ⌰共 − ⑀兲 = f 1共⑀,T = 0兲,
共⑀兲 = 兺 ␦共⑀ − ⑀k兲.
共63兲
k
In order to examine the 共q − 1兲 expansion of the generalized ˆ = nˆ in Eq. 共54兲, where nˆ denotes the distributions, we set O k k number operator of the state k. A simple calculation leads to the O共q − 1兲 expansion of the generalized distribution given by
冋 冋
册
f 1 1 2 2 f 1 +  +¯ f q共⑀, 兲 = f 1共⑀, 兲 + 共q − 1兲   2 2 = f 1共⑀, 兲 + 共q − 1兲 共⑀ − 兲
f1 1 2 f 1 + 共⑀ − 兲2 2 ⑀ 2 ⑀
共64兲
f q共⑀,  → 0兲 ⬀ 关1 + 共q − 1兲共⑀ − Eq兲兴关1/共1−q兲兴−1 = 关e−q 共⑀−兲兴q , 共68兲 exq
expressing the q-exponential function defined by
册
=
In deriving Eq. 共65兲, we have employed the relation 共Y 1 / 兲 / Y 1共兲 = −具H − N典1 + 共Nq − Nq兲 ⯝ O共q − 1兲. In Appendix A, we have made a similar analysis with the unnormalized MEM, showing that Eq. 共65兲 is consistent with Eq. 共A39兲 which agrees with the result in the AA 关21兴. 2. Properties of the generalized distribution
We will examine some limiting cases of the generalized distribution given by Eqs. 共61兲 and 共62兲. 共1兲 In the limit of q → 1.0, Eq. 共65兲 leads to f q共⑀, 兲 = f 1共⑀, 兲.
where ⌰共x兲 stands for the Heaviside function. Equation 共67兲 implies that the ground-state FD distribution is not modified by the nonextensivity. 共3兲 In the high-temperature limit of  → 0.0, where ⍀1 ⯝ −共1 / 兲兺ke−共⑀k−兲 with ln共1 ⫾ x兲 ⯝ ⫿ x for small x, we obtain 共 = 0.0兲
exq = expq共x兲
共65兲
+ ¯.
共67兲
共66兲
再
关1 + 共1 − q兲x兴1/共1−q兲 for 1 + 共1 − q兲x ⬎ 0,
共69兲
for 1 + 共1 − q兲x ⱕ 0,
共70兲
0
with the cutoff properties. Equation 共68兲 corresponds to the escort distribution, P q共 ⑀ 兲 =
p q共 ⑀ 兲 q ⬀ 关e−q 共⑀−兲兴q , cq
共71兲
with the q-exponential distribution pq共⑀兲 given by pq共⑀兲 = e−q 共⑀−兲 .
共72兲
Equations 共61兲 and 共62兲 show that the ⑀ dependence of f q共⑀ , 兲 arises from that of f 1共⑀ , 兲. In particular, the q-FDD preserves the same ⑀ symmetry as f 1共⑀ , 兲: 共a兲 f q共⑀ , 兲 = 1 / 2 for ⑀ = , 共b兲 f q共⑀ , 兲 has the antisymmetry
共2兲 In the zero-temperature limit of T = 0, the q-FDD becomes 011126-6
f q共− ␦⑀ + , 兲 − 21 = 21 − f q共␦⑀ + , 兲
for ␦⑀ ⬎ 0,
共c兲 f q共⑀ , 兲 / ⑀ is symmetric with respect to ⑀ = .
PHYSICAL REVIEW E 80, 011126 共2009兲
BOSE-EINSTEIN AND FERMI-DIRAC DISTRIBUTIONS… 0 -0.05
1. Model for electrons
共⑀兲 = 共1/2W兲⌰共W − 兩⑀兩兲,
⌶1共u兲 = e
⍀1共u兲 = −
−u⍀1共u兲
0.05 kB T/W
0.1
q=1.0 1.1 1.2 1.3
-0.2 -0.250
0.2
0.4 0.6 kB T/W
0.8
1
FIG. 1. 共Color online兲 The temperature dependence of Eq of the electron model for q = 1.0 共dashed curves兲, q = 1.1 共chain curves兲, q = 1.2 共dotted curves兲, and q = 1.3 共solid curves兲, the inset showing the enlarged plot for kBT / W ⱕ 0.1.
are in the linear and logarithmic scales, respectively. It is shown that with more increasing q from unity, f q共⑀兲 at ⑀ Ⰷ has a longer tail. The properties of f q共⑀兲 are more clearly seen in its derivative of − f q共⑀兲 / ⑀, which is plotted in Fig. 3 with the logarithmic ordinate. We note that − f q共⑀兲 / ⑀ is symmetric with respect of ⑀ = . With increasing q above unity, − f q共⑀兲 / ⑀ has a longer tail. Dotted and solid curves for q ⬍ 1.0 in Figs. 2 and 3 will be discussed in Sec. III C.
,
1 兵ln关1 + e−u共1−兲兴 − ln关1 + e−u共1+兲兴 2u
+ ln关1 + eu共1+兲兴 − ln关1 + eu共1−兲兴其 1 兵Li2共− e−u共1+兲兲 − Li2共− eu共1−兲兲其, 2u2
−
-0.25 0
-0.15
共73兲
where W denotes a half of the total bandwidth. We have performed numerical calculations of Eq and for q ⱖ 1.0 as a function of T for a given number of particles of N and the density of states 共⑀兲. We may obtain analytical expressions for ⌶1共u兲, N1共u兲, and E1共u兲 which are necessary for our numerical calculations. By using Eq. 共73兲 for Eqs. 共9兲–共12兲, we obtain 共with W = 1.0兲
-0.245
-0.1
Eq/W
For model calculations of electron systems, we employ a uniform density of state given by
-0.24
Eq/W
E. Numerical calculations
2. Debye model for phonons
We adopt the Debye model whose phonon density of states is given by N1共u兲 = 1 +
1 关ln共1 + e−u共1+兲兲 − ln共1 + eu共1−兲兲兴, 2u
1.2 1
+
0.6 0.4
1 关Li2共− e−u共1+兲兲 − Li2共− eu共1−兲兲兴, 2u2
0.2 0-4
where Lin共z兲 denotes the nth polylogarithmic function defined by
Lin共z兲 = 兺
k=1
1 0 -1
zk . kn
log10 fq
⬁
q 0.8 0.9 1.0 1.2 1.5 1.8
0.8
1 关ln共1 + e−u共1+兲兲 + ln共1 + eu共1−兲兲兴 2u
fq
E1共u兲 = −
(a)
-2 -3 -4
We adopt N = 0.5, for which = 0.0 independent of the temperature because of the adopted uniform density of states given by Eq. 共73兲. The temperature dependence of Eq calculated self-consistently from Eqs. 共18兲–共20兲, is shown in Fig. 1 whose inset shows the enlarged plot for low temperatures 共kBT / W ⱗ 0.1兲. We note that Eq at low temperatures is larger for larger q although this trend is reversed at higher temperatures 共kBT ⲏ 0.3兲. The calculated q-FDDs f q共⑀兲 for various q values for kBT / W = 0.1 are shown in Figs. 2共a兲 and 2共b兲 whose ordinates
-5 -6 -20
-2
0 β(ε-µ)
2
4
0 β(ε-µ)
10
20
(b)
q 0.8 0.9 1.0 1.2 1.5 1.8
-10
FIG. 2. 共Color online兲 The ⑀ dependence of the q-FDD of f q共⑀兲 for q = 0.8 共solid curves兲, q = 0.9 共dotted curves兲, q = 1.0 共dashed curves兲, q = 1.2 共double-chain curves兲, q = 1.5 共bold solid curves兲, and q = 1.8 共chain curves兲 with the 共a兲 linear and 共b兲 logarithmic ordinates; the results for q ⱖ 1.0 and q ⬍ 1.0 being calculated by the EA and IA, respectively 共kBT / W = 0.1兲.
