W(r) = (3 / 2 pi <r2>0)3/2 exp (-r2 / 2 <r2>0)
which is a monotonically decreasing function, as the length, r, of the vector increases.
The distribution for the length itself, without regard to direction, is
W(r) = 4 pi r2 (3 / 2 pi nl2)1/2 exp (- 3 r2 / 2 nl2)
which passes through a maximum at
rmost probable = (2n / 3)1/2l
The root-mean-square end-to-end distance is somewhat larger than the most probable value.
<r2>1/2 = n1/2l
Illustrative curves are depicted in PC Fig. 7.14.
This model is often used for the distribution functions for flexible chains. It becomes more accurate as n increases. It fails at very high extensions, where it predicts nonzero probabilities for extensions that are physically unattainable.
Return to the index