## Lattice Gas

The Ising model is very versatile. In addition to being a model of magnetic systems, it can also be a model of a fluid. The lattice gas version of the Ising model replaces the spins $s_i = \pm 1$ with occupation numbers $n_i = 0$ or $1$. If there are particles on both neighboring sites, there is an attractive interaction $-u_0$; otherwise, the interaction energy is 0. A hard core repulsion typical of fluid particles is maintained by the limitation that $n_i \le 1$.

In the lattice gas Monte Carlo simulation we exchange the values of $n_i$ for neighboring sites rather than flipping spins. Thus, the number of particles is conserved, and the chemical potential plays a role analogous the magnetic field. The program also includes a gravitational field which causes particles to sink toward the bottom. In addition to exploring the phase transition in this model, the program can be used to explore quenches from high to low temperatures.

Problem: Simulation of the lattice gas
1. What is the value of the critical temperature $T_c$ for the lattice gas in two dimensions in units of $u_0/k$?
2. Program LatticeGas simulates the lattice gas on a square lattice of linear dimension $L$. The initial state has all the particles at the bottom of the simulation cell. Choose $L=32$ and set the gravitational field equal to zero. Do a simulation at $T= 0.4$ with $N= 600$ particles. After a few Monte Carlo steps you should see the bottom region of particles (blue sites) develop a few small holes or bubbles and the unoccupied region contain a few isolated particles or small clusters of particles. This system represents a liquid (the predominately blue region) in equilibrium with its vapor (the mostly yellow region). Record the energy. To speed up the simulation set steps per display equal to 100.
3. Increase the temperature in steps of 0.05 until $T = 0.7$. At each value of $T$ run for at least $10,000\,$mcs to reach equilibrium and then press the Zero Averages button. Run for at least $20,000\,$mcs before recording your estimate of the energy. Describe the visual appearance of the positions of the particle and empty sites at each temperature. At what temperature does the one large liquid region break up into many pieces, such that there is no longer a sharp distinction between the liquid and vapor region? At this temperature there is a single fluid phase. Is there any evidence from your estimates of the energy that a transition from a two-phase to a one-phase system has occurred?
4. Repeat part (b) with $N=512$. In this case the system will pass through a critical point. The change from a one-phase to a two-phase system occurs continuously in the thermodynamic limit. Can you detect this change or does the system look similar to the case in part (b)?
5. If we include a gravitational field, the program removes the periodic boundary conditions in the vertical direction, and thus sites in the top and bottom rows have three neighbors instead of four. The gravitational field should help define the liquid and gas regions. Choose $g =0.01$ and repeat the above simulations. Describe the differences you see.
6. Simulate a lattice gas of $N = 2048$ particles on a $L=64 \times 64$ 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 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 for which there is one domain of mostly occupied sites in equilibrium with one domain of mostly unoccupied sites.

## Resource

Problem 5.24 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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