Correlation of the end-to-end vector

As an example of a dynamic characteristic of the Rouse chain we calculate $\langle \vec{R}(t)\cdot \vec{R}(0)\rangle$, i.e. the time correlation function of the end-to-end vector . First we notice that

$$\vec{R}=\vec{R}_{N}-\vec{R}_{0}=2\sum_{k=1}^{N}\vec{X}_{k}\{(-1)^{k}-1\}\cos \left( \frac{k\pi }{2(N+1)}\right) .$$ (6.35)

Our result will be dominated by k values which are extremely small compared to N. We therefore write

$$\vec{R}=-4\sum_{k=1}^{N} {}^{\prime }\vec{X}_{k}$$ (6.36)

where the prime at the summation sign indicates that only terms with odd kshould occur in the sum. Then

$$\langle \vec{R}(t)\cdot \vec{R}(0)\rangle =16\sum_{k=1}^{N}{}^{\prime }\langle \vec{X}_{k}(t)\cdot \vec{X}_{k}(0)\rangle$$ (6.37)

where we have used the fact that different modes are uncorrelated.

From Eqs. (6.32) to (6.33) we get

  

$$\vec{X}_k(t) \vec{X}_k(0) e^{-t/\tau_k} + \int_0^t d\tau e^{-(t-\tau)/\tau_k} \vec{F}_k(\tau)$$ = (6.38)

$$\langle \vec{X}_k(t)\cdot\vec{X}_k(0) \rangle \langle \vec{X}_k(0)^2 \rangle e^{-t/\tau_k}$$ = (6.39)

In Appendix A of this chapter it is shown that

$$\langle \vec{X}_k(0)\cdot\vec{X}_k(0) \rangle = \frac{3D}{N+1} \frac{\tau_k}{ 2} \ k \ne 0.$$ (6.40)

Introducing everything into Eq. (6.37) we get

$$\langle \vec{R}(t)\cdot\vec{R}(0) \rangle = \frac{8b^2}{\pi^2} (N+1) \sum_{k=1}^N {}^{\prime}\frac{1}{k^2} e^{-t/\tau_k} .$$ (6.41)

The characteristic decay time at large t is $\tau_1$, which is proportional to N².

Notice that in this derivation we have averaged over all initial values. We might also have calculated $\langle \vec{R}(t) \cdot \vec{R}(t^{\prime}) \rangle_{\vec{X}_k(0)}$, i.e. the time correlation function of $\vec{R}(t)$, given some initial configurations of the chain. For very large t and $t^{\prime}$ the result should be independent of $\vec{X}_k(0)$. Indeed using Eq. (6.38) twice, it is not difficult to find

$${ \langle \vec{X}_k(t)\cdot\vec{X}_k(t') \rangle_{\vec{X}_k(0)} = }$$

$$\vec{X}_k(0)^2 e^{-(t+t^{\prime})/\tau_k} + \frac{3D}{N+1} \frac{... ...left( e^{-\vert t^{\prime}-t\vert/\tau_k} - e^{-(t+t^{\prime})/\tau_k} \right).$$ (6.42)

which for very large t and $t^{\prime}$ leads to the desired result.