Download Revision of Boltzmann statistics for a finite number of particles

Revision of Boltzmann statistics for a finite number of particles
S. Kakorina兲
Biophysical Chemistry, Department of Chemistry, University of Bielefeld, P.O. Box 100 131,
D-33501 Bielefeld, Germany
共Received 19 January 2008; accepted 11 July 2008兲
The Stirling approximation, ln共N!兲 ⬇ N ln共N兲 − N, is used in the literature to derive the exponential
Boltzmann distribution. We generalize the latter for a finite number of particles by applying the
more exact Stirling formula and the exact function ln共N!兲. A more accurate and analytical
formulation of Boltzmann statistics is found in terms of the Lambert W-function. The
Lambert-Boltzmann distribution is shown to be a very good approximation to the exact result
calculated by numerical inversion of the Digamma-function. For a finite number of particles N the
exact distribution yields results that differ from the usual exponential Boltzmann distribution. As an
example, the exact Digamma-Boltzmann distribution predicts that the constant-volume heat
capacity of an Einstein solid decreases with decreasing N. The exact Digamma-Boltzmann
distribution imposes a constraint on the maximum energy of the highest populated state, consistent
with the finite total energy of the microcanonical ensemble. © 2009 American Association of Physics
Teachers.
关DOI: 10.1119/1.2967703兴
I. INTRODUCTION
The usual exponential Boltzmann distribution describes
the populations of 共non-degenerate兲 energy states ␧i, i
= 0 , 1 , . . . , m, for non-interacting particles in an isolated
system.1 A crucial step in the derivation is the application of
Stirling’s approximation:2,3
ln共n!兲 ⬇ n ln共n兲 − n
共Stirling ’ s approximation兲.
共1兲
For small values of n Eq. 共1兲 is very crude because it does
not contain the leading term, 共1 / 2兲ln共n兲, of the Stirling
series:4
冉 冊
ln共n!兲 = n +
+ ¯
1
1
1
1
−
ln共n兲 − n + ln共2␲兲 +
2
2
12n 360n3
共Stirling series兲.
共2兲
The Stirling formula gives a much better approximation to
ln共n!兲 because it includes all the leading terms in Eq. 共2兲:3
冉 冊
ln共n!兲 ⬇ n +
+
1
ln共n兲 − n
2
1
ln共2␲兲
2
共Stirling formula兲.
共3兲
II. REVISED BOLTZMANN DISTRIBUTION
The reciprocal terms 1 / 共12n兲 − 1 / 共360n3兲 + ¯. in Eq. 共2兲 are
not essential if n ⬎ 2 共see Fig. 1兲. Generally, the occupation
number ni of the highest energy states ␧i is very small if ␧i
→ ⬁. However, the range of applicability of Eq. 共1兲 is limited
to very large values of ni, where the condition ln共ni兲 / 2
Ⰶ ni ln共ni兲 − ni is satisfied. If ni is small, say ni = 1 or 2, Eq.
共1兲 yields negative values 共see Fig. 1兲. Therefore, the exponential Boltzmann distribution is limited strictly to ni Ⰷ 1 and
applies only for ␧i not too large. However, the distribution is
used in the literature even for ␧i → ⬁ and ni → 0, where Eq.
共1兲 does not apply. It is interesting that Stirling’s approximation, Eq. 共1兲, fails and the more accurate Stirling formula,
Eq. 共3兲, is required to derive the Gaussian distribution from
the binomial distribution.5
48
Am. J. Phys. 77 共1兲, January 2009
It is natural to ask if the use of the more exact Stirling
formula, Eq. 共3兲, and the exact Stirling series, Eq. 共2兲, leads
to a more accurate formulation of the Boltzmann variational
problem of the calculation of the most probable population
of energy states of the microcanonical ensemble, and if the
exact distribution is different from the exponential Boltzmann distribution. The solution of the Boltzmann variational
problem using Eq. 共3兲 leads to a transcendental equation. We
show that this equation can be solved in closed form by
using the Lambert W-function, also known as the ProductLog function or omega function.6 The Lambert W-function
W共x兲 is defined as the solution of the equation
W共x兲exp共W共x兲兲 = x. Several symbolic computation packages
contain routines for the Lambert W-function and recent articles discuss problems that can be solved analytically using
this function.6 The solution of the Boltzmann variational
problem using the exact function ln共n!兲 leads to a complicated transcendental equation, which must be solved numerically. In any case, it is worthwhile to obtain a more exact and
physically tractable solution of the Boltzmann variational
problem.