011126-7
PHYSICAL REVIEW E 80, 011126 共2009兲
HIDEO HASEGAWA
0.8 0.9 1.0
5 Eq/kB TD
log10 (-dfq/dε)
6
q 1.2 1.5 1.8
-2
4
1
3 2
-4
0.8
Eq/kB TD
q
0
0.6 0.4 0.2
q 1.0 1.1 1.2 1.3
0 0 0.1 0.2 0.3 0.4 0.5 T/TD
1 -6 -20
-10
0 β(ε-µ)
10
00
20
FIG. 3. 共Color online兲 The ⑀ dependence of the derivative of q-FDD, − f q共⑀兲 / ⑀, for q = 0.8 共the solid curve兲, q = 0.9 共the dotted curve兲, q = 1.0 共the dashed curve兲, q = 1.2 共the double-chain curve兲, q = 1.5 共the bold solid curve兲, and q = 1.8 共the chain curve兲 with the logarithmic ordinate; the results for q ⱖ 1.0 and q ⬍ 1.0 being calculated by the EA and IA, respectively 共kBT / W = 0.1兲.
共兲 = A2
for 0 ⬍ ⱕ D ,
共74兲
3 where A = 9Na / wD , Na denotes the number of atoms, is the phonon frequency, and D is the Debye cutoff frequency. By using Eq. 共74兲 to Eqs. 共9兲–共12兲, we may obtain 共with D = 1.0 and = 0兲,
⌶1共u兲 = e−u⍀1共u兲 ,
⍀1共u兲 =
冋
−
III. INTERPOLATION APPROXIMATION A. Analytic expressions of the generalized distributions
冉 冊
1 u共E −N 兲 e q q ⌶1共u兲 = 1 Xq
共75兲
in Eqs. 共61兲 and 共62兲, we obtain the approximate generalized distributions given by
A 3 关u − 3u2 ln共1 − eu兲 3u3
2 q 1.1 1.2 1.5 1.8
1 log10 fq
0
ln共1 − eu兲 3Li2共eu兲 6Li3共eu兲 − + E1共u兲 = A u u2 u3 +
2
logarithmic scale: they are indistinguishable in the linear scale. It is shown that with more increasing q, f q共⑀兲 at ⑀ Ⰷ has a longer tail. Dotted and solid curves for q ⬍ 1.0 will be discussed in Sec. III C.
− 6uLi2共eu兲 + 6Li3共eu兲 − 6共3兲兴,
冋
1.5
FIG. 4. 共Color online兲 The temperature dependence of Eq of the Debye phonon model for q = 1.0 共dashed curves兲, q = 1.1 共chain curves兲, q = 1.2 共dotted curves兲, and q = 1.3 共solid curves兲; the inset showing the enlarged plot for T / TD ⱕ 0.5.
册
A 2 关u Li2共eu兲 − 2uLi3共eu兲 + 2Li4共eu兲兴, u4
N1共u兲 = −
1 T/TD
In Sec. II, we have discussed the generalized distributions based on the exact representation given by Eqs. 共61兲 and 共62兲. It is, however, difficult to calculate them because they need self-consistent calculations of Nq and Eq. If we assume
A 44 + 15u − 60 ln共1 − eu兲 180u u3 + 60 ln共1 − cosh u + sinh u兲
0.5
-1 -2 -3
册
6Li4共eu兲 1 4 − − . 4 15u4 u4
-4 -50
We have performed numerical calculations with the Debye model for q ⱖ 1.0. The temperature dependence of selfconsistently calculated Eq is shown in Fig. 4 where inset shows the enlarged plots for low temperatures 共T / TD ⬍ 0.5兲. We note that Eq at low temperatures is larger for larger q. The calculated q-BEDs f q共⑀兲 for various q values for T / TD = 0.01 are shown in Fig. 5 whose ordinate is in the
q 0.8 0.9 1.0
2
4
β(ε-µ)
6
8
10
FIG. 5. 共Color online兲 The ⑀ dependence of the q-BED of f q共⑀兲 for q = 0.8 共the solid curve兲, q = 0.9 共the dotted curve兲, q = 1.0 共the dashed curve兲, q = 1.1 共the chain curve兲, q = 1.2 共the double-chain curve兲, q = 1.5 共the bold solid curves兲, and q = 1.8 共the thin solid curve兲 with the logarithmic ordinate, the results for q ⱖ 1.0 and q ⬍ 1.0 being calculated by the EA and IA, respectively 共T / TD = 0.01兲.
011126-8
PHYSICAL REVIEW E 80, 011126 共2009兲
BOSE-EINSTEIN AND FERMI-DIRAC DISTRIBUTIONS…
f IA q 共⑀, 兲
=
冦
冕 冉 冕 冉 ⬁
G u;
0
i 2
冊
1 q f 1共⑀,u兲du , q − 1 共q − 1兲
H t;
C
for q ⬎ 1.0,
共76兲
1 q f 1共⑀,− t兲dt for q ⬍ 1.0, , 1 − q 共1 − q兲
共77兲
where G共u ; a , b兲 and H共t ; a , b兲 are given by Eqs. 共14兲 and 共31兲, respectively. Equations 共76兲 and 共77兲 are referred to as the interpolation approximation 共IA兲 in this paper because they have the important interpolating character, as will be shown shortly 共Sec. III B兲. Note that calculations of f IA q 共⑀ , 兲 by Eqs. 共76兲 and 共77兲 do not require Nq and Eq. Equation 共76兲 may be regarded as a kind of the SA. One of advantages of the IA is that we can obtain the simple analytic expressions for the q-BED and q-FDD as follows. 共1兲 q -BED. We first expand the Bose-Einstein distribution f 1共⑀ , 兲 as
冊
f 1共 ⑀ ,  兲 =
f IA q 共⑀, 兲 −共n+1兲x
for x ⬎ 0,
共78兲
n=0
冋
关e−共n+1兲x 兴q q
1 共q − 1兲x
for 0 ⬍ q ⬍ 3,
Fq共x兲 =
冋
k=0
册 冉 q/共q−1兲
1 q , +1 q − 1 共q − 1兲x
冊 共80兲
x
共83兲
for x ⬍ 0,
共84兲
=
冦
Fq共x兲
for x ⬎ 0,
共85兲
1 2
for x = 0,
共86兲
1 − Fq共兩x兩兲 for x ⬍ 0,
共87兲
1 1 = 共k + a兲z ⌫共z兲
冕
冉
册 再冉 q/共q−1兲
for 0 ⬍ q ⬍ 3,
1 1 q , + q − 1 2共q − 1兲x 2
1 q , +1 q − 1 2共q − 1兲x
冊冎
共88兲
冊
for 1 ⬍ q ⬍ 3. 共89兲
Its derivative is given by ⬁
f IA q 兴共2q−1兲 = − 兺 共− 1兲nq共n + 1兲关e−共n+1兲兩x兩 q x n=0
for 0 ⬍ q ⬍ 3, 共90兲
⬁ z−1 −at
0
t e dt 1 − e−t
which is symmetric with respect to x = 0. The q-FDD given by Eqs. 共85兲–共88兲 reduces to f 1共⑀ , 兲 in the limit of q → 1.0. We may obtain a useful expression of the q-FDD for 兩x兩 ⬍ 1 given by 共see Appendix B 1兲
for Re z ⬎ 1.
