Appendices
Appendix A
There is a correspondence between a sufficiently long and flexible chain in a polymer melt, and
a random walk: the trajectory of the random walk corresponds to the coarse grained
configuration of the chain, and one step of the random walk corresponds to going along the
chain over the distance of one segment (see chapter 1). It was first shown by Flory 11 (and later
on confirmed experimentally, 24,25 theoretically 26 and by computer simulations 27 ) that the
probability to find the chain in a certain coarse grained configuration equals the probability that
the Brownian particle follows the corresponding trajectory.
Consider a random walk starting at the origin. Let  g ( , ) N x be the probability density to arrive
at the point x after N steps. In this appendix it is shown that in the limit of a large number of
steps  g ( , ) N x becomes Gaussian, irrespective of the single step probability density
g g ( ) ( , ) x x ≡ 1 , provided that it is isotropic. The probability density  g ( , ) N x satisfies the
recurrence relation
g g g ( , ) ( , ) ( ) N x dy N x y y + = +
∫
1 (A1)
Since for large values of N the characteristic length scale of  g ( , ) N x is much larger than the
characteristic length scale of  g ( ) x , the value of  g ( , ) N x will hardly change over the area
where the integrand gives the major contribution to the integral. Therefore, it makes sense to
expand  g ( , ) N x in a Taylor series around the point x. Since the first order term gives no
contribution because of the isotropy of  g ( ) x , it is necessary to expand till the second order.
The result is
g g
g
g ( , ) ( , ) ( )
( , )
N x N x
x x
dy y y y
i j
N x
i j
+ = +
∫
1
1
2
2


(A2)
Due to the symmetry of  g ( ) x the summation over i and j only gives a contribution if i j = , and
this contribution is independent of i. Moreover,