http://aapt.org/ajp
A. Formulation of the variational problem
We consider an isolated system of N indistinguishable
non-interacting particles with total constant energy E.2 Each
particle may exist in any of the energy states ␧0 ⬍ ␧1 ⬍ ¯
⬍ ␧m, where m is the total number of states. The population
is characterized by the set of numbers n0 , n1 , . . . , nm of the
particles in the energy states ␧0 , ␧1 , . . . , ␧m, respectively. The
probability of finding ni distinguishable particles in each of
the m states is given by pni i, where pi is the probability of the
occupation of energy state ␧i. The total probability of a configuration 共n0 , ␧0兲 , 共n1 , ␧1兲 , . . . , 共nm , ␧m兲 is given by the prodnm
. For indistinguishable particles the statistical
uct pn00 pn11 ¯ pm
weight of the same configuration is given by1
© 2009 American Association of Physics Teachers
48
冉 冊 冉 冊
⳵F
⳵n0
=
n*
0
⳵F
⳵n1
n*
= ¯
1
冉 冊
⳵F
⳵nm
共10兲
= 0.
n*
m
We also have ⳵N / ⳵ni = ⳵E / ⳵ni = 0 so Eq. 共10兲 can be expressed as
冏 冏 冏 冏
⳵F
⳵ni
=
n*
i
⳵P
⳵ni
n*
− ␣ − ␤␧i = 0
for i = 0,1, . . . ,m.
i
共11兲
Fig. 1. Comparison of Stirling’s approximation, Eq. 共1兲 共䊐兲, and the Stirling
formula, Eq. 共3兲 共〫兲, with the exact function ln共n!兲 共⫻兲 up to n = 5.
N!
w共n0,n1, . . . ,nm兲 =
.
n0!n1! ¯ nm!
共4兲
The
total
probability
of
the
共n0 , ␧0兲 , 共n1 , ␧1兲 , . . . , 共nm , ␧m兲 takes the form
P共n0,n1, . . . ,nm兲 = w共n0,n1, . . .
,nm兲pn00 pn11
The derivative ⳵ P / ⳵ni in Eq. 共11兲 contains the derivative
d共1 / ni!兲 / dni. In this form, d共1 / ni!兲 / dni cannot be used to
solve Eq. 共11兲 analytically. The usual approach is to replace
the function P in Eq. 共11兲 by ln共P兲. Because P ⬎ 0 and ln共P兲
is a monotonic function of P, both ln共P兲 and P have extremum for the same ni*. Hence the function
冉
configuration
¯
nm
pm
,
共5兲
where w共n0 , n1 , . . . , nm兲 is given by Eq. 共4兲. We adopt the
principle of equal a priori probabilities p0 = p1 = ¯ = pm = p,
nm
so that the product pn00 pn11 ¯ pm
in Eq. 共5兲 reduces to the
n0+n1+¯+nm
N
constant p
= p , where
冉
m
+ ␤ E − 兺 n i␧ i
i=0
冊
共6兲
i=0
The most probable configuration, corresponding to the maximum of the probability in Eq. 共5兲, is calculated given the
constraint of the fixed total number of particles, Eq. 共6兲, and
the total energy:
m
E = 兺 n i␧ i .
Because of the two constraints, two of the m variables in
Eq. 共5兲 are dependent. One way to find the extremum is the
Lagrange method of undetermined multipliers using two
multipliers, ␣ and ␤, in the form7
冉
m
F共n0,n1, . . . ,nm兲 = P共n0,n1, . . . ,nm兲 + ␣ N − 兺 ni
冉
m
冊
+ ␤ E − 兺 n i␧ i .
i=0
i=0
冊
also has an extremum at ni*. Because ⳵ ln共n j!兲 / ⳵ni = 0 for j
⫽ i, the derivative of ln共P兲 can be written as
冉
m
冊
⳵ ln共P兲
⳵
⳵
=
ln共N!pN兲 − 兺 ln共n j!兲 = −
ln共ni!兲.
⳵ni
⳵ni
⳵ni
j=0
共13兲
冏 兺 冉 冊冏
i=0
共8兲
⳵F
⳵ni
= 0,
共9兲
n*
i
where ⳵F / ⳵ni is the partial derivative of F with respect to ni
provided that all other variables n j with j ⫽ i are kept constant. Because all ⳵F / ⳵ni ⱕ 0, the only way to satisfy Eq. 共9兲
is to require that
49
We use Eq. 共13兲 and obtain the condition for an extremum of
⌽ in Eq. 共12兲 in the form
−
⳵
ln兩共ni!兲兩n* − ␣ − ␤␧i = 0.
i
⳵ni
共14兲
It remains to solve Eq. 共14兲.
