Exercise 4.3: Approach to equilibrium II

Consider $N=12$ particles in a box of linear dimensions $L_x=L_y=8$. Consider the initially the particles are placed in a rectangular grid and the velocities are random with $v_{\max }=1.0$.

1. Do a number of MD steps (50,100) and average the quantity $\left\vert \mathbf{v}^{*}\right\vert ^{2}$ to estimate the actual temperature. To adjust the temperature to a desired value, scale all velocity components for all particles in a suitable way. Repeat this procedure up to 10 times. After 500-1000 steps the fluid will be well equilibrated and the temperature will be steady (although fluctuating slightly). Use $\Delta t=0.02$.

2. Plot the temperature averaged over intervals of 5 time steps as a function of time for each of the previous temperatures. What is the qualitative dependence of the temperature fluctuations?

3. Calculate the kinetic and potential energies as a function of time. Are the kinetic and potential energies conserved separately?