Its derivative is given by
f IA q
兺 共− 1兲ne−n兩x兩
1 2共q − 1兲x
−
where 共z , a兲 denotes the Hurwitz zeta function:
共z,a兲 = 兺
for x = 0,
n=0
n=0
共79兲
for 1 ⬍ q ⬍ 3,
⬁
共82兲
⬁
n=0
f IA q 共⑀, 兲 =
1 2 ⬁
兴q Fq共x兲 = 兺 共− 1兲n关e−共n+1兲x q
⬁
=兺
for x ⬎ 0,
with
where x = 共⑀ − 兲. Substituting Eq. 共78兲 to Eqs. 共76兲 and 共77兲 and employing Eq. 共5兲 and 共27兲, we obtain the q-BED in the IA given by
f IA q 共⑀, 兲
兺 共− 1兲ne−共n+1兲x
n=0
where x = 共⑀ − 兲. Substituting Eqs. 共82兲–共84兲 to Eqs. 共76兲 and 共77兲 and employing Eq. 共5兲 and 共27兲, we obtain the q-FDD in the IA given by
⬁
f 1共 ⑀ ,  兲 = 兺 e
冦
⬁
f IA q ⯝
⬁
= − 兺 q共n + 1兲关e−共n+1兲x 兴共2q−1兲 q
for 0 ⬍ q ⬍ 3.
n=0
f IA q q共2q − 1兲共3q − 2兲 2 q ⯝− + x + ¯ 16 x 4
共81兲 We may easily realize that f q共⑀ , 兲 in Eq. 共79兲 reduces to f 1共⑀ , 兲 in the limit of q → 1.0 where exq → ex. 共2兲 q -FDD. The Fermi-Dirac distribution f 1共⑀ , 兲 may be expanded as
q共2q − 1兲共3q − 2兲 3 1 q − x+ x + ¯, 2 4 48
共91兲
for 0 ⬍ q ⬍ 3. 共92兲
In the case of q ⬍ 1.0, summations over n in the q-BED and q-FDD 关Eqs. 共79兲 and 共88兲兴 are terminated when the condition n + 1 ⬎ 1 / 共1 − q兲x is satisfied because of the cutoff prop-
011126-9
PHYSICAL REVIEW E 80, 011126 共2009兲
HIDEO HASEGAWA
λ
log10 fq
-2
1.02
-3 -4
1
2 log10 fq
0 -1
4
q 1.0 1.2 1.5 1.8
1
0.98 0
-5 -10
-5
β(ε-µ)
10
0 β(ε-µ)
5
-2
-60
10
0.5
0
-4
0.96 -10
λ
1
0 0
2
β(ε-µ) 4 6
8
10
q 1.0 1.1 1.2 1.5 1.8
2
4
β(ε-µ)
6
8
10
FIG. 6. 共Color online兲 The ⑀ dependence of the q-FDD of f q共⑀兲 calculated by the EA for q = 1.0 共dashed curves兲, q = 1.2 共chain curves兲, q = 1.5 共dotted curves兲, and q = 1.8 共solid curves兲 with the logarithmic ordinate; the inset showing the ratio of EA = f IA q 共⑀兲 / f q 共⑀兲 共kBT / W = 1.0兲.
FIG. 7. 共Color online兲 The ⑀ dependence of the q-BED of f q共⑀兲 calculated by the EA for q = 1.0 共dashed curves兲, q = 1.1 共doublechain curves兲 q = 1.2 共chain curves兲, q = 1.5 共dotted curves兲, and q = 1.8 共solid curves兲 with the logarithmic ordinate; the inset showing EA the ratio of = f IA q 共⑀兲 / f q 共⑀兲 共kBT / W = 0.1兲.
erties of the q-exponential function given by Eq. 共70兲. Then the q-FDD for q ⬍ 1.0 has the cutoff properties given by
Equations 共96兲 and 共97兲 agree with Eqs. 共67兲 and 共68兲, respectively, for the EA. Thus the generalized distributions in the IA have the interpolation properties, yielding results in agreement with those in the EA within O共q − 1兲 and in highand low-temperature limits.
f IA q 共⑀兲 =
再
0.0 for ⑀ − ⬎ 1/共1 − q兲 ,
共93兲
1.0 for ⑀ − ⬍ − 1/共1 − q兲 ,
共94兲
while the q-BED has the cutoff properties given by Eq. 共93兲. These are the same as the q-exponential distribution pq共⑀兲 given by Eq. 共72兲.
C. Numerical calculations IA Numerical calculations of f IA q 共⑀ , 兲 关⬅f q 共⑀兲兴 have been EA performed. Results of the FDD of f q 共⑀兲 in the EA for q ⬎ 1.0 and kBT / W = 1.0 are shown in Fig. 6. With more increasing q, the distributions have longer tails, as shown in Fig. 2 for kBT / W = 0.1. The result in the IA is in good agreement with the EA because the ratio defined by EA ⬅ f IA q 共⑀兲 / f q 共⑀兲 is 0.97ⱗ ⱗ 1.01 for −10⬍ ⑀ ⬍ 10 as shown in the inset. The ⑀ dependence of the BED of f EA q 共⑀兲 in the EA for q ⬎ 1.0 and T / TD = 0.1 is plotted in Fig. 7 which shows similar behavior to those for T / TD = 0.01 shown in Fig. 6. Its inset shows that the ratio of is 0.7ⱗ ⱗ 1.0 for 1.0⬍ q ⱕ 1.2. These calculations justify, to some extent, the distribution in the IA given by Eqs. 共80兲, 共85兲–共87兲, and 共89兲. We have calculated the q-BED and q-FDD also for q ⬍ 1.0 by using Eqs. 共79兲 and 共85兲–共88兲. Dotted and solid curves in Fig. 2 show the q-FDD of f IA q 共⑀兲 for q = 0.9 and q = 0.8, respectively. Their derivatives of − f IA q 共⑀兲 / ⑀ for q = 0.9 and q = 0.8 are plotted by the dotted and solid curves, respectively, in Fig. 3. Dotted and solid curves in Fig. 5 show
B. Comparison with the exact approach
From Eqs. 共48兲 and 共49兲 with Y 1共u兲 = 1.0, the q-BED and q-FDD for q ⯝ 1.0 in the IA become
冋
f IA q 共⑀, 兲 = f 1共⑀, 兲 + 共q − 1兲 共⑀ − 兲
f1 1 2 f 1 + 共⑀ − 兲2 2 ⑀ 2 ⑀
册
共95兲
+ ¯,
which is in agreement with those in the EA given by Eq. 共65兲 within O共q − 1兲. In the zero-temperature limit, the q-FDD reduces to 共96兲 f IA q 共⑀,T = 0兲 = ⌰共 − ⑀兲. In the opposite high-temperature limit, the q-BED and q-FDD become −x q f IA q 共⑀,  → 0兲 ⬀ 关eq 兴 .
