In a recent paper (Tran et al, Ann. Phys.311, 204 (2004)), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for the restricted or coloured partitions pks (n), which is the number of partitions of an integer n into the summand of sth powers of integers such that each power of a given integer may occur utmost k times. While the method is not rigorous, it reproduces the well-known asymptotic results for s = 1 apart from yielding more general results for arbitrary values of s

Bhaduri, R. K.

Murthy, M. V. N.

Srivatsan, C. S.

English

Publications of the IAS Fellows

PRAMANA c© Indian Academy of Sciences Vol. 66, No. 3
— journal of March 2006
physics pp. 485–494
Gentile statistics and restricted partitions
C S SRIVATSAN1, M V N MURTHY2 and R K BHADURI3
1Department of Physics, Indian Institute of Technology, Kanpur 208 016, India
2The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India
3Department of Physics, McMaster University, Hamilton, Ont. L8S 4M1, Canada
E-mail: murthy@imsc.res.in
MS received 19 August 2005; revised 13 December 2005; accepted 24 January 2006
Abstract. In a recent paper (Tran et al, Ann. Phys. 311, 204 (2004)), some asymptotic
number theoretical results on the partitioning of an integer were derived exploiting its
connection to the quantum density of states of a many-particle system. We generalise
these results to obtain an asymptotic formula for the restricted or coloured partitions
psk(n), which is the number of partitions of an integer n into the summand of sth powers
of integers such that each power of a given integer may occur utmost k times. While the
method is not rigorous, it reproduces the well-known asymptotic results for s = 1 apart
from yielding more general results for arbitrary values of s.
Keywords. Restricted partitions; colour partitions; quantum density of states; Gentile
statistics.
PACS Nos 03.65.Sq; 02.10.De; 05.30.-d
1. Introduction
It is well-known that in one dimension, the density of states of a system of ideal
bosons confined in a harmonic trap at a given energy E, in the N → ∞ limit,
is the same as the number of ways of partitioning an integer n = E into a sum
of integers [1]. The result is the celebrated Hardy–Ramanujan formula [2]. An
in-depth analysis of the fundamental additive decomposition of positive integers by
sums of other positive integers with various restrictions is given in ref. [3].
In a recent paper, Tran et al [4] exploited the connection between quantum
many-particle density of states and integer partitions in number theory to obtain
more general results for the additive decomposition or partitioning of an integer.
It was shown that the asymptotic density of states for ideal bosons in a system
with a power-law energy spectrum, is analogous to ps(n), that is, the number
of ways of partitioning an integer n as a sum of sth powers of integers. It was
further shown that distinct partitions ds(n) correspond to the asymptotic density
of states of a pseudofermion-like system in a power-law energy spectrum. The
name pseudofermion is coined to indicate that the system’s density of states was
485
C S Srivatsan, M V N Murthy and R K Bhaduri
constructed by applying the exclusion principle only to the particle distribution.
The hole distribution was neglected while allowing any number of particles in the
ground state, i.e., a collapse of the ground state, where the Fermi energy EF is set
to zero. For the rest of the paper we follow the same procedure as in [4] where only
particle distribution, relevant to partitions, is taken into account but not the hole
distribution. We note that the method is not rigorous but yields correct asymptotic
results in the well-known cases.
In this paper we extend the analysis to obtain an asymptotic formula for restricted
or coloured partitions. We define psk(n) as the number of partitions of an integer n
into a sum of sth powers of integers, where each power of a given number occurs
utmost k times. The method used here is similar to the one used in the previous
