Exact expression for the number of states in lattice models

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by  Agata Fronczak

Exact expression for the number of states in lattice models
Agata Fronczak and Piotr Fronczak
Faculty of Physics, Warsaw University of Technology,Koszykowa 75, PL-00-662 Warsaw, Poland
(Dated: February 7, 2013)
We derive a closed-form exact combinatorial expression for the number of states in canonical
systems with discrete energy levels. The expression results from the exact low-temperature power
series expansion of the partition function. The approach provides interesting insights into basis of
3 statistical mechanics. In particular, it is shown that in some cases the logarithm of the partition
1 function may be considered the generating function for the number of internal states of energy
0 clusters, which characterize system’s microscopic configurations. Apart from elementary examples
2 includingthePoisson, geometric andnegativebinomialprobabilitydistributionsfortheenergy,the
framework is also validated against the one-dimensional Ising model.
n
a
J PACSnumbers: 05.20.-y,05.20.Gg,02.10.Ox,02.30.Lt

3
2
It has long been recognizedthat the number of states, where

g(E), with a given value of the energy, E, is a key
]
h quantity in equilibrium statistical physics. In princi- x=e−βε, (2)
c ple, all energy-related thermodynamic properties of a
e and energy is considered to be discrete
classical canonical system can be calculated once g(E)
m
is known. In particular, in the case of a system with
- discrete energy levels, the canonical partition function E =εN, (3)
t
a is Z(β) = g(E)e−βE, where the summation is
t E with ε representing a portion of energy and N =
s over the allowed energy levels, and quantities such as
t. the HelmholtPz free energy, F(β) = −lnZ(β)/β, the 0,1,2,... (with the ground-state energy equal to zero).
a InEq.(1),g(εN)representsthenumberofmicroscopic
ensemble-averaged energy, hEi = −∂lnZ(β)/∂β, and
m configurations in which energy of the system is εN.
the specific heat(which is relatedto energyfluctuations,
From Eq. (1) it is apparent that the exact distribu-
- hE2i−hEi2 =−∂hEi/∂β) can be simply obtained.
d tion function for the energy, g(εN), can be determined
n Duetothecentralroleoftheenergydistributionfunc- from the coefficients of the partitionfunction. As a rule,
o
tion, a variety of theoretical and computational studies however, when the order of expansion is increased com-
c
have addressed the problem of how to obtain g(E), see plexity of contributing terms also increases rapidly. In
[
e.g. [1–6]. On the other hand, relatively little was done this context, a rule of thumb states that the work in-

1 to understand, how the state-space representation of a volved in direct calculating the next term, g(εN), is the
v
considered system relates to its energy distribution, and sameasthatneededtocalculateallthe precedingterms,
1
how to recover such an information from, for example, g(0),g(ε),...,g(εN −ε). For this reason, generation of
6
3 the partition function. lengthy low-temperature power-series for the partition
1 In this paper we derive a new and general combinato- functionis thoughtobe ahighly specializedart. Thatis
. why,eveninthe caseofthe two-dimensionalIsingmodel
2 rialformulaforthedensityofstates. Theformularesults
on a square lattice, for which the method was primary
0 from the exact low-temperature power-series expansion
3 ofthecanonicalpartitionfunction. Inourapproachg(E) advanced (see ch. 6 in [1], the exact distribution of en-
1 ergies was just obtained in 1996 [2, 9], nearly a half of a
isexpressedbytheso-calledBellpolynomials,whichhave
: century after Kaufman’s exact solution for the partition
v a very convenient combinatorialinterpretation: they de-
function [10].
i scribe, how equal energy portions can cluster together
X Theaimofthispaperistoshowthattheeffortneeded
within the studied system. This, in turn, gives insight
r to calculate the coefficients of the low-temperature
a into system’s microscopic details.
power-series expansion for the canonical partition func-
Inthelow-temperatureexpansionofthepartitionfunc- tionisgreatlyreducedbyusingthecombinatorialscheme
tion,Z(β)= e−βE(Ω),microstates,Ω,arecountedin described below. The approach is very general and ap-
Ω
orderoftheirimportanceasthetemperatureisincreased pliestoanylatticemodel. We alsoarguethatthe frame-
P
fromzero[7,8]. Ifthe groundstateofasystemisknown workprovidesinterestinginsightsinto basisofstatistical
and if the successive excitations from this state can be mechanics. In particular, it is shown that in some cases
classified in a simple way, one should, at least theoreti- the logarithm of the partition function, lnZ(x), may be
cally, be able to construct such a series, i.e. considered the generating function for the number of in-
ternal states of energy clusters, which characterize sys-
tem’s microscopic configurations. Apart from elemen-
∞
Z(x)= g(εN)xN, (1) taryexamplesincludingthePoisson,geometric,andneg-
ative binomial probability distributions for the energy,
N=0
X
2
the framework is validated against the one-dimensional and (9). For this reason, we follow by explaining their
Isingmodel,whichisnotatallatrivialexamplesinceits meaning.
