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## Lennard-Jones MD

Program LennardJonesMD simulates the behavior of particles moving in a two-dimensional box interacting through the Lennard-Jones potential. The velocities and positions evolve according to 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 so that particles that leave the box enter on the other side, much like many video games.

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

Problem: Different phases
1. Program LennardJonesMD simulates an isolated system of $N$ particles interacting via the Lennard-Jones potential. Choose $N=144$ and $L=18$ so that the density $\rho = N/L^2 \approx 0.44$. The initial positions are chosen at random except that no two particles are allowed to be closer than the length $\sigma$. Run the simulation and satisfy yourself that this choice of density and resultant total energy corresponds to a gas. What are your criteria?
2. Slowly lower the total energy of the system. (The total energy is lowered by rescaling the velocities of the particles.) If you are patient, you will be able to observe “liquidlike” regions. How are they different from “gaslike” regions?
3. If you decrease the total energy further, you will observe the system in a state roughly corresponding to a solid. What are your criteria for a solid? Explain why the solid that we obtain in this way will not be a perfect crystalline solid.
4. Describe the motion of individual particles in the gas, liquid, and solid phases.
5. Speculate why a system of particles interacting via the Lennard-Jones potential can exist in different phases. Is it necessary for the potential to have an attractive part for the system to have a liquid phase? Is the attractive part necessary for there to be a solid phase? Describe a simulation that would help you answer this question.

## 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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