Am. J. Phys., Vol. 77, No. 1, January 2009
III. APPLICATION OF STIRLING’S
APPROXIMATIONS
A. Crude Stirling’s approximation
If we apply Stirling’s approximation, Eq. 共1兲, or the
Stirling formula, Eq. 共3兲, to Eq. 共14兲, we find
− ln共ni*兲 −
At this stage of the analysis all the ni are free variables, so
m
m
ni and E ⫽ 兺i=0
ni␧i. The values ni* of an extrethat N ⫽ 兺i=0
mum of F can be calculated from the equation
dF =
冊
共7兲
i=0
m
i=0
共12兲
m
N = 兺 ni .
m
⌽共n0,n1, . . . ,nm兲 = ln共P共n0,n1, . . . ,nm兲兲 + ␣ N − 兺 ni
␮
− ␣ − ␤␧i = 0,
2ni*
共15兲
where ␮ = 0 or 1 represents the solution for Eq. 共1兲 or Eq. 共3兲,
respectively. If ␮ = 0, the solution of Eq. 共15兲 yields the exponential Boltzmann distribution:7
ni* = exp共− ␣ − ␤␧i兲,
共16兲
where ␤ = 1 / 共kT兲, k is Boltzmann’s constant, and T is the
thermodynamic temperature. The ratio Pi = ni* / N is the usual
exponential Boltzmann probability. The partition function is
given by
i=m
Z = 兺 exp共− ␤␧i兲.
共17兲
i=0
The Lagrange multiplier ␣ can be written as
S. Kakorin
49
␣ = − ln共N/Z兲.
共18兲
B. The more accurate Stirling formula
For ␮ = 1 the solution of Eq. 共15兲 is obtained in terms of
the Lambert W-function W共x兲:6
−1
,
1
2W − exp共␣W + ␤␧i兲
2
冊
冉
* =
ni,W
共19兲
where the subscript W refers to the values calculated with the
function W共x兲. Because the second derivative of ⌽ is negative,
冏 冏
⳵⌽
⳵n2i
2
n*
i,W
1
1
=− * +
* 兲2 ⬍ 0,
ni,W 2共ni,W
共20兲
* is a maximum, and the ratio P = n* / N
the solution ni,W
i,W
W
i,W
is the Lambert-Boltzmann probability, where N = 兺m n* .
W
i=0 i,W
In analogy to Eq. 共17兲 we calculate the Lambert W-partition
function ZW in the form
冉
m
冊
* 共␧ 兲
ni,W
1
i
= − 2NWW − exp共␣W兲 .
ZW = 兺 *
2
i=0 ni=0,W共␧0 = 0兲
共21兲
The solution of Eq. 共21兲 for the Lagrange multiplier ␣W
yields the form
冉 冊
␣W = − ln
NW
ZW
−
.
ZW
2NW
共22兲
In contrast to Eq. 共18兲 the value of ␣W is limited to a certain
range as is discussed in Sec. III C.
C. Range of ␣W
* the range of the
For the real and positive numbers ni,W
argument x = −exp共␣W + ␤␧i兲 / 2 of W共x兲 in Eq. 共19兲 is limited
to 0 ⱖ x ⱖ −1 / e. In the range 0 ⱖ x ⱖ −1 / e, W共x兲 is the
W−0 -branch of the Lambert W-function, −1 ⱕ W−0 ⱕ 0.6 Therefore the function W in Eqs. 共19兲 and 共21兲 has to be replaced
by W−0 . The range of validity of x imposes a constraint on the
parameters ␣W and ␧i:
0 ⱕ exp共␣W + ␤␧i兲 ⱕ 2/e.
共23兲
Equation 共23兲 implies that ␣W is related to the value ␧Wm of
the highest energy state by
␣W = ln共2/e兲 − ␤␧Wm .
共24兲
For a given ␣W Eq. 共23兲 requires that
共25兲
Equation 共25兲 suggests that the Lambert-Boltzmann distribution, Eq. 共19兲, applies only up to the maximum energy ␧Wm
共inclusive of the ␧Wm-value兲. If we substitute Eq. 共22兲 into
Eq. 共25兲, we can calculate ␧Wm explicitly,
冋冉 冊 册
1
2NW
ZW
ln
+
,
␤
eZW
2NW
or
50
Am. J. Phys., Vol. 77, No. 1, January 2009
共27兲
The constraints in Eqs. 共26兲 and 共27兲 define the functional
dependence of the parameters ␧Wm, NW, and ZW for which the
Lambert-Boltzmann distribution in Eq. 共19兲 applies.