paper [4]. The leading exponential approximation for the asymptotic partition of
an integer into integer summands, s = 1, has been discussed recently by Blencowe
and Koshnick [5].
We first construct the partition function for a statistical system in which the
maximal occupancy of each state is given by k. Obviously k = 1 corresponds to
a fermionic system whereas there are no restrictions on k for a bosonic system.
Indeed the partition function obtained by this minimal restriction is the same as
that obtained by Gentile [6] and the corresponding statistics is known as Gentile
statistics. The partition function of Gentile statistics also has the property that it
nicely interpolates between the Fermi and Bose statistics. While the asymptotic
formula, obtained using Gentile statistics, reduces to the result for distinct par-
titions in the fermionic, k = 1, limit, the unrestricted or bosonic partitions are
obtained by taking the k →∞ limit.
2. Many-particle density of states
Before we proceed with the main theme of this paper, we recall briefly the results
from the previous analysis for the sake of completeness. Consider a system of N
particles, where N is very large. In general, the particles may obey any arbitrary
statistics. The canonical N -particle partition function is given by
ZN (β) =
∫ ∞
0
ρN (E) exp(−βE)dE, (1)
where β is the inverse temperature, E(N)i are the eigenenergies of the N -particle
system and ρN (E) =
∑
i δ(E − E(N)i ) is the N -particle density of states. The
density of states ρN (E) may therefore be obtained through the inverse Laplace
transform of the canonical partition function
ρN (E) =
1
2pii
∫ i∞
−i∞
exp(βE)ZN (β)dβ. (2)
In general, it is not always possible to do this inversion analytically. As in the
case of single-particle density of states, the many-particle density of states may
be decomposed into an average (smooth) part and an oscillating part [7]. The
smooth part ρ¯N (E) may be obtained by evaluating eq. (2) using the saddle-point
486 Pramana – J. Phys., Vol. 66, No. 3, March 2006
Gentile statistics and restricted partitions
method [8]. However, unlike the one-particle case, where the oscillating part may be
obtained using the periodic orbits in a ‘trace formula’ [7], it remains a challenging
task to find an expression for the oscillating part δρN (E) [9]. In what follows we
shall use the saddle-point method to obtain the smooth part of the density of states,
ρ¯N (E), and identify it with the number of ways the energy E is partitioned, on the
average, among N particles.
Defining the entropy, S(β), as
S(β) = βE + logZN (3)
and expanding the entropy around the stationary point β0 and retaining only up
to the quadratic term in eq. (2) yields the standard result [8]
ρ¯N (E) =
exp[S(β0)]√
2piS′′(β0)
, (4)
where the prime denotes differentiation with respect to inverse temperature and
E = −
(
∂ lnZN
∂β
)
β0
. (5)
The above analysis holds for any N -particle system irrespective of the nature of
the single-particle spectrum. Hereafter we omit the subscript N since we are only
interested in the asymptotic limit N →∞.
3. Restricted partitions
We consider a statistical system in which a given quantum state has a maximal oc-
cupancy of k in each single-particle state. Thus when k = 1 the system corresponds
to a system of fermions whereas there are no such restrictions on k for a system
of bosons except for symmetry considerations. We now construct the partition
function for such a system which also incorporates the property of interpolation
between Bose and Fermi statistics as k takes on different values.
In order to relate to the problem of integer partitions, we consider a system with
a single-particle spectrum given by ²m = ms, where the integer m ≥ 1 and s > 0.
The energy is rendered dimensionless by appropriate choice of units. For example,
when s = 1 the spectrum can be mapped on to the spectrum of a one-dimensional
oscillator, where the energy is measured in units of h¯ω. For s = 2, it is equivalent
