Spatial distribution functions 153〈|r−r′|^2 〉=λ^3(
βhω
2 π) 3 N+3
1
h^6(
2 π
βhω) 3 N(
2 πm
β) 3
×
4 π
(N+ 1)^3 /^2∫∞
0ds s^4 e−βmω(^2) s (^2) /(N+1)
. (4.5.52)
A useful trick for performing integrals of the form
∫∞
0 dx x2 nexp(−αx (^2) ) is to express
them as
∫∞
0
dx x^2 ne−αx
2
= (−1)n
∂n
∂αn
∫∞
0dxe−αx2= (−1)n∂n
∂αn1
2
√
π
α=
(n+ 1)!!
2 · 2 nαn√
π
α. (4.5.53)
Applying eqn. (4.5.53) yields, after some algebra,
〈|r−r′|^2 〉=3
4
√
2
(N+ 1)
βmω^2. (4.5.54)
The mean-square end-to-end distance increases both with temperature and with the
number of repeat units in the polymer. Because of the latter, mean-square end-to-end
distances are often reported in dimensionless form as〈|r−r′|^2 〉/(Nd^20 ), whered 0 is
some reference distance that is characteristic of the system. Inan alkane chain, for
example,d 0 might be the equilibrium carbon–carbon distance.
4.6 Structure and thermodynamics in real gases and liquids from
spatial distribution functions
Characterizing the equilibrium properties of real systems is a significant challenge due
to the rich variety of behavior arising from the particle interactions. In real gases and
liquids, among the most useful properties that can be described statistically are the
spatial distribution functions. That spatial correlations exist between the individual
components of a system can be seen most dramatically in the exampleof liquid water
at room temperature. Because a water molecule is capable of forming hydrogen bonds
with other water molecules, liquid water is best described as a complexnetwork of hy-
drogen bonds. Within this network, there is a well-defined local structure that arises
from the fact that each water molecule can both donate and accept hydrogen bonds.
Although it might seem natural to try to characterize this coordination shell in terms
of a set of distances between the molecules, such an attempt misses something funda-
mental about the system. In a liquid at finite temperature, the individual atoms are
constantly in motion, and distances are constantly fluctuating, asis the coordination
pattern. Hence, a more appropriate measure of the solvation structure is a distribution
function of distances in the coordination structure. In such a distribution, we would