D. Exact solution of the Boltzmann problem
in terms of the inverse Digamma-function
The exact derivative of ln共n!兲 is given by the Digammafunction ␺0: d ln共n!兲 / dn = ␺0共n + 1兲, where the argument n in
␺0 can be any real number. Equation 共15兲 acquires the form
− ␺0共ni* + 1兲 − ␣␺ − ␤␧i = 0,
共28兲
where the subscript ␺ refers to the values calculated with the
Digamma-function. The solution of Eq. 共28兲 is obtained in
terms of the inverse Digamma-function, ␺−1
0 :
ni,*␺ = ␺−1
0 共− ␣␺ − ␤␧i兲 − 1.
共29兲
In contrast to W−0 , ␺−1
0 is not included in the generally available software packages and must be calculated numerically.
The Digamma-partition function Z␺ which is analogous to
Eq. 共21兲 has the form
Z␺ =
N␺
.
␺−1
共−
␣ ␺兲 − 1
0
共30兲
The correct range of the argument ␣␺ of ␺−1
0 is found by
requiring that ni,*␺ be non-negative. For ni,*␺ ⱖ 0 Eq. 共28兲 suggests that −␣␺ − ␤␧i ⱖ ␺0共1兲 = −␥, where ␥ = 0.577. . . is the
Euler-Mascheroni constant. Because ␤␧i ⱖ 0, we find that ␣␺
is related to the maximum value ␧␺m by
0 ⱕ ␧␺m = 共␥ − ␣␺兲/␤ .
共31兲
The constraint in Eq. 共31兲 is similar to Eq. 共25兲 of the
Lambert-Boltzmann distribution, except for the constant ␥,
which replaces ln共2 / e兲 ⬇ −0.307. The exact value of the
maximum energy ␧␺m is larger by the quantity of 共␥
− ln共2 / e兲兲 / ␤ ⬇ 0.884kT than the approximate value ␧Wm
given in Eq. 共25兲. For large energies, ␧␺m , ␧Wm Ⰷ kT, the difference ␧␺m − ␧Wm is not important.
If we apply Eq. 共31兲 to Eq. 共30兲, we can express the exact
Digamma-partition function Z␺ as
Z␺ =
N␺
−1
␺ 0 共 ␤ ␧ ␺m −
␥兲 − 1
共32兲
.
As in Eq. 共27兲 the exact partition function Z␺ in Eq. 共32兲 can
be expressed in closed form in terms of the N␺ and ␧␺m. Note
that the usual partition sum Z in Eq. 共17兲 is independent of N.
IV. RESULTS AND DISCUSSION
0 ⱕ ␧0 ⬍ ¯ ⬍ ␧i ⬍ ¯ ⱕ ␧Wm = 共ln共2/e兲 − ␣W兲/␤ .
␧Wm =
ZW = − 2NWW−0 共− exp共− ␤␧Wm − 1兲兲.
共26兲
A. Exponential Boltzmann versus the Lambertand the exact Boltzmann distribution
As an example, we consider a system with n0* = 600 particles in the ground state with ␧0 = 0. Equation 共15兲 with ␮
= 0 yields ␣ = −ln共n0*兲 = −6.397; Eq. 共15兲 with ␮ = 1 and the
exact ␺0-Boltzmann equation, Eq. 共28兲, yield ␣W = −ln共n0*兲
− 1 / 共2n0*兲 = ␣␺ = −␺0共n0* + 1兲 = −6.398. If we use Eqs. 共25兲 and
共31兲, we find ␤␧Wm = ln共2 / e兲 − ␣W = 6.091 and ␤␧␺m = ␥ − ␣␺
= 6.975, respectively. Within the range of applicability, the
Lambert-Boltzmann distribution in Eq. 共19兲 and the exact
S. Kakorin
50
Boltzmann distributions yield approximately the same total
m=6 *
numbers N of particles: Nw = 兺i=0
ni,w = 945 and N␺
m=7 *
= 兺i=0 ni,␺ = 946.
The
Lambert
and Digamma statistical sums are also identical: ZW
= −2NWW−0 共−exp共␣W兲 / 2兲 = Z␺ = N␺ / 共␺−1
0 共−␣␺兲 − 1兲 = 1.576. If
⬁ *
we apply Eq. 共15兲 with ␮ = 0, we obtain N = 兺i=0
ni = 949 and
Z = N exp共␣兲 = 1.582. The key parameters of the exponential,
Lambert- and Digamma-Boltzmann statistics are summarized in Table I. It is seen that the Lambert and DigammaBoltzmann distributions yield approximately equal results,
which differ from the exponential Boltzmann statistics. Fig* for ␤␧ ⬎ 0. The total
ure 2共b兲 shows that ni* ⱖ ni,*␺ ⱖ ni,w
i
⬁ *
probability of all three statistics is equal to unity: 兺i=0
ni / N
m=6 *
m=7 *
= 兺i=0 ni,w / Nw = 兺i=0 ni,␺ / N␺ = 1.
