## Hard Disks MD

One of the simplest models of particle interactions is the hard core interaction, for which the interactioninteraction is infinite if two particles are closer than their diameter $\sigma$ and is zero otherwise. Hard disks are the two dimensional version of this interaction. A molecular dynamics (MD) simulation keeps track of the location and velocity of each hard disk and determines when the next collision occurs. Then all the hard disks are moved during the time between the last collision and the next collision, and the new velocities of the colliding disks after the collision are determined.

This model is very versatile. In Chapter 4 we use Program HardDisksMD to explore the pressure dependence of the density and the possibility of a phase transition. In Chapter 8 we explore the structure of the fluid, and in Chapter 10 we explore the self-diffusion constant, as well as the mean free time and mean free path.

**Problem: Qualitative behavior of $g(r)$**

Use Programs `HardDisksMD` and `LennardJonesMD` to simulate a system of hard disks
and particles interacting with the Lennard-Jones potential
to determine the behavior of $g(r)$ for various densities and temperatures.
We consider two-dimensional systems because they are easier to visualize.

- Consider a system of hard disks and describe how $g(r)$ changes with the density. The program uses units such that diameter $\sigma=1$. Is the temperature of the system relevant?
- Consider a system of particles interacting with the Lennard-Jones potential at the same densities (and number of particles) as you considered in part (a). Compare $g(r)$ for the two systems at the same density.
- Consider either interaction and describe how $g(r)$ changes as the density is increased. What is the meaning of the peaks in $g(r)$?
- How does $g(r)$ change with temperature for a given $\rho$?

## Resources

Problems 4.33, 8.16, 10.1, 10.2, 10.4, 10.8, 10.11, and 10.12 in*Statistical and Thermal Physics: With Computer Applications*, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).