共97兲
TABLE I. Generalized distributions in the limits of q → 1, T → 0, and  → 0. f 1 = 1 / 共e共⑀−兲 ⫿ 1兲: ⌰共x兲—the Heaviside function; exq, q-exponential function. T → 0 共FDD兲
→0
f 1 + 共q − 1兲关共⑀ − 兲 ⑀ + 2 共⑀ − 兲2 ⑀2 兴 + ¯
⌰共 − ⑀兲
关e−q 共⑀−兲兴q
f 1 + 共q − 1兲关共⑀ − 兲 ⑀ + 2 共⑀ − 兲2 ⑀2 兴 + ¯
⌰共 − ⑀兲
关e−q 共⑀−兲兴q
q→1
Method EAa IAb FAc SAd
f1 f1
1 1
f 1 − 2 共q − 1兲共⑀ − 兲 1
f 1 + 2 共q − 1兲共⑀ − 兲 1
2 f1
2 f
1
2 f
1
⑀
+¯
⌰共 − ⑀兲
e−q 共⑀−兲
⑀2
+¯
⌰共 − ⑀兲
e−q 共⑀−兲
2 2 f1
a
The exact approach 共the present study兲. The interpolation approximation 共the present study兲. c The factorization approximation 关24兴. d The superstatiscal approximation 关49兴. b
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1 EA FA
1 0
0 -1
-1
log10 fq
log10 fq
SA q=1.2
-2 q-BED
-40
2
-2
-4
q=1.1 q=1.0
4
β(ε-µ)
6
8
the q-BED of f IA q 共⑀兲 for q = 0.9 and q = 0.8, respectively. With more decreasing q from unity, the curvatures of f q共⑀兲 in both q-BED and q-FDD become more significant. The cutoff properties in the q-FDD and q-BED given by Eqs. 共93兲 and 共94兲 are realized in Figs. 2 and 5. We expect that f IA q 共⑀兲 in the case of q ⬍ 1.0 is a good approximation of the q-BED and q-FDD as in the case of q ⬎ 1.0. IV. DISCUSSION A. Comparison with previous studies
It is interesting to compare our results to those previously obtained with some approximations. 共A兲 Factorization approximation. Büyükkilic et al. 关24兴 derived the q-BED and q-FDD given by 1 , 兵eq关− 共⑀ − 兲兴其−1 ⫿ 1
adopting the FA given by
冋
N
Q = 1 − 共1 − q兲 兺 xn n=1
册
q=1.1 q=1.0
q-FDD
-5 -10
10
FIG. 8. 共Color online兲 The ⑀ dependence of the q-BED of f q共⑀兲 for q = 1.1 and 1.2 calculated by the EA 共solid curves兲, FA 共chain curves兲, and SA 共dotted curves兲 with the logarithmic ordinate, f 1共⑀兲, for q = 1.0 being plotted by the dashed curve for a comparison 共T / TD = 0.01兲.
f FA q 共⑀, 兲 =
EA FA SA
-3
-5
0 β(ε-µ)
5
10
FIG. 9. 共Color online兲 The ⑀ dependence of the q-FDD of f q共⑀兲 for q = 1.1 and 1.2 calculated by the EA 共the solid curve兲, FA 共the chain curve兲, and SA 共the dotted curve兲 with the logarithmic ordinate, f 1共⑀兲 for q = 1.0 being plotted by the dashed curve for a comparison 共kBT / W = 0.1兲.
f SA q 共⑀, 兲 =
冕 冉 ⬁
G u;
0
冊
1 1 , f 1共⑀,u兲du, 共102兲 q − 1 共q − 1兲
which is similar to but different from f IA q 共⑀ , 兲 given by Eq. 共76兲. Recently the q-FDD equivalent to Eq. 共98兲 is obtained by employing the SA in a different way 关49兴. The properties of the generalized distributions of the EA, IA, FA, and SA in the limits of q → 1.0, T = 0, and  → 0.0 are compared in Table I. The result of the IA agrees with that of the EA within O共q − 1兲 as mentioned before. However, the O共q − 1兲 contributions in the FA and SA are different from 1.2 1
(a) FA
IA
0.8
共98兲
q 1.1 1.0 0.9
fq
-3
q=1.2
0.6 0.4 0.2
1/共1−q兲
共99兲
0-4
-2
0
2
4
0
2
4
β(ε-µ)
0.3 (b)
N
⯝ 兿 关1 − 共1 − q兲xn兴
1/共1−q兲
共100兲 0.2 - dfq/dε
n=1
to evaluate the grand-canonical partition function, the upper 共lower兲 sign in Eq. 共98兲 being applied to boson 共fermion兲. It is noted that if we assume the factorization approxima−x q n IA q 兴q ⯝ 共e−x tion, 关e−共n+1兲x q q 兲 关共eq 兲 兴 in f q 共⑀兲 关Eqs. 共79兲 and 共88兲兴, we obtain f q共 ⑀ ,  兲 ⯝
1 , 兵eq关− 共⑀ − 兲兴其−q ⫿ 1
共101兲
which is similar to Eq. 共98兲 关41,55兴. 共B兲 The superstatistical approximation. In the SA, the generalized distribution is expressed as a superposition of f 1共⑀兲 关8,9兴,
0.1
0 -4
-2
β(ε-µ)
FIG. 10. 共Color online兲 The ⑀ dependences of 共a兲 the q-FDDs of f q共⑀兲 and 共b兲 its derivative of − f q共⑀兲 / ⑀ calculated by the IA for q = 0.9 共solid curves兲 and 1.1 共bold solid curves兲 and those calculated by the FA for q = 0.9 共dashed curves兲 and 1.1 共bold dashed curves兲; results for q = 1.0 being plotted by chain curves for a comparison.
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HIDEO HASEGAWA
These lead to an overestimate of electron excitations across the Fermi level in the FA. Furthermore − f FA q 共⑀兲 / ⑀ in the FA is not symmetric with respect to ⑀ = in contrast to that in the IA. The FA was criticized in Refs. 关25,26兴 but justified in Ref. 关27兴. The dismissive study 关25兴 was based on a simulation with N = 2. In contrast, the affirmative study 关27兴 performed simulations with N = 105 and 1015. Lenzi et al. 关26兴 criticized the FA, applying the EA 关50,51兴 to independent harmonic oscillators with N ⱕ 100. Our results are consistent with Refs. 关25,26兴. The FA given by Eq. 共100兲 has been explicitly or implicitly employed in many studies not only for quantum but also for classical nonextensive systems. It would be necessary to examine the validity of these studies using the FA from the viewpoint of the exact representation 关50,51,60,61兴. By using Eqs. 共5兲 and 共27兲, we may rewrite Q in Eq. 共99兲 as
that in the EA. In the zero-temperature limit, all the q-FDDs reduce to ⌰共 − ⑀兲. In the opposite high-temperature limit, the generalized distributions in the FA and SA reduce to e−q ⑀, while those in the EA and IA become 关e−q ⑀兴q, where the power index q arises from the escort probability in the OLMMEM given by Eq. 共71兲 关5,6兴. Figure 8 shows q-BED for q = 1.1 and q = 1.2 calculated by the FA, SA, and EA with the logarithmic ordinate. For a comparison, we show f q共⑀兲 for q = 1.0 by dashed curves. The difference among f q共⑀兲’s of the three methods is clearly realized: tails in the q-BED of the FA and SA are overestimated. Figure 9 shows q-FDD for q = 1.1 and q = 1.2 calculated by the EA, FA, and SA with the logarithmic ordinate 共for more detailed f FA q 共⑀兲, see Fig. 1 of Ref. 关49兴兲. Tails in the FA and SA are larger than that in the EA, as in the case of the q-BED shown in Fig. 8. Figures 10共a兲 and 10共b兲 show the q-FDD and its derivative, respectively, calculated in the IA and FA. For q = 0.9, IA f FA q 共⑀兲 at ⑀ ⬍ is much reduced than f q 共⑀兲. For q = 1.1, on FA the contrary, f q 共⑀兲 at ⑀ ⬎ is much increased than f IA q 共⑀兲.
