Interactive demo
A molecular-dynamics simulation that runs in this page. Atoms interact through a real Lennard-Jones potential, the force field they generate is drawn as you watch, and temperature is a dial you turn.
Every pair of atoms feels one curve: hard repulsion when their electron clouds overlap, a shallow attractive well beyond it. That single shape, summed over neighbours, is enough to produce solids, liquids, gases, and the transitions between them.
The background wash and the arrows are not decoration. At every grid point the simulation drops in a test atom and measures what it would feel — the potential U and the force −∇U. That vector field is exactly what a machine-learned interatomic potential is trained to reproduce.
Temperature here is not a slider on an animation speed — it is the width of the velocity distribution, maintained by a Langevin thermostat. Raise it and the lattice melts; the speed histogram tracks the Maxwell–Boltzmann curve while it happens.
Pick a starting state, then push it around. Drag an atom and feel the neighbours resist; switch the pointer to the heat brush and melt a hole in a crystal. Everything on this page is measured from the same trajectory, live — nothing is pre-recorded.
Hover an atom for its speed, energy and net force; click to pin it, double-click to release. Keyboard: space run/pause · ↑ ↓ temperature · S single step · R rebuild.
Density, atom count and boundary rebuild the box; everything else takes effect on the running system.
Instantaneous T* from the kinetic energy. It fluctuates around the target by roughly 1/√N — that noise is physical, not numerical.
Kinetic, potential, and their sum, all in ε — one quantity, one axis. With the thermostat off, the total line should be flat.
Measured speeds of the majority species against the two-dimensional Maxwell–Boltzmann law p(v) ∝ v·exp(−mv²/2kBT) at the current temperature. Nothing fits it — they simply agree.
How much more likely you are to find an atom at distance r than in an ideal gas. Sharp repeating peaks mean a crystal; one broad peak decaying to 1 means a liquid; flat at 1 means a gas.
How far atoms wander from where they started. A straight line means diffusion, and its slope is 4D in two dimensions; a line that flattens means the atoms are trapped in a lattice.
U(r) and F(r) for the chosen pair, drawn as stacked panels on a shared r axis — two different units never share one y scale. The orange dot follows the nearest-neighbour distance of whichever atom you are hovering.
Method
This is a real molecular-dynamics code, only small and two-dimensional. It is worth being precise about what it computes and what it leaves out.
Lennard-Jones, U(r) = 4ε[(σ/r)12 − (σ/r)6], truncated at 2.5 σij and shifted so the energy is continuous there. Unlike pairs mix by Lorentz–Berthelot, σij = (σi+σj)/2 and εij = √(εiεj), with the cross term scalable so you can drive demixing. Molecules add harmonic bonds; the ionic preset adds damped shifted-force electrostatics rather than an Ewald sum, which is the standard cheap choice for a cutoff code.
BAOAB Langevin dynamics with one force evaluation per step; at zero friction it collapses to plain velocity Verlet, so "no thermostat" is genuinely constant-energy. Verified head-less: total energy drifts by 0.02 % of the kinetic energy over 8 000 NVE steps and the drift falls as Δt², the Langevin thermostat reproduces its target temperature to better than 1 %, and heavy and light species come out at the same temperature, as equipartition demands.
Reduced Lennard-Jones units — lengths in σ, energies in ε, masses in m, and kB = 1. The physical read-outs map those onto argon (ε/kB = 119.8 K, σ = 3.4 Å, m = 39.95 u), which fixes the time unit at τ = σ√(m/ε) = 2.15 ps and the force unit at ε/σ = 4.87 pN. Change the substance and only the labels change; the trajectory is identical.
Two dimensions, not three: the phase diagram is qualitatively right but the numbers are not argon's, and true long-range order is forbidden in 2D. There are no electrons, so no bond breaking, no polarisation, no chemistry. A few hundred atoms is far below what a finite-size study would need. It is a demonstration of mechanism, not a production calculation.
A machine-learned interatomic potential has exactly one job: given the positions of the atoms, return the energy and the forces — the two fields this page draws. Everything downstream, every melting point and diffusion coefficient and phase boundary, comes from integrating those forces. Watching a hand-written potential produce a melting transition makes it concrete what a learned potential has to get right, and why force errors of a few meV/Å are the currency of the field. The companion force-field family tree maps the methods that learn them.