### A Two-Dimensional Lattice Gas

Introduction

A lattice gas is an Ising model which has been interpreted to model a fluid instead of a magnetic system. In a lattice gas each site is assigned a value of 0 for an unoccupied site and 1 for a site occupied by a particle. The energy of interaction of two nearest neighbor occupied sites is -u0.

The program uses the Metropolis algorithm to simulate a system in equilibrium with a heat bath at temperature T. The number of particles and the volume or number of sites is fixed. The trial change consists of choosing a site and one of its nearest neighbors and exchanging the values at the two sites. In the language of magnetism this trial change is called spin-exchange dynamics or Kawasaki dynamics. The critical temperature Tc in this system is about 0.567 (one-quarter of the value of Tc in the Ising model); the critical density corresponds to 50% occupancy.

A gravitational field can also be added to the simulation such that each particle has an additional energy equal to gh, where g is the field and h is the vertical position of the particle. We assume the particles have unit mass and that the temperature is given in terms of u0.

Problems

The goal of the following problems is to observe two phase coexistence and spinodal decomposition.

1. Run the program at T = 0.4 with 600 particles on a 32 × 32 lattice in zero gravitational field. You should see one region of particles (green sites) with a few small holes or bubbles and another region with just a few isolated particles or small clusters of particles. This system represents a liquid (the predominately green region) in equilibrium with its vapor (the mostly white region). Record the energy. To speed up the simulation set `steps per display` to 100.
2. In steps of 0.05 increase the temperature until T = 0.7. Record the energy and describe the visual appearance of the configurations of sites at each temperature. Run for at least 10000 mcs to equilibrate the system and then press the `Zero Averages` button and run for at least 20000 mcs more before recording an estimate for the energy at each temperature. At what temperature does the large liquid region break up into many pieces, such that there is no longer a sharp distinction between the liquid and vapor region? Is there any evidence from your estimates of the energy that a transition from a two phase system to a one phase system has occurred?
3. Repeat Problem 2 with 200 particles.
4. Repeat Problem 2 with 512 particles. In this case you will be passing through a critical point. You will not see a big difference because of finite size effects.
5. If a gravitational field is added, the program removes the periodic boundary conditions in the vertical direction, and thus sites at the top and bottom only have three neighbors. Use g = 0.01 and repeat the above simulations. What differences do you see?
6. Run a simulation with 2048 particles on a 64 × 64 lattice at T = 2.0 with no gravitational field for 5000 mcs. Then change the temperature to T = 0.2. This process is called a temperature quench, and the resulting behavior of the system is called spinodal decomposition. The domains grow very slowly as a function of time. Discuss why it is difficult for the system to reach its equilibrium state where there is one domain of mostly occupied sites in equilibrium with one domain of mostly unoccupied sites.

References

- A discussion of the lattice gas can be found in Chapter 5 of Harvey Gould and Jan Tobochnik, Statistical and Thermal Physics.

Java Classes

- LatGas2D
- LatGas2DApp

Updated 4 March 2009.