Pressure in hard disk systems

We wish to study the pressure of the hard disk system as a function of the density. van der Waals proposed a simple equation of state which improves on the perfect gas law $$PV=Nk_BT$$, where $P$ is the pressure, $V$ the volume $T$ temperature and $N$ number of particles. For particles with only repulsive interactions, like the hard disk system he proposed $$P(V/N-b) = k_B T$$.

• Study the pressure of a hard sphere system as a function of $N$ and $V$.
• How well can you get agreement with the perfect gas law?
• Can you extract a value for the volume correction $b$. How well does the van der Waals expression work? Does it break down in certain limits?

You will need to initialize the state of the system for different average densities, what is the best strategy for placing $N$ particles in a box? To get reasonable results $N$ should be neither too small, where systematic errors creep in, nor too large when then dynamics becomes slow due to the simple algorithms that we are using.

Defining the temperature

In statistical mechanics you have perhaps seen that equipartition implies that the kinetic energy per particle is linked to the temperature $KE= \frac{3}{2} k_BT$, when working in three dimensions. The numerator 3 is just the dimension of space that we are working in. In two dimensions for our simulation we thus find that the kinetic energy per particle is linked to the temperature through: $$KE= k_BT $$

The Virial theorem

For a longer and more difficult introduction to pressure measurements in molecular simulation you might try to also determine the pressure using the "virial" theorem.

The pressure is evaluated from the statistics of collisions between pairs of particles via the Claussius virial theorem