to setting energy unit as h¯2/2M , whereM is the particle mass in a one-dimensional
square well with unit length. These are the only two physically interesting cases.
As in [4], we however keep s arbitrary even though for s > 2 there are no quadratic
Hamiltonian systems.
We define a restricted or coloured partition of an integer n as its additive decom-
position into integer powers where no integer is repeated more than k times. For
example, consider partitions of an integer, say N = 6 as a summand of integers:
Pramana – J. Phys., Vol. 66, No. 3, March 2006 487
C S Srivatsan, M V N Murthy and R K Bhaduri
6, 5 + 1, 4 + 2, 3 + 2 + 1, (p11(6) = 4)
3 + 3, 4 + 1 + 1, 2 + 2 + 1 + 1, (p12(6) = 7)
3 + 1 + 1 + 1, 2 + 2 + 2, (p13(6) = 9)
2 + 1 + 1 + 1 + 1, (p14(6) = 10)
1 + 1 + 1 + 1 + 1 + 1, (p16(6) = 11). (6)
We are interested in the relation between the asymptotic expression for such re-
stricted partitions and the density of states of the system which allows utmost k
occupancy of the single-particle states.
The relevant physical system for obtaining partitions is thus one in which at zero
temperature all the particles are in the ground state whose energy is set to zero. At
any excitation energy, particles are excited from the ground state and distributed
in the single-particle states with maximal occupancy given by k in each state. The
number of ways in which such a distribution can be achieved is the density of states
at that energy (as also the number of ways of partitioning the energy).
The partition function for such a system in the limit of the number of particles
N →∞ is given by
Z∞(β) =
∞∏
m=1
[1 + exp(−βms) + · · ·+ exp(−kβms)]
=
∞∏
m=1
k∑
n=0
exp(−nβms), (7)
where the single-particle spectrum is given by a power law. We note that the
partition function in eq. (7) is indeed the grand-canonical partition function with
the chemical potential µ = 0. In particular k = 1,∞ correspond to the two cases
considered in the earlier paper [4]. Note that with k arbitrary this is indeed the
partition function for the well-studied Gentile statistics [6].
By setting x = exp(−β), the partition function may be written as
Z∞(x) =
∞∑
n=1
psk(n)x
n =
∞∏
n=1
[1− x(k+1)ns ]
[1− xns ] . (8)
In the asymptotic limit of a large number of particles, psk(n) is the number of ways
of partitioning n.
Using eq. (3), we obtain
S = βE +
∞∑
n=1
ln[1− exp(−(k + 1)βns)]
−
∞∑
n=1
ln[1− exp(−βns)]. (9)
We now evaluate the sums approximately using the Euler–Maclaurin series. We
have
488 Pramana – J. Phys., Vol. 66, No. 3, March 2006
Gentile statistics and restricted partitions
−
∞∑
n=1
ln[1− exp(−βns)] = −
∞∑
n=0
ln[1− exp(−β(n+ 1)s)]
= −
∫ ∞
0
ln[1− exp(−β(x+ 1)s)]dx
−1
2
ln[1− exp(−β)] + s
∞∑
k=1
B2k
2k(2k − 1) , (10)
where B2k are the Bernoulli numbers. The integral in the Euler–Maclaurin expan-
sion may be evaluated as follows:
−
∫ ∞
0
ln[1− exp(−β(x+ 1)s)]dx
= −
∫ ∞
0
ln[1− exp(−βxs)]dx+
∫ 1
0
ln[1− exp(−βxs)]dx
=
C(s)
β1/s
+ ln[1− exp(−β)]− s
∫ 1
0
βxs
exp(βxs)− 1dx, (11)
where
C(s) = Γ
(
1 +
1
s
)
ζ
(
1 +
1
s
)
(12)
is given in terms of the Riemann zeta function. In the high-temperature limit, that
is β → 0, we have
−
∫ ∞
0
ln[1− exp(−β(x+ 1)s)]dx ≈ C(s)
β1/s
+ ln[1− exp(−β)]− s+O(β).
(13)
The series involving Bernoulli numbers may be evaluated approximately as fol-
lows: The Euler–Maclaurin series applied to logarithm of gamma function gives
ln(Γ(n+ 1)) = n ln(n)− n+ 1
2
ln(2pi) +
∞∑
k=1
B2k
2k(2k − 1)n2k−1 . (14)
We may consider this as the expansion of zero by putting n = 1 and get
s
∞∑
k=1
B2k
2k(2k − 1) = s−
s
2
ln(2pi). (15)
Thus keeping terms up to O(β), and using the above arguments, the last term
in eq. (9) may be written as
−
∞∑
n=1
[1− exp(−βns)] ≈ C(s)
β1/s
+
ln[1− exp(−β)]
2
− s
2
ln(2pi) +O(β).
(16)
Pramana – J. Phys., Vol. 66, No. 3, March 2006 489
C S Srivatsan, M V N Murthy and R K Bhaduri
One may also get the above result directly using the symbolic math program
MAPLE. The infinite sum in the second term of eq. (9) is also evaluated simi-
