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Figure 6.12: A schematic diagram of the coordinate transformation exploited to compute the circular convolution. The crosses represent chain ends and dots represent kinks. The center line represents a periodic coordinate system. For regular convolutions, we set the chain end to be the zero and we compute the convolution in the s coordinate system. For the circular convolution, it is more convenient to choose L1, the first kink arc length position as the zero and sum over the chain end positions as represented by thes0 coordinate system.
We define the 1D contour length Laplace Transform and inverse transform F(p)˜ ≡ L{F(L)}=
Z ∞ 0
dL F(L)e−pL, (6.80)
F(L) = L−1{F˜(p)}= 1 2πi
Z
L
dpF˜(p)epL, (6.81)
where R
L denotes a contour integral along the Laplace contour. The Faltung theorem states that Laplace Transform of a convolution is the product of the Laplace Transforms
L{F⊗G}= ˜FG,˜ (6.82)
for functionsF andG, where the arc length convolution is defined
F⊗G(L)≡ Z L
0
dL0 F(L0)G(L−L0). (6.83)
The generalization of the Faltung theorem is true formfunctions
L{F1⊗...⊗Fm}= ˜F1...F˜m. (6.84)
6.9.1 Circular convolutions
For the closed chain we need to evaluate a special type of convolution which is circular. By circular we mean that the end points are identified so that the arc-length position of the chain ends is not at an end point of the propagator. In this case it is convenient to redefine the arc-length coordinate system to be zero at the position of the first kink, L1, and sum over the position of the chain ends as depicted in Fig.6.9.1. Themkink contribution to the partition function is therefore
Zm∗ = ζm 4π
h(K⊗)m−1LKi
(0;L), (6.85)
where the factor ofLcomes from the integration over the end position and varies in the convolution.
To simplify this result it is convenient to go to the transformed partition function Z˜m∗ =−ζm
4π
K˜m−1 ∂
∂p
K˜ =− ζm 4πm
∂
∂p
K˜m, (6.86)
where the Lhas been transformed into a pderivative. We now return from Fourier-Laplace space, giving
Zm∗ =ζmL
4πm[K⊗]m(0;L), (6.87)
which is written in terms of convolutions of the spatial propagator only.
6.9.2 Numerical inverse transforms
To compute the numerical inversions of the Fourier and Laplace Transforms involving the spatial propagator, we first truncate the continued fraction [34] in Eq.6.49, then we compute the numerical
inversion with the built-in Mathematica functionsInverseLaplaceTransformandInverseFourierTransform.
In particular, the structure factor and partition function in an external field involve only a single numerical inverse Laplace Transform.
6.9.3 J factor computation
We have chosen to present most of our results in the last section as explicit series in kink number rather than writing them in the summed form (Eq.6.48and6.67). The purpose is two fold. First these expansions allowJ to be computed efficiently for many small values ofζ, because the K(m) are independent of ζ and only the first few must be computed explicitly for sufficient numerical accuracy. Furthermore the K(m) are simply related to the kink number distribution allowing the same computation to suffice for both results. Our computational discussion will mainly focus on the short contour length limit where these kink number expansions converge quickly. As we have already discussed at length, the large contour length limit can trivially be computed with the WLC results using the renormalized effective persistence length, ξ∗. This corresponds to the k→0 and p→0 limit where the theories are identical. In fact, in this limit, we can use the Gaussian chain to compute theJ factor. This computation appears briefly in Appendix6.11.
It is at short contour length where the two theories significantly diverge and kinking is induced.
As we have discussed above, in the limit as the contour length goes to 0, the polymer resembles a rigid rod. It is problematic to directly compute the inverse transforms of K∗ or K numerically in this limit since
K(0;L→0)∝ Z ∞
0
dk ksinkL, (6.88)
which, although it can be computed analytically by expanding sine into two exponentials then
integrating them on different contours, is problematic numerically. It is therefore convenient to use the expanded definition ofK∗in terms of theK(m)(eqn6.62). In the short contour length limit, an asymptotic expression already exists[4] forJ andK(0;L).
Convolutions of K must still be computed. Numerical computations of the inverse transforms of powers of ˜K are still problematic in the L→0 limit since, while they are more convergent than K, they must be integrated to largekwhere it is difficult to compute accurate Laplace transforms.
But this implies it is in precisely this limit that bending is really irrelevant and that the mechanics of these kinked chains becomes kink dominated. Once there are two or more kinks, the chain can now be closed without elastic bending. That is to say that, when these terms become difficult to calculate numerically, they can be well approximated by the kinkable rigid rod! The kinkable rigid rod is treated in Appendix6.10. As a practical matter there is a small contour length regime (L∼1), between the rigid rod limit and the contour length at which numerical transform inversion are rapidly convergent and where it is most convenient to use direct Monte Carlo integrations to compute the K(m). These direct Monte Carlo integrations serve as a useful check on our other numerical and analytic computations. For large dimensionless kink densities, the kink-number expansion is not rapidly converging and direct numerical inversions of the exactly summed transformed results are required. For most computations of theJ factor at small kink density, the rigid rod approximation suffices to compute the two kink term and the kink number sum can be truncated at this point as illustrated schematically by Fig.6.6.3.