Demo 1: LJ dimer — tracing a force-extension fold

Two Lennard-Jones atoms are pulled apart by fixing one atom and applying a force to the other. The force-extension curve has a fold point bifurcation at the inflection point of the LJ potential (||(r \approx 1.245\,\sigma||)). Beyond this point, the bond snaps and so any configuration where the two atoms are sufficiently apart is a degenerate equilibrium. Standard quasi-static loading can capture the snapping, but is unable to trace the unstable equilibrium. Arclength continuation parameterises the solution path by its arc length rather than by the applied force, letting it smoothly traverse the fold and trace the full S-curve onto the unstable segment. Given the simplicity of the system, we actually have an analytical solution tracing the stable and unstable equilibria. In this notebook, we will use LACT to trace it numerically using a LAMMPS instance and LACT continuation wrapper.