Consider the Rouse equation:
|
(1) |
where is arclength, time, is position, and is
random noise with zero mean and autocorrelation function
|
(2) |
Take to be a scalar. The extension to the vector case is straightforward.
This problem illustrates how to calculate
, and
.
- Define the Fourier transform with respect to both and of , Call it
. Write down the inverse, that is, how to express in terms
of integrals of . Do the same for . Call the result .
- Fourier transform equation 1 to obtain an algebraic equation.
- Fourier transform eqn. 2 with respect
to two sets of variables
and
and similarly for
and . Calculate
.
- Solve for
.
- Consider
, where
.
Relate this to
, through an integral formula.
- Relate the Fourier transform of to through the simple relation between
the Fourier transform of a function and its derivative.
- Because the Rouse equation is translationally invariant in time, we can set and
calculate instead
. Calculate this by expressing it in terms of integrals over
and .
- Transform back to obtain
.
- From this obtain
using part 5.
- Compare this with the prediction for
according to the equilibrium
statistical mechanics of a random walk.
- Follow a similar procedure to obtain
. Note that there
is translational invariance in arclength also so that you can set .