larly by scaling β appropriately. Note that the constant term does not get scaled.
Thus the entropy to the leading order in β is given by
S = βE +
αC(s)
β1/s
− 1
2
ln[1− exp(−(k + 1)β)]
+
1
2
ln[1− exp(−β)] +O(β), (17)
where
α = 1− 1
(k + 1)1/s
. (18)
We note that the expansion in powers of β is equivalent to taking the high-
temperature limit though this limit should be employed cautiously when β is
weighted by k which can in principle take large values. The last term of O(β)
in the RHS of eq. (17) causes a shift in the energy E by a constant (for example
1/24 in the s = 1 case) in S. In the asymptotic formula for the level density for
large E, obtained by the saddle-point method, this may be ignored. Note, however,
that such a shift was included by Rosenzweig [10] to improve the Bethe formula
[11] for the nuclear level density. We further note that the k →∞, or the bosonic
limit, is special since this limit involves precisely the sum given in eq. (16) and
not the difference of two series as in eq. (9). As a result the β independent term,
namely −(s/2) ln(2pi), appears only in this limit and not for any finite k. Indeed,
as we shall see below, this term is crucial in getting the prefactor correctly in the
Hardy–Ramanujan Formula.
The saddle point on the real axis [11a] is evaluated by taking the derivative of
the entropy,
S′(β) = E − 1
s
αC(s)
β(1+1/s)
, (19)
where only the leading terms in β are retained. Equating the derivative to zero,
the saddle point in β is given by
β0 =
(
αC(s)
sE
)s/(1+s)
= κsE−(s/(1+s)), (20)
where
κs =
(
αC(s)
s
)s/(1+s)
. (21)
Substituting this in the saddle-point expression for the density of states in eq.
(4), the asymptotic density of states is given by
ρ¯
(s)
k (E) ≈ κs
√
s
× exp[κs(s+ 1)E
1/(1+s)]√
2pi(s+ 1)E(3s+1)/(s+1)[1− exp(−(k + 1)κsE−s/(s+1))]
, (22)
490 Pramana – J. Phys., Vol. 66, No. 3, March 2006
Gentile statistics and restricted partitions
where the subscript k in ρ¯ indicates the main property of the statistical system
under consideration. We have kept the exponential term under the square-root sign
in the denominator to indicate the interpolation property of the density of states
between the Bose (k → ∞) and the Fermi (k = 1) limits even though keeping the
exponential may not be consistent with the order of the expansion. The correct
expression is given below.
For finite k there always exists an energy E which is large enough such that the
following approximation is useful:
ρ¯
(s)
k (E) ≈
√
sκs
exp[κs(s+ 1)E1/(1+s)]√
2pi(s+ 1)(k + 1)E(2s+1)/(s+1)
. (23)
Note that the above expression is identical to eq. (23) for ds(n) in [4] corresponding
to the special case of k = 1. The degeneracy or k dependence in the above equation
is also hidden in the parameter κs. An asymptotic formula derived in ref. [5] based
on Gentile statistics for the special case of s = 1 is the same as the one given in
eq. (23). However, to the best of our knowledge, no general formula for arbitrary
s and k exists.
In particular for s = 1, k = 1 the above equation reduces to the well-known
formula [3,12] for distinct partitions of an integer into a set of integers
ρ¯(E) ≈ d(n = E) =
exp
[
pi
√
E/3
]
4× 31/4E3/4 . (24)
The bosonic limit may be obtained from eq. (22) by taking the limit k → ∞.
However, we also note that this limit should be taken in such a way that the β
independent factor in eq. (16) is also included. With this proviso we obtain
ρ¯
(s)
k (E) ≈ κs
√
s
exp[κs(s+ 1)E1/(1+s)]
(2pi)(s+1)/2
√
(s+ 1)E(3s+1)/(s+1)
, (25)
which is the celebrated Hardy–Ramanujan asymptotic formula for integer parti-
tions. In particular for s = 1, we have the well-known result (see for example ref.
[12]).
ρ¯(E) ≈ p(n = E) =
exp
[
pi
√
2E/3
]
4
√
3E
. (26)
In figure 1 we show a comparison between the exact pk(n) (continuous line),
and the asymptotic density of states, ρ¯(1)k (n) (dashed line) obtained from eq. (23)
for various values of k when the power s = 1. The asymptotics is reached earlier
for smaller k than for larger k as can be seen easily from the figures. In fact for