ordering temperature is zero, and the series, Eq. (1), is Suppose that N distinguishable particles are parti-
known to diverge at x=0 as expected for a model with tioned into k non-empty and disjoint clusters of n > 0
i
a zero temperature phase transition [8]. elements each, where k n =N. There are exactly
i=1 i
Derivation of the main result. To derive the main re-
sultofthispaper,whichisthegeneralcombinatorialfor- N Pk 1 N−k+1 1 cn
mula for the number of states, we begin with the well- =N! =N! (12)
n ,...,n n ! n!
known relation between the canonical partition function (cid:18) 1 k(cid:19) i=1 i n=1 (cid:18) (cid:19)
Y Y
and the Helmholtz free energy, i.e.
ofsuchpartitions,wherec ≥0standsforthenumberof
n
Z(x)=eA(x), (4) clustersofsizen,withthelargestclustersizebeingequal
to N−k+1, and where Eqs. (10) are satisfied. Suppose
wherexistheexpansionvariableofthelow-temperature further that in such a composition clusters of the same
series, Eq. (2), and size are indistinguishable from one another, and each of
c clusters of size n can be in any one of a ≥0 internal
∞ xn n n
states. Then the number of partitions becomes, cf. with
A(x)=lnZ(x)= a =−βF(β), (5)
n
n! Eq. (12),
n=0
X
with an = ∂nA(x)/∂xn|x=0. Then, using the generat- N−k+1 1 an cn
ing function for Bell polynomials, Y ({a }), which are N! . (13)
N n c ! n!
the polynomials with a sequence of parameters {an} = nY=1 n (cid:16) (cid:17)
a ,a ,...a ,givenbythe formalpowerseriesexpansion
1 2 N
Summing Eq. (13) over all integers c ≥ 0 specified
[11], Eq. (4) can be written as n
by Eqs. (10), one gets the partial Bell polynomial,
∞ xn ∞ 1 BN,k({an}), which is defined by Eq. (9). If for all n≥1
Z(x)=exp ann! =ea0 N!YN({an})xN. (6) the coefficients an ≥ 0, the polynomial describes the
"n=0 # N=0 number of partitions of a set of size N with exactly k
X X
subsets, where each coefficient, a , represents the num-
In the last formula, the complete Bell polynomials, n
ber of internal states of a cluster of size n. Finally, sum-
Y ({a }), are defined as follows
N n mingthepartialpolynomialsoverkonegetsthecomplete
Bellpolynomial,Y ({a }),thecombinatorialmeaningof
Y ({a })=1, (7) N n
0 n
which is obvious.
and for all N ≥1 According to the explanations above the main result
ofthispaper,Eq.(11),showshowequalenergyportions,
N
ε, are distributed, and how they cluster together within
Y ({a })= B ({a }), (8)
N n N,k n the system whose structural details are hidden in the
Xk=1 coefficients {an}. If the coefficients are non-negative,i.e.
∀ a ≥ 0, they can be interpreted as thermodynamic
where B ({a }) represent the so-called partial (or in- n≥1 n
N,k n
probabilitiesofenergyclustersofagivensize. Otherwise,
complete)Bellpolynomials,whichcanbecalculatedfrom
their meaning is not clear.
the expression below
Inwhatfollows,thePoissonandthenegativebinomial
N−k+1 1 an cn distributions are discussed as elementary mathematical
BN,k({an})=N! , (9) examples of application of the combinatorial approach.
c ! n!
X nY=1 n (cid:16) (cid:17) The advantages of the new framework with respect to
traditionalmethodsofstatisticalmechanicsareenhanced
wherethesummationtakesplaceoverallintegersc ≥0,
n with reference to the one-dimensional Ising model.
such that
Poisson distribution. As the first example, the Pois-
N−k+1 N−k+1 son distribution for the energy of a canonical system is
cn =k and ncn =N. (10) considered,
n=1 n=1
X X
e−hEihEiE
Finally comparing Eqs. (1) and (6), one gets the exact P(E)= , (14)
E!
expression for the number of states:
ea0 where, see Eq. (3), ε = 1 and E = N = 0,1,2,.... In
g(εN)= YN({an}). (11) order to bring Eq. (14) into the canonical form,
N!
