Lennard-Jones Molecular Dynamics

300 disks in a periodic box interacting through the Lennard-Jones potential $U(r)=4\varepsilon[(\sigma/r)^{12}-(\sigma/r)^{6}]$ (reduced units $\sigma=\varepsilon=m=k_B=1$ ), integrated by velocity-Verlet (the verified shared symplectic engine) with a shifted-force cutoff so energy is conserved. Particles are coloured by kinetic energy. The temperature is read from the kinetic energy by equipartition, the pressure from the virial, and the structure from the radial distribution $g(r)$ : a flat $g(r)\to1$ for a gas, a strong first peak near $r=2^{1/6}\sigma$ for a dense liquid. Set it cold and dense and it freezes into a triangular lattice; heat it and it melts.

Figure 1. Lennard-Jones molecular dynamics: particles coloured by kinetic energy (left), beside the live radial distribution function g(r) (right), the ratio of the local neighbour density at separation r to the bulk density, whose peaks are the coordination shells visible in the box and whose r to 1 tail is the structureless ideal gas. Method: shifted-force LJ, velocity-Verlet (shared symplectic engine), virial pressure, g(r) from pair histograms.

temperature T1.00

density rho0.55

steps / frame4

WHAT TO TRY

Raise the temperature: the disks rattle harder, the pressure climbs, and the radial distribution g(r) washes out as liquid order melts toward a gas.
Increase the density: neighbours pack in, g(r) grows sharp shells, and the virial pressure swings as the repulsive core of the Lennard-Jones potential takes over.
Read g(r) below the critical point: a tall first peak and decaying oscillations are the signature of a liquid, structured up close but disordered far away.