Q=
冦
冕 冉 冕 冉 ⬁
0
1 1 G u; , q−1 q−1
i 2
H t;
C
Q = 关1 − 共1 − q兲x1兴1/共1−q兲 丢 q ¯ 丢 q关1 − 共1 − q兲xN兴1/共1−q兲 , 共103兲
冊兿 冊兿 N
1 1 , 1−q 1−q
N
etxndt for q ⬍ 1.0,
共105兲
I=
冕
共107兲
=
冕
n=1
where 丢 q denotes the q product defined by 关62兴 x 丢 qy ⬅ 关x1−q + y 1−q − 1兴1/共1−q兲 .
共106兲
Equations 共104兲 and 共105兲 are the integral representations of the q product given by Eq. 共103兲. The result of the FA in Eq. 共100兲 is derived if we may exchange the order of integral and product in Eqs. 共104兲 and 共105兲, which is of course forbidden.
共⑀兲f q共⑀兲d⑀
⬁
共⑀兲d⑀ + 兺 cn,q共kBT兲n共n−1兲共兲,
共108兲
n=1
with
B. Generalized Sommerfeld expansion
cn,q = −
We will investigate the generalized Sommerfeld expansion for an arbitrary function 共⑀兲 with the q-FDD of f q共⑀兲 given by 关49兴
共104兲
e−uxndu for q ⬎ 1.0,
n=1
n n!
冕
共⑀ − 兲n
f q共 ⑀ 兲 d⑀ . ⑀
共109兲
Substituting f q共⑀兲 in the EA given by Eq. 共65兲 to Eq. 共109兲 and using integrals by part, we obtain cn,q for even n,
冦
n共n − 1兲 共q − 1兲 + ¯ for even n, 1+ EA 2 cn,q = 1 + 共q − 1兲 + ¯ for n = 2, cn,1 1 + 6共q − 1兲 + ¯ for n = 4,
共110兲 共111兲 共112兲
while cn,q = 0 for odd n, where cn,1 denotes the relevant expansion coefficient for q = 1.0: c2,1 = 2 / 6共=1.645兲 and c4,1 = 74 / 360共=1.894兲. Equation 共110兲 shows that cn,q is increased with increasing q. 011126-12
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TABLE II. O共q − 1兲 contributions to cn,q 共n = 1 – 4兲 of the generalized Sommerfeld expansion coefficients. Method EAa IA
c2,q
c1,q 2
0
b
0 2
6 共q − 1兲
SAd
6
关1 + 共q − 1兲兴
6
关1 + 共q − 1兲兴
2
2
FAc
c3,q 0 0 74
2 6 兵1 + O关共q − 1兲 兴其 2
0
6
60 共q − 1兲
关1 + 3共q − 1兲兴
0
c4,q 74
360 关1 + 6共q − 1兲兴
74
360 关1 + 6共q − 1兲兴 2 360 兵1 + O关共q − 1兲 兴其 74 360 关1 + 10共q − 1兲兴
77
a
The exact approach 共the present study兲. The interpolation approximation 共the present study兲. c The factorization approximation 关24兴. d The superstatiscal approximation 关49兴. b
By using f IA q 共⑀兲 in the IA, we may obtain cn,q given by 共for details, see Appendix B 2兲
IA cn,q
cn,1
=
冦
冉
冊
1 +1−n q−1 1 共q − 1兲n⌫ +1 q−1 ⌫
冉
共113兲
for even n,q ⬍ 1,
共114兲
1 2−q
for n = 2,
共115兲
1 共2 − q兲共3 − 2q兲共4 − 3q兲
for n = 4.
共116兲
冉
冊
q +1 1−q q +1+n 共1 − q兲n⌫ 1−q ⌫
冉
It is easy to see that Eqs. 共115兲 and 共116兲 are in agreement with Eq. 共111兲 and 共112兲, respectively, of the EA within O共q − 1兲. A simple calculation using f SA q 共⑀兲 leads to SA cn,q cn,1
=
冦
冉
冊
1 −n q−1 1 共q − 1兲n⌫ q−1
for even n,q ⬎ 1
1 共2 − q兲共3 − 2q兲
for n = 2
⌫
冉 冊
冊
for even n,q ⬎ 1
1 for n = 4, 共2 − q兲共3 − 2q兲共4 − 3q兲共5 − 4q兲
冧
共117兲
which are similar to but different from those given by Eqs. 共115兲 and 共116兲.
冊
The Sommerfeld expansion coefficients in the FA may be calculated with the use of f FA q 共⑀兲 关49兴. A comparison among the O共q − 1兲 contributions to cn,q 共n = 1 – 4兲 in the four methods of EA, IA, FA, and SA is made in Table II. The results of the IA coincide with those of the EA. The O共q − 1兲 contributions to c2,q and c4,q in the SA are three and 5/3 times larger, respectively, than those in the EA. The O共q − 1兲 contributions FA to c2,q and c4,q in the FA are vanishing. It is noted that c1,q FA ⫽ 0 and c3,q ⫽ 0 in contrast with the results of c1,q = c3,q = 0 in the EA, IA, and SA. This is due to a lack of the symmetry in − f FA q 共⑀兲 / ⑀ with respect to ⑀ = as shown in Fig. 10共b兲. Figure 11共a兲 shows the q dependence of coefficients of cn,q / cn,1 for n = 2 and 4 calculated by the four methods. EA for n = 2 and 4, respectively, Circles and squares express cn,q calculated by the EA for kBT / W = 0.1 共Fig. 1兲. Solid curves IA in the IA. The coefficient for n = 2 共n = 4兲 in express cn,q the IA is in good agreement with the result in the EA SA shown by chain curves for 1.0ⱕ q ⱗ 1.5 共1.0ⱕ q ⱗ 1.2兲. cn,q EA IA and cn,q . Dashed are overestimated compared to cn,q FA curves denoting cn,q 关49兴 are plotted only for 0.8ⱕ q ⱕ 1.2 because the FA is considered to be valid for a small 兩q − 1兩 FA is qualitatively different from 关23兴. The q dependence of cn,q
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HIDEO HASEGAWA 2
C. Low-temperature phonon specific heat
log10 [cn,q/cn,1]
(a)
n=4
1
We consider the phonon specific heat at low temperatures. By using Eqs. 共60兲 and 共65兲, we obtain
0 EA IA FA SA
n=2
-1 -20
0.5
1 q
2
1.5
log10 [αq/α1]
共122兲
␣EA q = 1 + 6共q − 1兲 + ¯ , ␣1
共123兲
冉 冊
共124兲
with
2
␣1 =
(b)
1 0
0.5
1 q
1.5
2
FIG. 11. 共Color online兲 共a兲 The q dependence of cn,q / cn,1 for n = 2 and 4 of the generalized Sommerfeld expansion coefficients 关Eq. 共108兲兴 with the q-FDD and 共b兲 the q dependence of ␣q / ␣1 of the coefficients in the low-temperature phonon specific heat with the q-BED, calculated by the EA 共circles and squares兲, IA 共solid curves兲, FA 共dashed curves兲 关49兴, and SA 共chain curves兲: the result of the SA is indistinguishable from that of the FA in 共b兲 共see text兲. FA those of the EA, IA, and SA: cn,q is symmetric with respect to q = 1.0 whereas those in other three methods are monotonously increased with increasing q. The energy of electron systems at low temperatures may be calculated with the use of the generalized Sommerfeld expansion. By using Eqs. 共108兲 and 共110兲 for Eq. 共73兲 with 共⑀兲 = ⑀共⑀兲, we obtain the energy given by
