## Lennard-Jones MD

Program LennardJonesMD simulates the behavior of particles in a two-dimensional box interacting through a Lennard-Jones potential. The motion evolves using 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.

The output includes the velocity distribtion function, the pair distribution function $g(r)$, and the self-diffusion of a tagged particle.

Problem: The mean square displacement of particles in a fluid
1. Why is the mean displacement, $(1/N)\sum_{i=1}^N \langle {\bf r}_i(t)\rangle$, of little interest?
2. 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.
3. 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?
4. How does $r^2(t)$ for the two systems compare at the same density?

## 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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