🔵 Lennard-Jones Molecular Dynamics

The attractive r⁻⁶ term models London dispersion forces (van der Waals attraction): temporary fluctuating dipoles in one atom induce a dipole in a neighbouring atom, creating a weak but universal attractive force. The repulsive r⁻¹² term is a numerical approximation to the much stronger Pauli exclusion repulsion that prevents electron clouds from overlapping; the r⁻¹² form is computationally convenient but physically less rigorous than exponential repulsion.

The equilibrium separation (where force is zero and potential is minimum) is r_min = 2^(1/6)×sigma ≈ 1.122 sigma. At this distance, the energy is -epsilon (the well depth). For argon, sigma ≈ 3.4 Angstroms and epsilon/k_B ≈ 120 K, so the equilibrium separation is about 3.82 Angstroms and the binding energy per pair corresponds to about 120 K (very weakly bound, consistent with argon's low boiling point of 87 K).

Molecular dynamics (MD) integrates Newton's equations of motion numerically: given positions and velocities, forces are computed, and positions are updated step by step. It follows the true dynamical trajectory and gives time-dependent information (diffusion, velocity autocorrelations, spectra). Monte Carlo (MC) generates equilibrium configurations by proposing and accepting/rejecting random moves based on the Boltzmann factor; it samples equilibrium properties efficiently but contains no physical dynamics or time information.

Depending on temperature (relative to epsilon/k_B) and density, a Lennard-Jones system exhibits gas (low density, particles freely moving), liquid (intermediate density, short-range order, diffusive motion), and solid (close-packed FCC crystal, long-range order). The gas-liquid critical point, the triple point (where solid, liquid, and gas coexist), and melting/solidification can all be observed by varying temperature and pressure in the simulation.

The pair correlation function g(r) measures how particle density varies as a function of distance from a reference particle, relative to the average density. In a gas, g(r) ≈ 1 everywhere (no structure). In a liquid, g(r) shows peaks at preferred separations (first, second coordination shells) that decay to 1 at large r. In a crystal, sharp peaks persist at large r reflecting the regular lattice. The positions and heights of peaks in g(r) reveal bond lengths, coordination numbers, and structural order in the material.