ThebasicdifficultywithEq.(11)mayarisefromanun- e−βE
P(E)=g(E) , (15)
acquaintance with Bell polynomials as given by Eqs. (8) Z(β)
3
one has to assume that hEi = e−β. Then, the Poisson and
distribution can be written as
∞ xn
1 e−βE A(x)=−rln(1−x)=r (n−1)! . (23)
P(E)= , (16) n!
E! ee−β nX=1
where (compare Eqs. (15) and (16)) The series expansion of the logarithm of the partition
function, A(x), gives the following sequence of the coef-
1
g(E)= and lnZ(β)=e−β. (17) ficients {an}:
E!
To verify the combinatorial approach introduced in a0 =0 and ∀n≥1 an =r(n−1)!. (24)
this paper, let us first notice that in the case of the
Poisson distribution x = e−β, Eq. (3). Therefore, the The sequence provides a meaningful microscopic infor-
mation about the considered canonical system. First, it
logarithmof the partition function is simply, cf. Eqs.(5)
appears that the parameter r describes the degeneracy
and (17),
of internal states of the energy-clusters, and the lack of
A(x)=x. (18) degeneracy, r = 1, results in the geometric/exponential
distribution. Second,since (n−1)!is the number ofper-
TheidentityfunctionforA(x)providesaverysignificant
mutations of n objects where the first object is fixed,
sequence of the coefficients {a } for the energy distribu-
n the only reasonable explanation behind a ∝ (n−1)! is
n
tion function, Eq. (11), i.e.
thelinearorderingofenergyportionswithintheclusters.
At first glance the idea of ordering of indistinguishable
a =0 and {a }=1,0,0,...,0. (19)
0 n
energyportionsmayseemunacceptable,butitgetsintel-
Thesequencestatesthatinthecaseofacanonicalsystem ligibilityifonerealizesthatinsomesenseenergyportions
with the Poissondistribution of energy, energy portions, become distinguishable once they, for example, become
ε = 1, are independent from each other. In some sense interaction energies between given pairs of particles.
it means that such a system is structureless (it may, for To validate the main result of this paper, Eq. (11),
example, consists of noninteracting parts). one has to insert the coefficients given by Eqs. (24) into
Finally,insertingtheobtainedcoefficientsintoEq.(11) the mentioned expression. After some algebra one gets
one can easily get the energy distribution function, Eq. (22):
Eq. (17),
1
1 1 g(N) = YN({r(n−1)!}) (25)
g(N)= Y (1,0,0,...,0)= , (20) N!
N
N! N! N
1
= rkB (0!,1!,2!,...) (26)
where YN(1,0,0,...,0) = 1, because there is only one N! N,k
composition of a set of size N in which only clusters of Xk=1
sizNeenga=tiv1embianyomexiiasltd(ias1tr=ibu1tiaonnd. aAns=th0esfoecroanldl nex>am1)p.le = 1 N rk N = N +r−1 , (27)
N! k N
the negative binomial distribution for the energy of a k=1 (cid:20) (cid:21) (cid:18) (cid:19)
X
canonical system is studied,
where basic combinatorial identities have been used, in-
E+r−1 cluding [11, 12]: i. the generating function for the un-
P(E)= E (1−x)rxE, (21) signed Stirling numbers of the first kind, Nk , and ii.