B. Population of energy states of a quantum oscillator
Fig. 2. 共a兲 The solid curve is the exponential Boltzmann distribution n*, Eq.
* , Eq. 共19兲; and 䊐, the exact
共16兲; ⫻, the Lambert-Boltzmann distribution nW
Digamma-Boltzmann distribution n*␺, Eq. 共29兲 as a function of ␤␧, where ␧
is the energy of the states and ␤ = 1 / kT. The arrows show the maximum
* and n*.
energies ␤␧Wm = ln共2 / e兲 − ␣W ⬇ 6.1 and ␤␧␺m = ␥ − ␣␺ ⬇ 7.0 of nW
␺
* and n*. The distributions shown
Note that ni* in Eq. 共16兲 disagrees with nW
␺
* = n* = 600 for the ground state of the zero energy ␧ = 0 and
are for n*0 = n0,W
0
0,␺
T = 300 K. 共b兲 The same as in 共a兲 in the range 5 ⱕ ␤␧ ⱕ 7, where the discrep*
*
*
*
ancies between n*, nW, and n␺ are most pronounced; nW and n␺ were calculated using Mathematica.
The effect of the constraints in Eqs. 共25兲 and 共31兲 on the
population of the energy states can be shown, for instance,
by determining the heat capacity of a system of oscillators.
In the harmonic approximation the vibrational energy of a
quantum oscillator is given by ␧i = ⌬␧共i + 1 / 2兲, where i
= 0 , 1 , . . . , m and ⌬␧ = h␯0 is the separation between the adjacent states, h is Planck’s constant, and ␯0 is the characteristic frequency. If the usual Boltzmann distribution is used to
calculate the population of the energy states, the partition
function is given by
i=m
Z共m兲 = 兺
i=0
Digamma-Boltzmann distribution in Eq. 共29兲 are close to the
exponential Boltzmann distribution in Eq. 共16兲 关see Fig.
2共a兲兴. However, Eq. 共16兲 applies up to ␤␧m → ⬁, whereas
Eqs. 共19兲 and 共29兲 apply up to ␤␧Wm ⬇ 6 and ␤␧␺m ⬇ 7, respectively 关see Fig. 2共b兲兴. For example, for a system of equidistant energy states, ␧i = ⌬␧i, where i = 0 , 1 , . . . , m and ⌬␧
= kT, T = 300 K, the Lambert and the exact Digamma-
冉
冊
冉 冊
冉 冊 冉 冊
⌬␧共m + 1/2兲
kT
␧i
exp −
=
.
kT
⌬␧
⌬␧
exp
− exp −
2kT
2kT
1 − exp −
共33兲
The maximum energy of the harmonic oscillator is not
bounded from above and therefore the number of levels can
be infinite, m → ⬁. However, for an isolated system of N
oscillators of total energy E there is a constraint on the energies: 0 ⬍ ␧0 ⬍ ¯ ⬍ ␧i ⬍ ¯ ⬍ ␧m ⱕ E. In equilibrium and
Table I. Comparison of the key parameters of the three forms of the Boltzmann distribution for a system with
* = n* = 600 particles in the ground state with the zero energy ␧ = 0 and equidistant energy states: ␧
n*0 = n0,W
0
i
0,␺
= ⌬␧i, where i = 0 , 1 , . . . , m and ⌬␧ = kT , T = 300 K; W−0 is the negative branch of the Lambert W-function, ␺0 is
the Digamma-function, ␥ ⬇ 0.577 is the Euler-Mascheroni constant, ln共2 / e兲 ⬇ −0.307, ␤ = 1 / kT, k is the Boltzmann constant, and N is the total number of particles.
51
Exponential Boltzmann
distribution, Eq. 共16兲
Lambert-Boltzmann
distribution, Eq. 共19兲
Exact Digamma-Boltzmann
distribution, Eq. 共29兲
␣ = −ln共n*0 兲 = −6.397
Eq. 共15兲 with ␮ = 0
* 兲 − 1 / 共2n* 兲=−6.398
␣W = −ln共n0,W
0,W
Eq. 共15兲 with ␮ = 1
␣␺ = −␺0共n*0,␺ + 1兲 = −6.398
Eq. 共28兲
␤␧m ⬍ ⬁
␤␧Wm = ln共2 / e兲 − ␣W = 6.091
Eq. 共25兲
␤␧␺m = ␥ − ␣␺ = 6.975
Eq. 共31兲
m⬍⬁
mW = ␧Wm / ⌬␧ = 6
m␺ = ␧␺m / ⌬␧ = 7
⬁ *
ni = 949
N = 兺i=0
6
* = 945
Nw = 兺i=0
ni,w
7
N␺ = 兺i=0
ni,*␺ = 946
Z = N exp共␣兲 = 1.582
Eq. 共17兲
ZW = − 2NWW−0 −
Eq. 共21兲
Am. J. Phys., Vol. 77, No. 1, January 2009
冉 冊
e ␣W
= 1.576
2
N␺
= 1.576
␺−1
共−
␣ ␺兲 − 1
0
Eq. 共30兲
Z␺ =
S. Kakorin
51
tion Z共m → ⬁兲 is independent of N. Equations 共25兲 and 共31兲
define the range of validity of the exponential Boltzmann
distribution and assist in the calculation of the correct values
of the partition function.