Eq共T兲 ⯝ Eq共0兲 + c2,q共kBT兲2共兲 + ¯ ,
共118兲
from which the low-temperature electronic specific heat is given by Cq共T兲 ⯝ ␥qT + ¯ ,
124 N ak B , 5
where ␣1 is the relevant coefficient for q = 1.0. The coefficients of low-temperature phonon specific heat ␣q in the IA, SA, and FA are given by 共for details, see Appendix B 3兲
EA IA FA SA
-1 -20
C q ⯝ ␣ qT 3 + ¯ ,
共119兲
␣IA 1 q = , ␣1 共2 − q兲共3 − 2q兲共4 − 3q兲
共125兲
␣SA 1 q , = ␣1 共2 − q兲共3 − 2q兲共4 − 3q兲共5 − 4q兲
共126兲
␣FA q = 1 + O关共q − 1兲2兴, ␣1
共127兲
where the O共q − 1兲 contribution to ␣FA q is vanishing 关49兴. EA within Equation 共125兲 shows that ␣IA q agrees with ␣q IA IA O共q − 1兲 and that the ␣q is related with c4,q as ␣IA q / ␣1 IA = c4,q / c4,1. Coefficients of ␣q / ␣1 calculated by the four methods are plotted as a function of q in Fig. 11共b兲. Squares denote the result of numerical calculation by the EA for T / TD = 0.01 共Fig. 4兲. The solid curve expresses ␣IA q , which is in good agreement with the result of the EA for 1.0ⱕ q ⱗ 1.2 but deviates from it at q ⲏ 1.2. Dashed and chain curves show ␣q calculated by the FA and SA, respectively. It is interesting that the result of the SA nearly coincides with that of the FA for 1.0ⱕ q ⱗ 1.2, where both the results of the SA and FA are overestimated compared to the EA. The inset of Fig. 4 shows that the energy Eq at low temperatures in the Debye model is larger for larger q, which is consistent with the q dependence of ␣q shown in Fig. 11共b兲.
with
␥q c2,q = , ␥1 c2,1 ␥1 =
2 2 k 共兲, 3 B
V. CONCLUDING REMARKS
共120兲
共121兲
where ␥1 is the linear-T expansion coefficient for q = 1.0. The inset of Fig. 1 shows that the calculated energy Eq at low temperatures in the electron model is larger for a larger q, which is consistent with larger ␥q and c2,q for a larger q as shown in Fig. 11共a兲.
It is well known that in nonextensive classical statistics, the nonextensivity arises from the long-range interaction, the long-time memory, and a multifractal-like space time 关2兴. The metastable state or quasistationary state is characterized by long-range interaction and/or fluctuations of intensive quantities 共e.g., the inverse temperature兲 关10兴. For example, in the long-range-interacting gravitating systems, the physical quantities are not extensive: the velocity distribution obeys the power law and the stable equilibrium state is lacking, which lead to negative specific heat 关63兴. The situation is the same also in nonextensive quantum statistics. It has been
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As for item 共iv兲, the q-BED and q-FDD in the IA are expected to be useful and to play important roles in the nonextensive quantum statistics 关61兴.
reported that the observed black-body radiation may be explained by the nonextensivity of the order of 兩q − 1兩 ⬃ 10−4 – 10−5, which is attributed to the long-range Coulomb interaction 关21兴. Memory effect and long-range interaction cannot be neglected in weakly nonideal plasma of stellar core 关64兴. In addition to the large systems where the interactions may be truly long range, one should consider small systems where the range of the interactions is of the order of the system size. Small-size systems would not be extensive, and many similarities with the long-range case will be realized. Indeed, the negative specific heat is observed in 147 sodium clusters 关65兴. Magnetic properties in nanomagnets may be different from those in large-size ones 关66兴. Small drops of quantum fluids may undergo a Bose-Einstein condensation. Thanks to recent development in the evaporation cooling technique, it becomes possible to study BoseEinstein condensation in an extremely diluted fluid where the long-range interactions play essential roles in the condensate stability. Artificial sonic or optical black hole 关67,68兴 represents an intrigue quantum catastrophic phenomenon. Only little is known about the thermodynamics of these quantum systems. Experimental and theoretical studies on these subjects deepen our understanding of basic quantum phenomena. To summarize, we have discussed the generalized distributions of q-BED and q-FDD in nonextensive quantum statistics based on the EA 关50,51兴 and IA. Results obtained are summarized as follows: 共i兲 with increasing q above q = 1.0, the q-BED and q-FDD have long tails, while they have compact distributions with decreasing q from unity, 共ii兲 the coefficients in the generalized Sommerfeld expansion, the linear-T coefficient of electronic specific heat, and the T3 coefficient of phonon specific heat are increased with increasing q above unity, whereas they are decreased with decreasing q below unity, 共iii兲 the O共q − 1兲 contributions in the EA agree with those in the AA based on the OLM-MEM 关5兴 as well as the unnormalized MEM 关3兴, and 共iv兲 the generalized distributions given by simple expressions in the IA proposed in this study yield results in agreement with those obtained by the EA within O共q − 1兲 and high- and low-temperature limits.
