Introduction

A Monte Carlo simulation of particles on a square lattice with a maximum of one particle per site.

Algorithm

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 <R
^{2}> 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.)

Problems

- Run the simulation with the default parameters and compute the slope of the plot of <R
^{2}> 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

<R ^{2}> = 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/L
^{2}. Discuss the dependence of D on ρ. - Is <R
^{2}> always approximately linear in time for any density?

Java Classes

- LatticeGas
- LatticeGasApp

Updated 2 March 2009.