C. Exact value of the Einstein heat capacity
According to the Einstein model of the three-dimensional
quantum oscillator, the constant-volume heat capacity is expressed as7
CV = 3kNT2
冉
⳵2 ln共Z共T兲兲
⳵T2
冊
共34兲
.
V,N
To calculate Eq. 共34兲 with Z = Z␺ we apply Eq. 共33兲 and
␧␺m共T兲 = ⌬␧共m + 1 / 2兲 = kT共␥ − ␣␺兲 to Eq. 共32兲 and obtain a
transcendental equation for ␣␺:
冉
␣␺ = ␺0 N
Fig. 3. The partition function Z, the maximum energy ratio ␧m / ⌬␧ of the
highest populated state 共here ⌬␧ = 2.3⫻ 10−21 J兲, and the reduced constantvolume heat capacity CV / 共3Nk兲, calculated with the exact DigammaBoltzmann distribution for the Einstein Ag-solid as a function of the temperature T 共left column兲 and the total number of particles N 共right column兲.
共a兲 The solid lines are the exact Digamma-partition function Z␺共␧␺m兲, Eq.
共32兲, of the oscillator with energy states ␧i = ⌬␧共i + 1 / 2兲, where i
= 0 , 1 , . . . , m. The points represent the conventional partition sum, Z共m兲,
truncated at m = ␧␺m / ⌬␧ − 1 / 2, Eq. 共33兲, at N = 300 共1兲, N = 50 共2兲, and N
= 5 共3兲. 共b兲 The solid line represents Z␺共␧␺m兲, Eq. 共32兲; the points refer to
Z共m兲, Eq. 共33兲, at T = 1200 K 共1兲, T = 300 K 共2兲, and T = 100 K 共3兲. 共c兲 The
solid line is the exact maximum energy ratio ␧␺m / ⌬␧ of the oscillator, Eq.
共31兲; the points represent the approximate values ␧Wm / ⌬␧, Eq. 共37兲, corrected by 共␥ − ln共2 / e兲兲kT at N = 300 共1兲, N = 50 共2兲, and N = 5 共3兲. 共d兲 The
solid lines are the exact energy ratio ␧␺m / ⌬␧, Eq. 共31兲, and the points are the
approximate values ␧Wm / ⌬␧, Eq. 共37兲, corrected by 共␥ − ln共2 / e兲兲kT at T
= 1200 K 共1兲, T = 300 K 共2兲, and T = 100 K 共3兲. 共e兲 The points represent the
exact reduced heat capacity CV / 共3Nk兲 Eq. 共34兲, versus T for N = 300 共1兲,
N = 50 共2兲 共d兲, and N = 5 共3兲. The solid line corresponds to the usual Einstein
equation, Eq. 共36兲, for CV / 共3Nk兲 at ␧m → ⬁. 共f兲 The points are the exact
values of CV / 共3Nk兲, Eq. 共34兲, for T = 1200 K 共1兲, T = 300 K 共2兲, and T
= 100 K 共3兲. The lines 共---兲, 共¯兲, and 共—兲 represent the usual Einstein
equation, Eq. 共36兲, at T = 50, 300, and 1200 K, respectively. Note that the
Einstein values CV / 共3Nk兲 are independent of N. The Dulong-Petit limiting
law predicts CV / 共3Nk兲 = 1.
