1. Consider the Langevin equation for a particle with strong damping under the influence of a
random force:
Where is a random function with mean zero and . Calculate . The average is over all possible realizations of the random function . Hint: Solve for to get an answer in terms of . 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 and time
that satisfies the diffusion equation
Assume the particle starts out at the origin, that is , solve for and compare your answer to problem 1. Generalize this to where the particle starts out at position .
Hint: Fourier transform with respect to giving a function . This will give a first order ordinary differential equation. The solution has one arbitrary constant that depends on , . Obtain using the initial condition and back transform.
3. The current of particles going past a point x satisfies
4. Instead of 3(b), we have a totally absorbing boundary at . In this case, all particles hitting this boundary vanish. The appropriate boundary condition here is . Calculate the probability distribution for position and time , in this case.
5. Consider a polymer chain in one dimension of length . Its path is a random walk as in problem 1, but with time replaced with arclength, that is . Assume that the polymer is held fixed at and assume that there is an impenetrable barrier at .
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.