Polymers in Confined Geometry.pdf

78 APPENDIX A. DISCRETE WORM-LIKE CHAIN CORRELATIONS
The persistence length lp was defined through the spatial tangent-tangent
correlation function in eq. (2.12), which is for the discrete model
ˆtm · ˆtn = exp −|m − n| l
. (A.5)
To derive the relation between k and lp the correlation function is calculated
using eq. (A.4)
1 N
ˆtm · ˆtn =
dˆti ˆtm · ˆtn
−βH({ki})
e
Z({ki})
i=1
{ki=k}
1 N
=
dˆti ˆtm · ˆtm+1
ˆtm+1 · ˆtm+2 · · ·
Z({ki})
i=1
· · ·
ˆtn−1 · ˆtn
−βH({ki})
e
{ki=k}
1 ∂
=
Z({ki})
|m−n|
Z({ki})
∂km · · · ∂kn−1 {ki=k}
(A.3)
(A.4) 1
=
I0(k) N−1
∂ |m−n|
N−1
I0(ki) , (A.6)
∂km · · · ∂kn−1
i=1 {ki=k}
where in the second line
ˆti · ˆti+2 = ˆti · ˆti+1
ˆti+1 · ˆti+2 has been used, which
can be seen by writing the product of tangents as the cosines, using trigonometric
theorems and the properties of the averages. For the 3D case below it is shown
in [24, p. 377].
Now using [1, (9.6.28)] in the form I ′ 0(k) = I1(k) gives
|m−n|
I1(k)
ˆtm · ˆtn =
I0(k)
which results in the desired relation in 2D
−1 I1(k)
2D: lp = −l ln
I0(k)
lp
!
= (A.5), (A.7)
. (A.8)
For a given value of lp, k has to be determined numerically, since this formula is
not invertible in a closed form.
A similar calculation, following the above lines, can be done for three dimension.
By substituting dˆti = dφi dcos θi, we obtain for the partition function
Z3D({ki}) = (4π) N
N−1
i=1
sinh ki
, (A.9)
ki