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214 Bosons

distinguishableparticles:
1
Z N = Z P (4.47)
N!
P

1 N dist
= d xρ {x 1 ,... , x N }, {x P (1) ,...,x P (N) },β . (4.48)
N!
1 2 3 4 P
τ = β Forideal particles, the distinguishable-particledensity matrix separates
into aproduct ofsingle-particledensity matrices, but the presence of
τ = 0
permutations implies that these single-particledensity matrices are not
1 2 3 4
necessarily diagonal. Forconcreteness, weconsider, for N = 4particles,
the permutation P =( 14 23 ), which in cyclerepresentationis written
12 34
` ´ as P =(1)(243) (see Fig. 4.17). This permutationconsists of one cycle of
Fig. 4.17 The permutation 1423
1234
represented as a path. length 1 and one cycle of length 3. The permutation-dependent partition
function Z (1)(243) is

Z (1)(243) = dx 1 ρ(x 1 ,x 1 ,β) dx 2

× dx 3 dx 4 ρ(x 2 ,x 4 ,β) ρ(x 4 ,x 3 ,β) ρ(x 3 ,x 2 ,β) . (4.49)

ρ(x 2 ,x 2,3β)
The last line ofeqn (4.49) contains a doubleconvolutionand can be
written as a diagonal single-particledensity matrix at temperature T =
1/(3β). Thisisanelementary application ofthe matrix squaring de-
scribed in Chapter 3. After performing the last two remaining integra-
tions, over x 1 and x 2 , we find that the permutation-dependent partition
function Z (1)(243) is the product ofsingle-particle partitionfunctions,
one at temperature 1/β and the other at 1/(3β):
Z (1)(243) = z(β)z(3β). (4.50)
Here, and in the remainder of the present chapter, wedenote the single-
particle partitionfunctions with the symbol z(β):

single-particle
: z(β)= dxρ(x, x, β)= e −βE σ . (4.51)
partitionfunction
σ
Equation (4.50) carries the essential message that—forideal bosons—
the N-particle partitionfunction Z(β) can be expressed as a sum of
products ofsingle-particle partitionfunctions.However, this sum of N!
termsisnontrivial, unlike the one for the gas ofideal distinguishable
particles. Only forsmall N can we think of writing outthe N! permu-
tations and determining the partitionfunction via the explicit sumin
eqn (4.47). It is better to adapt the recursionformula of Subsection4.2.2
to the ideal-boson partitionfunctions.Now,the cycle weights are given
by the single-particledensity matrices at temperature kβ.Taking into
account that the partitionfunctions carry afactor 1/N!(see eqn (4.48)),
we find
N
1
Z N = z k Z N−k (with Z 0 =1). (4.52)
N
k=1