## Lennard-Jones MD

Program LennardJonesMD simulates behavior of particles moving in a two-dimensional box interacting through a Lennard-Jones potential. The program implements the numerical solution of the differential equations in Newton's second law. The velocity Verlet algorithm is used to update the positions and velocities of the particles. Periodic boundary conditions are used.

One goal of this simulation is to explore how the same inter-particle potential can lead to different macroscopic behavior such as gas, liquid, or solid phases. Other information that can be obtained includes the velocity distribtion function, the pair distribution function $g(r)$, which provides information about the spatial correlations structure of the system, and the diffusion coefficient of a tagged particle.

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.

1. 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?
2. 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.
3. Consider either interaction and describe how $g(r)$ changes as the density is increased. What is the meaning of the peaks in $g(r)$?
4. How does $g(r)$ change with temperature for a given $\rho$?

## Resources

Problems 1.9, 6.14, 8.16, and 10.2 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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