## Hard Disks MD

One of the simplest models of particle interactions is the hard core interaction, for which the interaction is infinite when 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.

A system of hard disks has many applications. 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: Simulation of hard disks**

The dynamics of a system of hard disks are straightforward in
principle because the particles move in straight lines in between
collisions. Program
`HardDisksMD` finds when the next
two particles are going to collide and moves the particles accordingly.

- Run the simulation with the default parameters and record the density and the results for $PA/NkT$ and the temperature. Is the temperature a useful quantity for hard disks? Does it fluctuate during the simulation. If not why not? Does the pressure $P$ fluctuate?
- Compute $PA/NkT$ as a function of density for a fixed number of particles. Is there any evidence of a phase transition where the slope of the pressure with respect to density changes abruptly? An abrupt change in the slope might indicate the existence of a phase transition.

## 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).