冦
Zq =
Oq =
冦
冕 冉 冕 冉 ⬁
G u;
0
i 2
冕 冉 冕 冉 ⬁
1 Zqq
G u;
0
i
2Zqq
APPENDIX A: THE (q − 1) EXPANSION IN THE UN-NORMALIZED MEM
Tsallis et al. 关21兴 developed the AA to investigate the nonextensivity in the observed black-body radiation by using the un-normalized MEM 关3兴. We will show that the EA with the un-normalized MEM yields the result in agreement with the AA within O共q − 1兲. Calculations of the q-BED and q-FDD for q ⯝ 1.0 are presented. 1. Un-normalized MEM
An application of the un-normalized MEM to the Hamilˆ yields the optimized density matrix given by 关3兴 tonian H
ˆ q =
1 ˆ 兴1/共1−q兲 , 关1 − 共1 − q兲H Zq
ˆ 兴1/共1−q兲其. Zq共兲 = Tr兵关1 − 共1 − q兲H
共A1兲 共A2兲
ˆ is given by The expectation value of the operator O ˆ 典 = Tr兵ˆ qO ˆ Oq共兲 ⬅ 具O q q 其 =
1 Zqq
ˆ 兴q/共1−q兲O ˆ 其. Tr兵关1 − 共1 − q兲H
共A3兲 共A4兲
2. Exact approach
With the use of the exact representations given by Eqs. 共5兲 and 共27兲, Eqs. 共A2兲 and 共A4兲 are expressed by
冊
for q ⬎ 1,
共A5兲
1 1 Z1共− t兲dt for q ⬍ 1, , 1 − q 共1 − q兲
共A6兲
冊
冊
1 1 + 1, Z1共u兲O1共u兲du q−1 共q − 1兲
H t;
C
This work was partly supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
1 1 , Z1共u兲du q − 1 共q − 1兲
H t;
C
ACKNOWLEDGMENT
for q ⬎ 1,
共A7兲
1 1 − 1, Z1共− t兲O1共− t兲dt for q ⬍ 1, 1−q 共1 − q兲
共A8兲
冊
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HIDEO HASEGAWA
with
ˆ 2典 = 2共⑀ − 兲2 f 3 − 2共⑀ − 兲f 2E + f 共E2 + E + E 兲, 具nˆkH 1 k k 1 1 2 3 1 1 1 ˆ
O1共u兲 =
ˆ其 Tr兵e−uHO , Z1共u兲
共A9兲
ˆ
Z1共u兲 = Tr兵e−uH其,
共A18兲 with E1 = 兺 共⑀k − 兲f 1 ,
共A10兲
k
where C denotes the Hankel contour and G共u ; a , b兲 and H共t ; a , b兲 are given by Eqs. 共14兲 and 共31兲, respectively. In order to evaluate Eqs. 共A5兲–共A8兲, we expand their integrands around u =  and −t =  as is made in Sec. II C. By using Eqs. 共22兲, 共23兲, 共35兲, and 共36兲, we obtain
1 Zqq
冋
O1 + 共q − 1兲
共Z1O1兲 
k
E3 = 兺 共⑀k − 兲2 f 21 . k
1 2Z 1 Zq = Z1 + 共q − 1兲2 2 + ¯ , 2  Oq =
E2 = 兺 共⑀k − 兲2 f 1 ,
共A11兲
Substituting Eqs. 共A15兲–共A18兲 to Eq. 共A13兲, we obtain f q ⯝ f 1 + 共1 − q兲兵f 1 ln Z1 + 关共⑀k − 兲ex f 21 + f 1E1兴其
册
1 2 + 共q − 1兲2 2 共Z1O1兲 + ¯ . 2 
− 共A12兲
共1 − q兲2 关共⑀k − 兲2ex共ex + 1兲f 31 + 2共⑀k − 兲ex f 21E1兴 2 共A19兲
+ ¯.
Tsallis et al. 关21兴 employed a one-component boson Hamiltonian given by
By using the relations given by
Z1 ˆ典 Z , = − 具H 1 1  Z1 ˆ 2典 Z , = 具H 1 1 2
具nˆk典1 =
O1 ˆO ˆ典 , = 具H典1具O典1 − 具H 1  2O 1 ˆ 2O ˆ 典 − 具H ˆ 2典 具O ˆ 典 + 2关具H ˆ 典2具O ˆ 典 − 具H ˆO ˆ 典 具H ˆ 典 兴, = 具H 1 1 1 1 1 1 1 2 we finally obtain the O共q − 1兲 expansion of Oq given by
共
which agrees with Eq. 共7兲 in Ref. 关21兴 derived by the AA. 共1兲 q -BED. In order to calculate the q-BED, we consider ˆ = nˆ with the Hamiltonian for bosons given by O k
ˆ 典 = ⑀共ex + 1兲f , 具nˆH 1 1
共A22兲
ˆ 2典 = ⑀2共ex + 1兲f 2 , 具H 1 1
共A23兲
ˆ 2典 = ⑀2共e2x + 4ex + 1兲f 3 . 具nˆH 1 1
共A24兲
共A14兲
共A25兲
+ ¯,
which is different from Eq. 共A19兲 with = 0 because of the difference in the adopted Hamiltonians given by Eqs. 共A14兲 and 共A20兲. ˆ = nˆ with the Hamiltonian for 共2兲 q -FDD. We consider O k fermions given by ˆ = 兺 共⑀ − 兲nˆ , H k k
k
where nˆk and ⑀k stand for the number operator and the energy of the state k. We obtain 具nˆk典1 =
1 = f 1共 ⑀ k兲 ⬅ f 1 ex − 1
共A21兲
共x = ⑀兲,
f q ⯝ f 1 + 共1 − q兲关 f 1 ln Z1 + x共ex + 1兲f 21 − 21 x2ex共ex + 3兲f 31兴
共A13兲
ˆ = 兺 共⑀ − 兲nˆ , H k k
1 ⬅ f1 ex − 1
A substitution of Eqs. 共A21兲–共A24兲 to Eq. 共A13兲 leads to
ˆ 2典 O ˆO ˆ 典 + 1 2关具H Oq ⯝ O1 + 共1 − q兲 O1 ln Z1 + 具H 1 1 1 2
兲
共A20兲
which yields
2
ˆ 2O ˆ典 兴 + ¯ , − 具H 1
ˆ = បnˆ ⬅ ⑀nˆ , H
共A26兲
k
which leads to
关x = 共⑀k − 兲兴, 共A15兲
具nˆk典1 =
1 = f 1共 ⑀ k兲 ⬅ f 1 e +1 x
关x = 共⑀k − 兲兴, 共A27兲
ˆ 典 = 共⑀ − 兲ex f 2 + f E , 具nˆkH 1 k 1 1 1
共A16兲
ˆ 典 = 共⑀ − 兲f 共1 − f 兲 + f E , 具nˆkH 1 k 1 1 1 1
共A28兲
ˆ 2典 = E 2 + E + E , 具H 1 2 3 1
共A17兲
ˆ 2典 = E 2 + E − E , 具H 1 2 3 1
共A29兲
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ˆ 2典 = 共⑀ − 兲2 f 共1 − f 兲共1 − 2f 兲 + 2共⑀ − 兲f 共1 − f 兲E 具nˆkH 1 k 1 1 1 k 1 1 1 + f 1共E21 + E2 − E3兲.
共A30兲
Substituting Eqs. 共A27兲–共A30兲 to Eq. 共A13兲, we obtain
1. Analytic expressions of q-FDD for 円(⑀ − )円 ™ 1
We may obtain an expression of the q-FDD for small x关=共⑀ − 兲兴 with the use of an expansion for f 1共⑀兲 given by
f q ⯝ f 1 + 共1 − q兲兵f 1 ln Z1 + 关共⑀k − 兲f 1共1 − f 1兲 + f 1E1兴其 −
APPENDIX B: SUPPLEMENT TO THE INTERPOLATION APPROXIMATION
共1 − q兲2 关共⑀k − 兲2 f 1共1 − f 1兲共1 − 2f 1兲 2
⬁
1 + 兺 dn,1xn 2 n=1
f 1共 ⑀ ,  兲 =
+ 2共⑀k − 兲f 1共1 − f 1兲E1兴 + ¯ .