for T ⬍ ⬁ the inequality N ⱖ n0 ⬎ ¯ ⬎ ni ⬎ ¯ ⬎ nm holds, so
that ␧m Ⰶ E. Nevertheless, Eq. 共33兲 is erroneously applied in
the literature without any limitation on ␧m up to m → ⬁:
Z共␧m → ⬁兲 = exp共−⌬␧ / 2kT兲 / 共1 − exp共−⌬␧ / kT兲兲.7 The constraints of Eqs. 共25兲 and 共31兲, which arise naturally from the
solvability conditions of the Boltzmann variational problem,
truncate the sum in Eq. 共33兲 at the limiting number m. Because m depends on N and T, the exact Digamma-partition
function in Eq. 共30兲 increases nonlinearly with N and T 关see
Figs. 3共a兲 and 3共b兲兴. Recall that the classical partition func52
Am. J. Phys., Vol. 77, No. 1, January 2009
冊
exp共⌬␧/共2kT兲兲 − exp共− ⌬␧/共2kT兲兲
+ 1 . 共35兲
1 − exp共␣␺ − ␥兲
Then we apply ␣␺共T兲 to Z␺ in Eq. 共30兲 and calculate CV in
Eq. 共34兲 numerically. For silver the Einstein temperature is
␪E = 168 K, and hence ⌬␧ = ␪Ek = 2.32⫻ 10−21 J. Calculations
with Eqs. 共31兲 and 共35兲 show that the maximum energy ␧␺m
increases with increasing T and N 关see Figs. 3共c兲 and 3共d兲兴.
For example, at T = 293 K and N = 100, Eq. 共31兲 yields: ␧m
= 4.7 kT 共0.19⫻ 10−19 J兲 and m = ␧␺m / ⌬␧ − 1 / 2 = 8. For T
= 293 K and N = 6.022⫻ 1023 we obtain ␧␺m = 55kT and m
= 95. Because of the constraints on ␧␺m and m, CV / 共3Nk兲 is
smaller than the Dulong-Petit limiting value CV / 共3Nk兲 = 1
even if T Ⰷ ␪E 关see Figs. 3共e兲 and 3共f兲兴. For example, for N
= 50 and T = 293 K, Eq. 共34兲 yields CV / 共3Nk兲 = 0.937, which
is smaller by 3.7% than the value CV / 共3Nk兲 = 0.973 of the
usual Einstein formula for ␧m → ⬁:7
CV共␧m → ⬁兲 =
3N⌬␧2 exp共⌬␧/kT兲
.
kT2共exp共⌬␧/kT兲 − 1兲2
共36兲
At very large values of N, say N = 6.022⫻ 1023, the exact
Digamma-Einstein heat capacity, Eq. 共34兲, approaches Eq.
共36兲, for example, at T = 293 K Eqs. 共34兲 and 共36兲 yield the
same value CV / 共3Nk兲 = 0.973.
D. Approximate value of the heat capacity
If we substitute Eq. 共33兲 into Eq. 共26兲 and solve for ␧Wm
relative to ␧Wm = ⌬␧共m + 1 / 2兲 for the case ZW / 共2NW兲
Ⰶ ln共2NW / 共eZW兲兲, we obtain the Lambert W maximum energy ␧Wm in the form
冋 冉 冊 冉 冉 冊冊册
␧Wm共T兲 ⬇ kT ln exp
−
2NW
⌬␧
⌬␧
−
1 − exp
2kT
e
kT
⌬␧
.
2
共37兲
The sum ␧Wm + 共␥ − ln共2 / e兲兲kT is close to the exact value ␧␺m
except for NW ⱕ 5 and T ⱖ 1200 K 共the melting temperature
of Ag is Tm = 1235.1 K兲 关see Figs. 3共c兲 and 3共d兲兴. Recall that
the quantity 共␥ − ln共2 / e兲兲kT ⬇ 0.884kT corrects the discrepancy between ␧Wm and ␧␺m. In the limiting case, T → 0,
Eq. 共37兲 reduces to the physically reasonable value:
S. Kakorin
52
Table II. Three forms of the Boltzmann distribution derived using different approximations for ln共n!兲.
