Exercise 4.2: Approach to equilibrium I

Write the code for simulating a two dimensional system of particles interacting via a Lennard-Jones potential. Consider $N=12$ particles in a box of linear dimensions $L_x=L_y=8$. For this choice of $N$ and $L$ the density $\rho =12/64=0.19$. Suppose that the particles are initially constrained to be in the left part of the box and placed on a rectangular grid. At $t=0$ the constraint is removed and the particles move freely throughout the entire box. Use $v_{\max }=1.0$, the maximum initial speed, and $\Delta t=0.02$.

1. Observe a sufficient number of snapshots for the particles to move significantly from their original positions (It will take of the order of 100-200 time steps). Does the system become more or less random?

2. From the visual snapshots of the trajectories, estimate the time for the system to reach equilibrium. What is your qualitative criterion for equilibrium?

3. Compute $n(t)$ the number of particles on the left hand side of the box. Plot its value as a function of time. What is the qualitative behavior of $n(t)$? What is the mean number of particles on the left side?