Diffusion on a lattice
A Monte Carlo simulation of particles on a square lattice with a maximum
of one particle per site.
The model is summarized by the following steps:
- A particle is chosen at random.
- One of the four nearest neighbor sites of the particle is chosen at random.
- If the site is empty,
the particle is moved to that site; otherwise it remains where it is. In either case the time advances by 1/N.
- The mean square displacement <R2> is computed after one Monte Carlo per particle, that is, after N particles have been chosen at random. (In one Monte Carlo step per particle, some particles may be chosen
more than once and some not chosen at all.)
- Run the simulation with the default parameters and compute the slope of the plot of <R2> versus time. You can use the Data Tool under the Tools menu to do a Curve Fit. The diffusion coefficient D is defined by the relation
<R2> = 2dDt,
where d is the spatial dimension (two in this case). Determine the value of D.
- Repeat the simulation for different densities and plot the diffusion
coefficient D as a function of the density ρ = N/L2. Discuss the dependence of D on ρ.
- Is <R2> always approximately linear in time for any density?
Updated 2 March 2009.