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The Rouse chain

We now concentrate on one bead, say bead no. n, while keeping all the other beads fixed. Its equilibrium distribution is given by

\begin{displaymath}G_{\mathrm{eq}}(
\vec{R}_{n})=C\exp \{-\beta \frac{3k_{B}T}{2...
...\beta \frac{3k_{B}T}{2b^{2}}(\vec{R}_{n+1}-\vec{R}_{n})^{2}\}.
\end{displaymath} (6.1)

According to Eq. (4.30) and Eq. (4.27) the Langevin equation  describing the motion of the bead then is, with $
\gamma =m\xi $

\begin{displaymath}\frac{d\vec{R}_{n}}{dt}=-\frac{3k_{B}T}{\gamma b^{2}}(2\vec{R}_{n}-\vec{R}
_{n-1}-\vec{R}_{n+1})+\vec{f}_{n}
\end{displaymath} (6.2)

where we have assumed that $D=kT/\gamma $ is independent on $\vec{R}_{n}$. The same reasoning may be applied to all other beads, leaving us with the equations of motion
    
$\displaystyle \frac{d\vec{R}_{0}}{dt}$ = $\displaystyle -\frac{3k_{B}T}{\gamma b^{2}}(\vec{R}_{0}-\vec{R}
_{1})+\vec{f}_{0}$ (6.3)
$\displaystyle \frac{d\vec{R}_{n}}{dt}$ = $\displaystyle -\frac{3k_{B}T}{\gamma b^{2}}(2\vec{R}_{n}-\vec{R}
_{n-1}-\vec{R}_{n+1})+\vec{f}_{n}$ (6.4)
$\displaystyle \frac{d\vec{R}_{N}}{dt}$ = $\displaystyle -\frac{3k_{B}T}{\gamma b^{2}}(\vec{R}_{N}-\vec{R}
_{N-1})+\vec{f}_{N}$ (6.5)
$\displaystyle \langle \vec{f}_{n}(t)\cdot \vec{f}_{m}(t^{\prime })\rangle$ = $\displaystyle 6D\delta
_{n,m}\delta (t-t^{\prime })$ (6.6)

Eq. (6.4) applies when $n=1,\ldots ,N-1$.

Before starting to analyse these equations in the next section, let us derive one simple result:

$\displaystyle \frac{d \vec{R}_G}{dt}$ = $\displaystyle \frac{1}{N+1} \sum_{n=0}^N \vec{f}_n$ (6.7)
$\displaystyle \vec{R}_G(t)$ = $\displaystyle \vec{R}_G(0) + \int_0^t d\tau
\frac{1}{N+1} \sum_n \vec{f}_n(\tau)$ (6.8)


 
$\displaystyle \langle (\vec{R}_G(t) - \vec{R}_G(0))^2 \rangle$ = $\displaystyle \langle \int_0^t d\tau \int_0^t d\tau ^{\prime}\left( \frac{1}{N+...
...ht) \cdot \left( \frac{1}{N+1} \sum_m \vec{f}_m
(\tau^{\prime}) \right) \rangle$  
  = $\displaystyle \frac{6D}{N+1}t = 6D_G t$ (6.9)

So $D_G = D/N = k_BT /N\gamma$, which is perfectly understandable.


next up previous contents index
Next: Normal mode analysis Up: The Rouse chain Previous: Introduction
W.J. Briels