k = 1 the formula is almost exact even for n small. We note that the numerical
estimates for the density based on eq. (22) are usually higher than the exact values
and asymptotically reach the exact value later than the density calculated based
on eq. (23) which is shown in the figure. The saddle point evaluated by keeping
Pramana – J. Phys., Vol. 66, No. 3, March 2006 491
C S Srivatsan, M V N Murthy and R K Bhaduri
100
101
102
103
104
105
106
107
108
109
 0  10  20  30  40  50  60  70  80  90  100
p1 k
( n )
 a n
d  ρ
1 k( E
)
n(or E)
k=3
k=4
k=2
k=1
k=10
Figure 1. Comparison of the exact p1k(n) (solid line) and the asymptotic
ρ¯1k(n) (dashed line) for s = 1
100
101
102
103
104
105
106
107
108
 200  300  400  500  600  700  800  900  1000
p2 k
( n )
 a n
d  ρ
2 k( E
)
n(or E)
k=1
k=2
k=3
k=4
k=10
Figure 2. Comparison of the exact pk2(n) (solid line) and the asymptotic
ρ¯2k(n) (dashed line) for s = 2.
only the two leading terms in β ensures that eq. (23) is more accurate for small
values of k though asymptotically both equations yield the same result.
Similarly, in figure 2, the computed p2k(n) is compared with ρ¯
2
k(n). We note that
in this case the fluctuation about the mean is larger for small values of k. In general
the fluctuations decrease with increasing k for a given s but increase with increasing
s for a given k as shown in figure 3 where we have plotted the asymptotic density
of states for s = 3 and various values of k.
492 Pramana – J. Phys., Vol. 66, No. 3, March 2006
Gentile statistics and restricted partitions
100
101
102
103
 400  500  600  700  800  900  1000
p3 k
( n )
 a n
d  ρ
3 k( E
)
n(or E)
k=3
k=4
k=6
k=10
Figure 3. Comparison of the exact pk3(n) (solid line) and the asymptotic
ρ¯2k(n) (dashed line) for s = 3.
4. Summary
To summarise, this paper extends the results of ref. [4] to the case of restricted
or coloured partitions. We identify the relevant generating function for coloured
partitions as the partition function of Gentile statistics. The resulting asymptotic
or smooth density states interpolates between the Bose (arbitrary partitions) and
Fermi (distinct partitions) systems as a function of the maximal occupancy of the
quantum state.
Acknowledgements
We thank the referee for perceptive comments. This work was started when one of
us (CSS) was visiting IMSc as a summer research student in 2004. RKB would like
to acknowledge financial support from NSERC (Canada).
References
[1] S Grossman and M Holthaus, Phys. Rev. E54, 3495 (1996); Phys. Rev. Lett. 79, 3557
(1997)
[2] G H Hardy and S Ramanujan, Proc. London Math. Soc. XVII 2, 75 (1918)
[3] G E Andrews, The theory of partitions (Addison-Wesley Publishing Company, Read-
ing, Mass, 1976) p. 1
[4] Muoi N Tran, M V N Murthy and R K Bhadhuri, Ann. Phys. 311, 204 (2004)
[5] Miles P Blencowe and Nicholas C Koshnick, J. Math. Phys. 42, 5713 (2001)
[6] G Gentile, Nuovo Cimento 17, 493 (1940)
Pramana – J. Phys., Vol. 66, No. 3, March 2006 493
C S Srivatsan, M V N Murthy and R K Bhaduri
[7] M Brack and R K Bhadhuri, Semiclassical physics (Westview Press, Boulder, 2003)
pp. 111, 213
[8] R Kubo, Statistical mechanics (North Holland, Amsterdam, 1965) p. 104
[9] J Sakhr and N Whelan, Phys. Rev. E67, 066213 (2003)
[10] N Rosenzweig, Phys. Rev. 108, 817 (1957)
[11] H A Bethe, Phys. Rev. 50, 332 (1936)
[11a] Note that we evaluate the saddle point only along the real axis. In general there may
be other complex saddle points that are ignored. We assume, without proof, that this
is the only one that gives the correct asymptotic behaviour as it is known to yield
rigorous answers for ordinary partitions
[12] M Abramowitz and I A Stegun, Handbook of mathematical functions (Dover Publica-
tions Inc., New York, 1964) p. 826
494 Pramana – J. Phys., Vol. 66, No. 3, March 2006


Gentile statistics and restricted partitions

http://repository.ias.ac.in/25566/1/348.pdf

Gentile statistics and restricted partitions

Abstract

Similar works

Full text

Available Versions

Publications of the IAS Fellows