Polymers in Confined Geometry.pdf

2.2. IDEAL PHANTOM CHAIN 7
configuration
R =
N
ti, (2.1)
i=1
with N ≫ 1. We will be mainly interested in the limit of long chains.
The first two moments of the end-to-end vector distribution are obtained
quite readily. The mean of the distribution 〈R〉 must be zero because of spacial
isotropy. The mean square end-to-end distance 〈R 2 〉 is obtained in a straightforward
calculation due to the absence of correlations
R 2 =
N
i=1
N
〈ti · tj〉 =
j=1
N
i=1
N
j=1
δij
t 2 = N t 2 = Nl 2 , (2.2)
where we have used the fact that subsequent segments are uncorrelated. In
summary we have:
〈R〉 = 0, (2.3)
R 2 = Nl 2 . (2.4)
From eq. (2.4) we note that a proper continuum limit of the FJC needs to take
the limits N → ∞, l → 0 such that Nl 2 stays finite.
Once we know these moments we can already write down the full probability
distribution of the end-to-end distance by exploiting the central limit theorem.
It states that the limiting distribution of a random variable, which is the sum of
independent random variables, is a Gaussian 1 with mean 〈R〉 and width 〈R 2 〉
p(R) =
3
2π 〈R2 3/2
exp −
〉
3R2
2 〈R2
. (2.5)
〉
This result brings us back to the random walk since eq. (2.5) may be interpreted
as the Greens function of the diffusion equation with the initial condition p(R, t =
0) = δ(R).
Due to the central limit theorem, universal behavior for a larger class of
flexible polymers is expected. As long as correlation between the segments along
the chain decay on a scale small compared to the total length these correlation
can be subsumed in an effective segment length or Kuhn segment b, which is
defined by the universal relation to hold:
R 2 = b 2 N ⇒ R ∼ bN 1/2 . (2.6)
For the FJC, b = l the Kuhn length is identical to the segment length. For an
actual chain the correlations are renormalized by looking at a larger scale at the
1 Of course the probability distribution function can also be obtained by calculation from
statistical mechanics.