共A31兲
When assuming a one-component fermion Hamiltonian given by ˆ = 共⑀ − 兲nˆ , H k k
⬁
f IA q 共⑀, 兲 =
1 + 兺 dn,qxn 2 n=1
with 1 ⬅ f1 具nˆk典1 = x e +1
关x = 共⑀k − 兲兴,
共q − 1兲n⌫
冉
dn,q = dn,1 共A34兲
ˆ 2典 = 共 ⑀ − 兲 2 f , 具H 1 k 1
共A35兲
ˆ 2典 = 共 ⑀ − 兲 2 f . 具nˆkH 1 k 1
共A36兲
共1 − q兲n⌫
f FA q ⯝ f1 −
冦
共A37兲
共1 − q兲 2  共⑀ − 兲2e共⑀−兲 f 21 + ¯ , 2
共A38兲
冋
再
再
f1 1 2 f 1 − 共⑀ − 兲 + 共⑀ − 兲2 2 ⑀ 2 ⑀
冎册
f1 ⑀
+ ¯,
冎
for 1 ⬍ q ⬍ 3,
冉
q +1 1−q
q +1−n ⌫ 1−q
冊
冊
for 0 ⬍ q ⬍ 1, 共B4兲
for n = 1,
共B5兲
for n = 2, dn,q = dn,q = q共2q − 1兲dn,1 dn,q = q共2q − 1兲共3q − 2兲dn,1 for n = 3,
共B6兲
qdn,1
共B7兲
where dn,1 = 共1 / n!兲n f 1共⑀ , 兲 / xn at x = 0: d1,1 = −1 / 4, d2,1 = 0, d3,1 = 1 / 48, etc. Equations 共B2兲–共B7兲 lead to f IA q 共⑀, 兲 ⯝
q共2q − 1兲共3q − 2兲 3 1 q − x+ x + ¯ 2 4 48
for 兩x兩 ⬍ 1. 共B8兲
whose O共q − 1兲 term corresponds to the last term of Eq. 共A37兲 derived by the un-normalized MEM. This is due to the fact that to adopt the one-component Hamiltonian given by Eq. 共A32兲 means to use the factorization approximation from the beginning. Equation 共A19兲 for q-BED and Eq. 共A31兲 for q-FDD are expressed in a unified way as f q ⯝ f 1 + 共1 − q兲 f 1 ln Z1 + E1 f 1 + 共⑀ − 兲
冉
dn,q = dn,1
f q ⯝ f 1 + 共1 − q兲关 f 1 ln Z1 + 共⑀ − 兲f 1 The difference between Eqs. 共A31兲 and 共A37兲 is due to the difference in the adopted Hamiltonians given by Eqs. 共A26兲 and 共A32兲. It is noted that the 共q − 1兲 expansion of q-FDD in the FA is given by
冊
冊
共B2兲
共B3兲
Substituting Eqs. 共A33兲–共A36兲 to Eq. 共A13兲, we obtain − 21 2共⑀ − 兲2e共⑀−兲 f 21兴 + ¯ .
冉
for 兩x兩 ⬍ 1,
1 +1+n q−1 1 ⌫ +1 q−1
共A33兲
ˆ 典 = 共⑀ − 兲f , 具nˆkH 1 k 1
共B1兲
Substituting Eq. 共B1兲 to Eqs. 共76兲 and 共77兲 and employing Eqs. 共5兲 and 共27兲, we obtain
共A32兲
we obtain
for 兩x兩 ⬍ 1.
2. Generalized Sommerfeld expansion in the IA
In the case of q ⬎ 1.0, Eq. 共61兲 yields
f q共 ⑀ 兲 =− ⑀
冕 冉 ⬁
G u;
0
冊
1 共⑀ − 兲eu共⑀−兲 q , du. q − 1 共q − 1兲 关eu共⑀−兲 + 1兴2 共B9兲
共A39兲
where f 1 = 1 / 共ex ⫿ 1兲. We note that the O共q − 1兲 term of the generalized distribution in Eq. 共65兲 derived by the OLMMEM corresponds to the last term in the bracket of Eq. 共A39兲.
Substituting Eq. 共B9兲 to Eq. 共109兲 and changing the order of integrations for ⑀ and u, we obtain cn,q =
n n!
冕 冉 ⬁
G u;
0
冊 冕
1 q , u−ndu q − 1 共q − 1兲
x ne x dx. 共e + 1兲2 x
共B10兲 At low temperatures, Eq. 共B10兲 reduces to
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HIDEO HASEGAWA
cn,q =
2共1 − 21−n兲共n兲 共q − 1兲n
冉
冕 冉 ⬁
G u;
0
冊
1 ⌫ +1−n q−1 cn,q = cn,1 1 +1 共q − 1兲n⌫ q−1
冉
cn,q = 0
冊
3. Low-temperature phonon specific heat in the IA
q ,1 u−ndu, 共B11兲 q−1
In the case of q ⬎ 1.0, Eqs. 共60兲 and 共76兲 yield C q ⯝ k B 2
冊
for even n, 共B12兲
⫻
冕
冕 冉 ⬁
G u;
0
⬁
0
共B13兲
for odd n.
=
q ,1 q−1
冊
共兲共q − 1兲共ប兲2ue共q−1兲បu ddu 关e共q−1兲បu − 1兴2
冉 冊冕 冉
冊 冕
⬁
3
9NakB T 共q − 1兲4 ⌰D
q ,1 u−4du q−1
G u;
0
⬁
0
x 4e x dx 共e − 1兲2 x
The ratio of cn,q / cn,1 is given by
cn,q = cn,1
冦
冉
冊
1 +1−n q−1 1 +1 共q − 1兲n⌫ q−1 ⌫
冉
共B23兲
冊
for even n,
共B14兲
for n = 2,
共B15兲
1 for n = 4. 共2 − q兲共3 − 2q兲共4 − 3q兲
共B16兲
In the case of q ⬍ 1.0, Eqs. 共62兲 and 共109兲 yield cn,q =
2共1 − 2 兲共n兲 i 2 共1 − q兲n
冕 冉
H t;
C
冊
q ,1 共− t兲−ndt, 1−q 共B17兲
冉
冊
q ⌫ +1 1−q cn,q = cn,1 q +1+n 共1 − q兲n⌫ 1−q
冉
冉 冊 T TD
3
共B24兲
,
with
1 2−q
1−n
= ␣q
冊
冉 冊
for 1 ⬍ q ⬍ 3,
for even n,
⫻
冕
冉 冊冕 冉
⬁
0
i 2
H t;
C
q ,1 1−q
冊
共兲共1 − q兲共ប兲2共− t兲e−共1−q兲បt ddt 关e−共1−q兲បt − 1兴2 共B26兲
共B18兲 cn,q = 0 leading to
cn,q = cn,1
冦
冉
for odd n,
冊
q +1 ⌫ 1−q q +1+n 共1 − q兲n⌫ 1−q
冉
1 2−q 1 共2 − q兲共3 − 2q兲共4 − 3q兲
共B19兲
=
冉 冊 冉 冊冕 冉
9NakB T 共1 − q兲4 TD ⫻共− t兲4dt
冊
冕
⬁
0
for even n,
共B20兲
for n = 2,
共B21兲
for n = 4.
from which we obtain
冉
3
i 2
冉
H t;
C
q ,1 1−q
冊
x 4e x dx, 共ex − 1兲2
冊
q +1 1−q ␣q = ␣1 q +5 共1 − q兲4⌫ 1−q ⌫
共B22兲
Equation 共B20兲 for q ⬍ 1.0 is the same as Eq. 共B14兲 for q ⬎ 1.0 if we employ the reflection formula of the gamma function,
共B25兲
where TD 共=បD / kB兲 stands for the Debye temperature and ␣1 is the T3 coefficient of the low-temperature specific heat for q = 1.0. In the case of q ⬍ 1.0, a similar analysis with the use of Eqs. 共60兲 and 共77兲 leads to C q ⯝ k B 2
冊
冉
1 −3 q−1 ␣q = ␣1 q 共q − 1兲4⌫ q−1 ⌫
冊
共B27兲
for 0 ⬍ q ⬍ 1. 共B28兲
Equations 共B16兲, 共B22兲, 共B25兲, and 共B28兲 yield
. ⌫共z兲⌫共1 − z兲 = sin共z兲
1 ␣q c4,q = = ␣1 共2 − q兲共3 − 2q兲共4 − 3q兲 c4,1
for 0 ⬍ q ⬍ 4/3. 共B29兲
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