Physics 120/240

Homework 4

Due 5/24/06

1. Consider the Rouse equation:
   

   $${\partial r\over \partial t} - {\partial^2 r\over \partial s^2} = \eta(s,t)$$ (1)

   
   
   where $s$ is arclength, $t$ time, $r(s,t)$ is position, and $\eta$ is random noise with zero mean and autocorrelation function
   

   $$\langle \eta(s,t)\eta(s',t')\rangle = \delta(s-s')\delta(t-t') ~.$$ (2)

   
   

   Take $r$ to be a scalar. The extension to the vector case is straightforward. This problem illustrates how to calculate $\langle (r(s,t)-r(0,t))^2\rangle$, and $\langle (r(s,t)-r(s,0))^2\rangle$.

   1. Define the Fourier transform with respect to both $s$ and $t$ of $r(s,t)$, Call it ${\hat r}(k,\omega)$. Write down the inverse, that is, how to express $r(s,t)$ in terms of integrals of ${\hat r}(k,w)$. Do the same for $\eta$. Call the result ${\hat \eta}$.
   2. Fourier transform equation 1 to obtain an algebraic equation.
   3. Fourier transform eqn. 2 with respect to two sets of variables $s \rightarrow k$ and $s' \rightarrow k'$ and similarly for $t$ and $t'$. Calculate $\langle {\hat \eta}{(k,\omega)}{\hat \eta}{(k',\omega')}\rangle$.
   4. Solve for $\langle {\hat r}{(k,\omega)}{\hat r}{(k',\omega')}\rangle$.
   5. Consider $\langle r'(s,t) r'(s',t)$, where $r' = \partial r/\partial s$. Relate this to $\langle (r(s,t)-r(0,t))^2\rangle$, through an integral formula.
   6. Relate the Fourier transform of $r'(s,t)$ to ${\hat r}$ through the simple relation between the Fourier transform of a function and its derivative.
   7. Because the Rouse equation is translationally invariant in time, we can set $t=0$ and calculate instead $\langle r(k,t=0) r(k',t=0)\rangle$. Calculate this by expressing it in terms of integrals over $\omega$ and $\omega'$.
   8. Transform back to obtain $\langle r'(s,t=0)r'(s',t=0)\rangle$.
   9. From this obtain $\langle (r(s,t=0)-r(s,t=0))^2\rangle$ using part 5.
   10. Compare this with the prediction for $\langle (r(s)-r(0)^2\rangle$ according to the equilibrium statistical mechanics of a random walk.
   11. Follow a similar procedure to obtain $\langle (r(s,t)-r(s,0))^2\rangle$. Note that there is translational invariance in arclength also so that you can set $s=0$.