E2: Random walks

Week 6. Due Wed 17 Feb 23:59, reviewed Thu 18 Feb

About 2 hours. Theory: Random walks.

1. A biased random walk

A walker in 1D takes \(n\) steps of length \(d\): to the right with probability \(p\), to the left with probability \(1-p\). The steps are independent.

  1. Show that \(\langle x_n - x_0\rangle = n d (2p-1)\).
  2. Show that the variance is \(\langle (x_n - x_0)^2\rangle - \langle x_n - x_0\rangle^2 = 4np(1-p)d^2\).
  3. Check that you recover \(\langle (x_n-x_0)^2\rangle = nd^2\) for \(p = 1/2\). What kind of motion does a bias correspond to physically?

2. From steps to a diffusion coefficient

A random walker takes a step of length \(d\) in a random direction every \(\Delta t\). Show that in 1D this corresponds to \(D = d^2/(2\Delta t)\). A 0.5 µm bead in water has \(D\approx 0.9\) µm²/s. What step length corresponds to \(\Delta t = 1/15\) s (one video frame)? Is the bead really taking steps of this size?

3. Simulation

Run randomwalk_exercise.py.

  1. What slopes do you expect for the three MSD curves (in units of stepsize²)? Why?
  2. Change numpart and timesteps so that the statistics are similar to your experiment (about 50 particles visible, 300–4500 frames). How large is the scatter of the fitted slope between runs?