Push atoms with forces, step them in time, and watch matter behave. Pairwise Lennard-Jones forces integrated by velocity Verlet — a symplectic map that keeps total energy bounded, so a box of classical particles obeys statistical mechanics you can simulate. Rendered, not quoted.
SOURCE Alder & Wainwright, “Studies in Molecular Dynamics. I. General Method,” J. Chem. Phys. 31(2), 459–466 (1959) — doi:10.1063/1.1730376
Each atom is a classical point mass. Every pair feels the Lennard-Jones potential
V(r) = 4ε[(σ/r)12 − (σ/r)6]
repulsive core, attractive tail, minimum −ε at exactly r = 21/6σ. The force on i is the sum of pair forces along each separation. Time advances by velocity Verlet: half-kick the velocity, drift the position, recompute forces, half-kick again. In the NVE ensemble — no thermostat — the map conserves total energy and (from equal-and-opposite pair forces) total momentum.
Atoms stepped by forces. Alder & Wainwright (1959) ran the first molecular-dynamics simulation on a computer. The forces here are the-lennard-jones pair potential; the stepper is the-symplectic-integrator (velocity Verlet). Together they turn a force law into statistical mechanics you can simulate — temperature, pressure, phase behaviour emerging from Newton alone.
Re-runs selfcheck() against the live engine: energy bounded under Verlet, momentum conserved, the map time-reversible, temperature = equipartition. It also arms the planted void in window 6 and confirms the tamper is caught.
Six atoms near an FCC-corner spacing (r ≈ 21/6σ), seeded random nudges and velocities, net momentum removed. Reduced units: ε=σ=m=kB=1, dt=0.004.
Live velocity-Verlet MD. Positions projected x–y; total energy tracked every step.
Over the fixed selfcheck run: Verlet keeps ΔE/E bounded while Euler drifts — the proven separation.
“Just step it however — a smaller dt fixes any integrator.” False. Non-symplectic steppers (forward Euler, plain RK4) have no bounded energy invariant; error accumulates secularly and the NVE ensemble is destroyed at every dt. Shrinking dt only delays the blow-up; it does not conserve energy. The wall is that structure, not step size, buys conservation.
Total energy is conserved because forces are computed exactly.
→ Forces being exact does not conserve energy — the integrator does. Verlet conserves a shadow Hamiltonian; Euler does not.
Temperature is an input you set.
→ In NVE, T is read out from motion: T = (2/DIM)·KE/(N·kB) by equipartition.
The LJ minimum sits at r = σ.
→ V(σ)=0. The minimum is at r = 21/6σ, depth −ε.
Disclosed sabotage: swap velocity Verlet for the-forward-euler stepper. Energy then drifts secularly and NVE is no longer conserved — the energy-bound check in window 7 fails and flips red.