Polymer Physics

Wðx;y;zÞ¼WðxÞWðyÞWðzÞ¼ð

b^2

p

Þ^32 =eb

(^2) ðx (^2) þy (^2) þz (^2) Þ
(3.3)
As illustrated in Fig.3.2, the radial distribution of the end-to-end distances can
be expressed as
WðRÞ¼Wðx;y;zÞ 4 pR^2 (3.4)
Here,R^2 ¼x^2 þy^2 þz^2. Accordingly,
WðRÞ¼ð
b^2
p
Þ^32 =eb
(^2) R 2
 4 pR^2 (3.5)
As demonstrated in Fig.3.3, with the increase ofRfrom zero,W(R)reaches a
maximum value, which is called the most probable end-to-end distanceR*. From
@WðRÞ
@R

¼ 0 (3.6)

we obtain

R¼b^1 (3.7)

Fig. 3.1 Illustration of a
Gaussian distribution
function

Fig. 3.2 Illustration of a
spherical surface formed by
the dots (x, y, z) with fixed
distancesRfrom the same
starting point at the central

34 3 Conformation Statistics and Entropic Elasticity