World 1 · States of Matter · 9 minInside view: molecular dynamics
Lennard-Jones atoms and g(r)
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The box runs molecular dynamics: Newton’s laws for every atom, in small time steps. The force comes from the Lennard-Jones potential
U(r) = 4ε [ (σ/r)¹² − (σ/r)⁶ ]
The r⁻¹² term is a steep wall (atoms cannot overlap); the r⁻⁶ term is the weak van der Waals attraction. The minimum sits at r = 2^(1/6) σ ≈ 1.12 σ, with depth ε. With ε/k_B ≈ 120 K and σ ≈ 0.34 nm this describes argon well.
Each step uses velocity Verlet: half-kick the velocities with the force, move the atoms, recompute the forces, half-kick again. It conserves energy very well over long runs, which is why almost every MD code uses it. A thermostat gently rescales velocities to hold the target temperature.
The radial distribution function g(r) answers: standing on one atom, how much more likely am I to find another at distance r than in a uniform gas? A crystal gives sharp peaks at its lattice distances; a liquid gives one strong peak, a few fading ripples, then 1; a gas is close to 1 everywhere beyond the wall.
Where is the first peak of g(r) for these atoms, and why?
Why does velocity Verlet need a small time step?