mutation within the set {k} produces an identical (except for the sign) wave

function. Therefore sequences obtained by permutation should be omitted

from the set {k}, which can be done, for example, by only allowing se-

quences for which the component k™s are ordered in non-decreasing order. If

we allow all sequences, as we do, we overcount the sum by N ! and we should

therefore divide the sum by N !.18 Note that this N ! has nothing to do with

17 For a mixture of di¬erent types of particles, the equations are trivially modi¬ed.

18 This is a rather subtle and usually disregarded consideration. The reader may check the

17.6 The classical approximation 475

√

the 1/ N ! in the de¬nition of ψ(k), which is meant to normalize the wave

function consisting of a sum of N ! terms.

Since the box is large, the distance ”k = 2π/a between successive levels

of each component of k is small and the sum over k can be replaced by an

integral. When we also write p for k, we can replace the sum as:

VN VN

’ dk = 3N dp, (17.90)

(2π)3N h

k

and obtain

1 1

(“1)P dr φ— (P) e’β H(r) φ0 (P),

ˆ

+P

Q= dp (17.91)

0

N !h3N N !

P ,P

pj ·r j ] .

def

e(i/ )P[

φ0 (P) = (17.92)

j

The problem is to evaluate the function

u(r; P) = e’β H(r) φ0 (P),

ˆ

def

(17.93)

which is in general impossible because the two constituents of the hamil-

ˆ

tonian K = ’( 2 /2m) j ∇2 and V (r) do not commute. If they would

j

commute, we could write (see Section 14.6 on page 385)

e’β H = e’βV e’β K ,

ˆ ˆ

(17.94)

and evaluate u as

u(r; P) = e’βV (r) e’β

2

j (pj /2m) φ0 (P). (17.95)

In (17.91) only the N ! terms with P = P survive and we would obtain the

classical limit

1

dp dr e’βH(p,q) .

Qcl = (17.96)

3N

N !h

In general, we need to solve u(r; P) (see (17.93)). By di¬erentiating with

respect to β, it s found that u satis¬es the equation

‚u ˆ

= ’Hu, (17.97)

‚β

which is the Schr¨dinger equation in imaginary time: it/ being replaced

o

by β.

correctness for the case of two one-dimensional particles with plane waves k1 , k2 and observe

that not only k1 k2 |k1 k2 = 1, but also k1 k2 |k2 k1 = “1. In fact, Kirkwood™s original article

(1933) omitted this N !, which led to an incorrect omission of 1/N ! in the ¬nal equations.

Therefore his equations were still troubled by the Gibbs paradox. In a correction published in

1934 he corrected this error, but did not indicate what was wrong in the original article. The

term is properly included by Huang (1987).

476 Review of statistical mechanics

Kirkwood proceeded by writing u as

u = w φ0 (P)e’βH(p,r) , (17.98)

formulating a di¬erential equation for w and then expanding w in a power

series in . This yields

w2 + O( 3 ),

2

w = 1 + w1 + (17.99)

⎡ ¤

iβ 2 ⎣

pj · ∇j V ¦ ,

w1 = ’ P (17.100)

2m

j

⎡ ¤

β2 β3 ⎣ 1

pj · ∇j )2 V ¦

w2 = ’ ∇2 V ’ (∇j V )2 + (P

j

4m 6m m

j j j

β4

pj · ∇j V )2 .

+ 2 (P (17.101)

8m

j

Inserting this in (17.91) and integrating all correction terms over dp, ¬rst

the terms in the sum over permutations for which P = P (N ! terms) can

be separated. All odd-order terms, which are antisymmetric in p, disappear

by integration over p. Then the 1 N ! terms for which P and P di¬er by

2

the exchange of one pair of particles can be integrated. This is the leading

exchange term; higher-order exchange terms will be neglected. The end

result is

2πmkB T 3N/2

1

dre’βV (r) (1 + fcor ),

Q= 2

N! h

2β2 1 β

=’ ∇2 V ’ (∇j V )2 + O( 4 )

fcor j

12 mj 2

j

β

e’mj rjk /β

2 2

“ r jk · (∇j V ’ ∇k V ) + . . . . 17.102)

1+ (

2

j=k

We note that the 2 correction term was earlier derived by Wigner (1932);

the exchange term had been derived by Uhlenbeck and Gropper (1932) in

the slightly di¬erent form an ensemble average of the product over particle

pairs (i, j) of (1 “ exp(’mkB T rij / 2 ), and without the ¬rst-order term in β

2

in (17.102).

What does this result mean in practice? First we see that the classical

canonical partition function is given by

3N/2

1 2πmkB T

dre’βV (p,r)

cl

Q = (17.103)

h2

N!

17.6 The classical approximation 477

1

dre’βH(p,r) .