Approximations
for ln共n!兲
Most probable
population numbers
T→0
冉 冊
ni* = e−␣−␤␧i
Eq. 共16兲
The partition
function
i→⬁ −␤␧i
e
Z = 兺i=0
Eq. 共17兲
The first Lagrange
multiplier
␣ = − ln
The maximum
energy of states
lim ␧Wm共T兲 =
Stirling’s formula, Eq. 共3兲
Stirling’s approximation,
1
1
Eq. 共1兲
ln共n!兲 ⬇ n + ln共n兲 − n + ln共2␲兲
ln共n!兲 ⬇ n ln共n兲 − n
2
2
* =
ni,W
冉
1
exp共␣W + ␤␧i兲
2
Eq. 共19兲
2W−0 −
冊
ZW = −2NWW−0 共−exp共−␤␧Wm − 1兲兲
Eq. 共27兲
冉冊
冉 冊
N
Z
Eq. 共18兲
NW
ZW
−
ZW
2NW
Eq. 共22兲
␣W = − ln
␧i ⱕ ␧Wm = 共ln共2 / e兲 − ␣W兲 / ␤
Eq. 共25兲
␧i ⬍ ⬁
⌬␧
,
2
−1
共38兲
which implies that all particles are in the ground state. The
Lagrange multipliers ␣W and ␣␺ are almost equal, for
example, ␣W = ln共2 / e兲 − ␤␧Wm = −4.086 and ␣␺ = ␥ − ␤␧␺m
= −4.081 for T = 293 K and NW = 100. Equations 共21兲 and 共32兲
also yield equal values of the partition sums: ZW = Z␺
= 1.715. If we apply Eq. 共37兲 to Eq. 共34兲, we obtain
CWV / 共3Nk兲 = 0.944, which is very close to the exact Digamma value C␺V / 共3Nk兲 = 0.954. For N = 5 the discrepancy
becomes larger, CWV / 共3Nk兲 = 0.760 compared to C␺V / 共3Nk兲
= 0.602, even if the condition ZW / 共2NW兲 Ⰶ ln共2NW / 共eZW兲兲
for Eq. 共37兲 is satisfied: 0.134Ⰶ 1.01. In summary, the Lambert W approximation, Eq. 共37兲, yields a reliable analytical
estimate of the maximum energy ␧m in terms of the experimentally accessible values T, N, and ⌬␧.
Exact function
ln共n!兲
ni,*␺=
=␺−1
共−
␣ ␺ − ␤ ␧ i兲 − 1
0
Eq. 共29兲
Z␺ =
N␺
␺−1
共
␤
␧
␺m − ␥ 兲 − 1
0
Eq. 共32兲
冉
N␺
+1
Z␺
Eq. 共30兲
␣␺ = − ␺0
冊
␧ i ⱕ ␧ ␺m = 共 ␥ − ␣ ␺兲 / ␤
Eq. 共31兲
Digamma- and the Lambert-Boltzmann distributions extend
the range of applicability of the exponential Boltzmann distribution to finite values of N and ni. The exact Boltzmann
distribution naturally imposes a constraint on the maximum
energy ␧m of the highest populated state, depending on T and
N 关see Eqs. 共25兲, 共31兲, and 共37兲兴. The constraints of Eqs. 共25兲
and 共31兲 confine the total number m of thermally available
energy states and truncate the partition sum in Eq. 共17兲 at i
= m. The smaller value of the partition sum results in a
smaller value for the heat capacity compared to the usual
heat capacity in Eq. 共36兲. The discrepancy between the exact
and the usual heat capacity calculated with the exponential
Boltzmann distribution increases with decreasing N. Measurements of the heat capacity of nanosystems could provide
experimental evidence for the exact Boltzmann statistics.
a兲
Electronic mail: [email protected]
Sean A. C. McDowell, “A simple derivation of the Boltzmann distribution,” J. Chem. Educ. 76, 1393–1394 共1999兲.
2
D. K. Russel, “The Boltzmann distribution,” J. Chem. Educ. 73, 299–300
共1996兲.
3
A. S. Wallner and K. A. Brandt, “The validity of Stirling’s approximation,” J. Chem. Educ. 76, 1395–1397 共1999兲.
4
G. Marsaglia, and J. C. W. Marsaglia, “A new derivation of Stirling’s
approximation to n!,” Am. Math. Monthly 9, 826–829 共1990兲.
5
H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, 2nd ed. 共Van Nostrand, New York, 1965兲, pp. 438–441.
6
D. A. Barry, J. Y. Parange, L. Li, H. Prommer, C. J. Cunningham, and F.
Stagnitti, “Analytical approximations for real values of the Lambert
W-function,” Math. Comput. Simul. 53, 95–103 共2000兲.
7
P. Atkins and J. De Paula, Physical Chemistry, 8th ed. 共Oxford University
Press, Oxford, 2006兲, pp. 560–576.
1
V. CONCLUSION
If we use the Stirling formula, Eq. 共3兲, or the exact function ln共n!兲 to solve the Boltzmann variational problem, Eq.
共14兲, we obtain the Boltzmann distribution in terms of either
the Lambert W-function or the Digamma-function 共see Table
II兲. The approximate Lambert-Boltzmann distribution in Eq.
共19兲 was shown to be a very good estimate of the exact
Boltzmann statistics in Eq. 共29兲 given in terms of the inverse
Digamma-function. Because the Lambert-W function is included in most symbolic packages, the Lambert-Boltzmann
distribution, Eq. 共19兲, can be implemented more easily than
the exact Digamma-Boltzmann distribution. Both the
53
Am. J. Phys., Vol. 77, No. 1, January 2009
S. Kakorin
53