## Hard Disks MD

Program `HardDisksMD`
keeps track of the location and velocity of each hard disk and determines when the next collision occurs. Then 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.

In Chapter 10 we explore the self-diffusion constant, as well as the mean free time $\tau$ and mean free path $\lambda$.

**Problem: The motion of particles in a fluid**ß

Program `HardDisksMD` simulates a system of hard disks
and tags one of the particles blue so that it can be distinguished from the other particles.
The default density is $\rho=0.16$, a density that is far from the density at the fluid-solid transition.
How would you describe the motion of the tagged particle? Does its motion differ from that of the other particles?
Does its motion look random? Are the collisions of the particles well defined?

**Problem: The mean square displacement of particles in a fluid**

- Why is the mean displacement, $(1/N)\sum_{i=1}^N \langle {\bf r}_i(t)\rangle$, of little interest?
- Program
`HardDisksMD`computes the mean square displacement $r^2(t)$ of a system of hard disks. The displacements in the $x$ and $y$ directions are computed separately to give you an idea of the statistical error. Describe the behavior of $r^2(t)$ for very short times and for long times. - Program
`LennardJonesMD`simulates a system of particles interacting with the Lennard-Jones potential. Is the motion of the tagged particle similar to what you observed for hard disks? - How does $r^2(t)$ for the two systems compare at the same density?

**Problem: The self-diffusion coefficient of hard disks**

Program `HardDisksMD` computes the mean square displacement by averaging over all particles and over time
origins chosen at $10\,\tau$ apart. The time is measured in units of $\tau = (m \sigma^2/kT)^{1/2}$,
where $\sigma$ is the hard disk diameter, $m$ is the mass of a disk, and $T$ is the
temperature.

- What is the time-dependence of the mean square displacement $r^2(t)$ for long times?
- Do you expect $D$ to increase, decrease, or remain unchanged as the density is increased at fixed temperature?
- Do you expect $D$ to increase, decrease, or remain unchanged as the temperature is increased at fixed density? Explain your reasoning for both cases.
- Choose several densities in the range $0.10 \leq \rho \leq 0.30$ and determine the qualitative dependence of $D$ on $\rho$.
- What is the approximate time dependence of $r^2(t)$ for short times?

**Problem: Distribution of times between successive collisions**

We can gain insight into the dynamics of a dilute gas by looking at the statistics of the collisions in a system of hard disks.

- Let $P(t)$ be the distribution of times $t$ between successive collisions of a particular particle with any other particle in the system. What do you expect the form of $P(t)$ to be for low densities?
- Program
`HardDisksMD`computes the histogram of collision times for a system of hard disks at density $\rho=0.16$ (and $T=1$). The results are averaged over all particles to obtain better results. The histogram is an estimate of $P(t)$ if the data is normalized. Does the qualitative behavior of the histogram agree with your expectations? - Fit your numerical data for the histogram to your choice of functional form and use your fit to determine the mean time between collisions.

**Problem: Tests of simple kinetic theory arguments**

Compare the results from Program `HardDisksMD` for the mean collision time and mean free path
to the predictions $\tau = 1/\nu = 1/\rho \sigma_c \overline{v}$
and $\lambda \approx 1/\sqrt 2 \rho \sigma_c$.

**Problem: Self-diffusion coefficient for hard disks**

- Use arguments similar to the ones used in the text to determine the temperature and density dependence of $D$ for a dilute system of hard disks. You will need to calculate $\overline{v}$ using the two-dimensional version of the Maxwell velocity distribution function.
- Compare your results from Program
`HardDisksMD`for the self-diffusion coefficient to the predictions from part (a) and the Boltzmann equation given by $D_B= \dfrac{1}{2 \rho \sigma } \Big (\dfrac{kT}{\pi m} \Big)^{\!1/2}$.

## 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).