Exercise 11.4: Individual particle diffusion in a lattice gas

Consider a nonzero concentration of $\rho$ random walkers (particle) on a square lattice. Each particle moves at random to empty nearest-neighbor sites but double occupancy of a site is forbidden; otherwise, the particles are non-interacting. Such model is an example of a ``lattice gas''. Note that the motion of an individual particle is correlated with the motion of the other particles. The physical motivation of this model arises from metal physics where diffusion is caused by thermal vacancies with a concentration that depends on the temperature. The main quantity of interest is the self-diffusion constant $D$ of an individual particle. The algorithm for a Mont Carlo simulation can be stated as follows:

1. Occupy the lattice sites at random with a concentration of $\rho$ particles. We have to follow the motion of each particle individually, so we identify them as the elements of some array.

2. At each step, pick a particle at random, and choose a neighboring site. if the neighboring site is empty, the particle is moved to this site, otherwise, the particle remains in the present position.

The measure of time in this context is arbitrary. The usual definition is that one unit of time corresponds to one Monte Carlo step per particle. During each of these steps, each particle attempts one jump on the average. The diffusion constant $D$ is obtained as the limit $t \rightarrow \infty$ of $D(t)$, where $D(t)$ is given by

$$D(t)=\frac{1}{2dt}\langle \Delta R(t)^2 \rangle$$

and $\langle \Delta R(t)^2 \rangle$ is the net mean-square displacement per particle after $t$ units of time.