(cid:18) (cid:19) properties of Bell polynomials, i.e. B ruka =
where x ∈ (0,1) and r > 0, while ε = 1 and E = N = rkuNB ({a }) and B ({(n−1)!})N=,kN(cid:2) [1(cid:3)3].n
0,1,2,...,seeEq.(3). Itisworthnotingthatamongthe N,k n N,k (cid:0)k(cid:8) (cid:9)(cid:1)
One-dimensional Ising model. The Hamiltonianofthe
discretedistributions,thenegativebinomialdistribution, (cid:2) (cid:3)
closedchainofV Isingspins,{s },withnearest-neighbor
NB(r,x), Eq. (21), is considered the discrete analogue i
interactionsinthe absenceofexternalmagneticfieldcan
of the Gamma distribution, Γ(r,β), whose probability
be written as
density function is P(E) = βrEr−1e−βE/Γ(r), where r
is the so-called shape parameter. The analogy is even V
more apparent if one realizes that, in the case of r = 1 H({s })=−J s s , (28)
i i i+1
thenegativebinomialdistributionbecomesthegeometric
i=1
X
distribution, NB(1,x), which is the discrete analogue of
the exponential distribution, Γ(1,β). where J > 0, si = ±1, and the periodic boundary con-
Bringing the negative binomial distribution, Eq. (21), dition is imposed by assuming that sV+1 = s1. In the
into the canonical form, Eq. (15), for x=e−β one gets: thermodynamic limit, V ≫ 1, the partition function of
the model is given by
E+r−1
g(E)= , (22)
E Z(β)=(eβJ +e−βJ)V. (29)
(cid:18) (cid:19)
4
To use Eq. (11) to calculate the number of states, do not satisfy the condition of non-negativity, which (if
g(E), characterizing the model, one must first rescale satisfied) allows to conclude on state space representa-
its energy, E, because in the present form of the Hamil- tion of the considered model. Nevertheless, the knowl-
tonian, Eq. (28), the ground state energy is negative, edge of {a } (regardless of their sign) enables a direct
n
H({−1}) = H({+1}) = −JV, whereas it should be at validation of the combinatorial approach. By inserting
least equal to zero, see Eqs. (1), (3) and (2). The prob- the coefficients into the expression for the energy distri-
lem can be solvedby adding to Eq. (28) a constant, JV, bution function, Eq. (11), after some algebra one gets,
so that for all the spin configurations, {s }, the modified for N =E∗,
i
Hamniltonian,
V
1
H∗({s })=−J s s +JV, (30) g∗(N) = Y ({V(−1)n−1(n−1)!}) (36)
i i i+1 N
N!
i=1
X N
1 N V
results in non-negative energy = Vk(−1)N−k = , (37)
N! k N
k=1 (cid:20) (cid:21) (cid:18) (cid:19)
E∗ =E+JV ≥0. (31) where (−1)N−k Nk iXs the signed Stirling number of the
firstkind. Fromthelastexpressiononeimmediatelygets
(cid:2) (cid:3)
With this change, the partition function of the model
becomes
V
Z∗(β)=e−βJVZ(β)=(1+e−2βJ)V, (32) g(E)= , (38)
E+V/2
(cid:18) (cid:19)
but statistical properties of the chain of spins, {s },
i
remain the same, as compared with the original Ising
model. In particular, the probability distributions for where E is the energy of the original Ising model de-
spin configurations, P({s }) = exp[−βH({s })]/Z(β) scribed by Eqs. (28) and (29).
i i
and P∗({s }) = exp[−βH∗({s })]/Z∗(β), coincide with
i i
The obtained result, Eqs.(37) and (38), is exactly the
one another. The same applies to probability distri-
expected one: the number of microscopic configurations
butions for the energy, P(E) = g(E)e−βE/Z(β) and
P∗(E∗)=g∗(E∗)e−βE∗/Z∗(β). oftheclosedchainofV Isingspinsisequaltothenumber
of ways to choose positions for E∗ pairs of equal spins,
In view of the above considerations one can see that
{+1,+1} and {−1,−1},from the available V positions.
g(E)=g∗(E∗)=g∗(E+JV). (33)
Summary. Our approach differs crucially from previ-
ousworksonthe numberofstates. We use the combina-
Thus, to determine the number of states for the one-
torialmathematicsofexponentialgeneratingfunctionsto
dimensionalIsingmodel,g(E),onecancalculateg∗(E∗),
obtaintheexactlow-temperaturepower-seriesexpansion
and then change the variables E∗ and E according to
ofthecanonicalpartitionfunction. Theexpansionresults
Eq. (31).
in the exact expressionfor the energy distribution in ar-
Inthefollowing,forsimplicityoneassumesinEq.(32)
bitrary lattice models. Apart from elementary examples
that J =1/2. Then, the logarithmof the partitionfunc-
including the Poisson, geometric, and negative binomial
tion, lnZ∗(β) can be written as, Eq. (5),
probabilitydistributionsfortheenergy,theframeworkis
∞ xn validated against the one-dimensional Ising model. The
A∗(x)=V ln(1+x)=V (−1)n−1(n−1)! , (34) work on the application of the formalism to the Ising
n!
n=1 model on a square lattice is in progress and will be re-
X
ported in a future article to follow.
where ε = 1 and x =e−β. From the series expansion, it
is obvious that the coefficients {a }, i.e. Acknowledgments. The work has been supported
n
from the National Science Centre in Poland (grant no
a =0 and ∀ a =V(−1)n−1(n−1)!, (35) 2012/05/E/ST2/02300).
0 n≥1 n
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