Homework 3

Due 5/10/06

1. Consider the Langevin equation for a particle with strong damping under the influence of a random force:

$${d x \over dt} = f(t)$$

Where $f(t)$ is a random function with mean zero and $\langle f(t) f(t') \rangle = \delta(t-t')$. Calculate $\langle (x(t)-x(0))^2\rangle$. The average is over all possible realizations of the random function $f$. Hint: Solve for $x(t)-x(0)$ to get an answer in terms of $f(t)$. Then square and average the result. Interchange order of integration of average.

2. The Langevin equation in problem 1 has a probability distribution for position $x$ and time $t$ that satisfies the diffusion equation

$$2 {\partial P(x,t)\over \partial t} = {\partial^2 P(x,t)\over \partial x^2}$$

Assume the particle starts out at the origin, that is $P(x,0) = \delta(x)$, solve for $P(x,t)$ and compare your answer to problem 1. Generalize this to where the particle starts out at position $x=l$.

Hint: Fourier transform with respect to $x$ giving a function ${\hat P}(k,t)$. This will give a first order ordinary differential equation. The solution has one arbitrary constant that depends on $k$, $a(k)$. Obtain $a(k)$ using the initial condition and back transform.

3. The current of particles going past a point x $j(x)$ satisfies

$${\partial j \over \partial x} = -{\partial P \over \partial t}$$

This is just a consequence of conservation of mass.

(a) Show that $j = -2(\partial P/\partial x)$, gives the diffusion equation of problem 2.

(b) Suppose there is an impenetrable boundary at $x=0$. What is boundary condition at this point?

(c) A particle starts out at $x=l$ satisfying the Langevin equation of problem 1. There is an impenetrable wall at $x=0$. Using the boundary condition from part (b) and the solution to the last problem, calculate the probability distribution for position $x$ and time $t$. Hint: use the ``method of images" to solve the diffusion equation so that you have appropriate boundary condition at $x=0$.

4. Instead of 3(b), we have a totally absorbing boundary at $x=0$. In this case, all particles hitting this boundary vanish. The appropriate boundary condition here is $P(x=0,t) = 0$. Calculate the probability distribution for position $x$ and time $t$, in this case.

5. Consider a polymer chain in one dimension of length $L$. Its path is a random walk as in problem 1, but with time replaced with arclength, that is $t\rightarrow s$. Assume that the polymer is held fixed at $x=l$ and assume that there is an impenetrable barrier at $x=0$.

(a) Calculate the probability distribution of the end to end distance. You may leave the normalization as an integral.

(b) The answer simplifies when $L$ becomes very large. Calculate the limiting form in this case.

Hint. The probability distribution is given by statistical mechanics. If the chain goes back and forth with fixed step length, then all configurations are equally likely except those that cross the origin. Paths that cross the origin have a statistical weight of zero.