Exercise 4.5: Equation of state of a non-ideal gas

In this problem we compute the pressure and hence the ``equation of state'' (the relation between pressure, temperature and volume) of a gas

1. Consider $L_x=L_y=6,N=16$. Choose the initial configuration to be an equilibrium configuration. Compute the mean pressure using the net momentum transfer method and the virial theorem. Allow at least 100 time steps for equilibration and 200 to compute averages. Is the pressure constant or fluctuates? How does the pressure compare with the ideal gas result? Which method of computing the pressure is more accurate?

2. An approximate equation of state for dense gases and liquids is the ``van der Waals'' equation:
   
   $$P=kT\frac{\rho }{1-b\rho }-a\rho ^{2}.$$
   
   
   The phenomenological parameters $a$ and $b$ are related to the repulsive and attractive parts of the interaction respectively and are approximately independent of the temperature.. Use the same configuration to find the dependence of $P$ on $T$. Plot $P$ versus $T$ and use the van der Waals equation to obtain an estimate for the parameters $a$ and $b$.