Exercise 4.7: The solid state and melting

1. Choose the parameters $N=16$, $\Delta t=0.01$, $L_x=4$, and $L_y=2\sqrt{% 3}$. Place the particles on a triangular lattice which fills the container and give each particle initial zero velocity. Measure the temperature and pressure as a function of time and plot several ``snapshots'' of the system. What is the total energy of the system? Does it remain a solid?

2. Give the particles a random initial velocity with $v_{\max }=0.1$. What is the total energy? Equilibrate the system for approximately 50 time steps and average the pressure and temperature over 100 time steps.

3. Pick an equilibrium configuration from the previous item and rescale all velocities by a factor 2. What is the new total energy? Take snapshots of the system every 25 time steps and describe the behavior of the motion of the particles. What is the equilibrium temperature and pressure of the system? After equilibrium is reached, rescale the velocities again in the same way. repeat this rescaling and measure $P(T)$ and $E(T)$ for size different temperatures.

4. Plot $E(T)-E(0)$ and the pressure $P(T)$ as a function of $T$. What is the dependence of difference $\Delta E$ with $T$?

5. Decrease the density by multiplying $L_x$, $L_y$ , and the particle coordinates by a factor $1.1$. What are your results for temperature and pressure? What is the nature of the snapshots? Continue rescaling the density and the positions until the system ``melts''. What is your qualitative criterion for melting?