Lennard-Jones Singularity: 1/r^12
Atoms collide in a periodic box. Every close encounter flashes as the 1/r^13 force spikes, and the solver timestep shrinks to survive it.
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The Lennard-Jones potential describes interatomic forces with a steep 1/r^12 repulsive term. When atoms approach closely, this creates a numerical singularity that forces classical solvers to use extremely small timesteps.
The Physics
Lennard-Jones 12-6 Potential
The LJ potential models van der Waals forces between neutral atoms:
V(r) = 4*eps * [(sig/r)^12 - (sig/r)^6]The r^-12 term (Pauli repulsion) diverges extremely fast as atoms approach. The r^-6 term (dispersion attraction) is weaker.
Force Divergence
The force is the negative gradient of the potential:
F(r) = 4*eps * [12*sig^12/r^13 - 6*sig^6/r^7]The 1/r^13 term dominates at close range. The pair stiffness grows like 1/r^14, so a stable timestep has to shrink roughly like r^7 during a hard collision.
Force vs Weighted View
Force (k=0 readout)
|F| ~ 1/r^13 (diverges at r=0)- - This is the force both panels integrate
- - The solver dt shrinks during close approach
- - dt targets a 1e-5 per-step energy error (a target, not a bound: overshooting steps are kept)
Weighted view (k=-1 readout)
r * |F| ~ 1/r^12- - Weight r^(-k) = r applied to the reported number only
- - Still singular: one power weaker, not bounded
- - Same atoms, same trajectories, same dt
Note: an earlier version of this demo multiplied the force itself by r. That is a different force law (no longer the gradient of V), and it broke energy conservation by about 20%. It was a model change, not a change of calculus, so the browser demo now keeps one force law for both panels.
Benchmark Context
The openmm_lennard_jones row of the local CASCADE suite is a synthetic toy: one 2-D test function with a singular point, relabelled per domain, scored by how close the optimizer got to that point. It is a local benchmark, not citable, and it is not reproduced by this browser demo.
Why the Weighting Cannot Fix This
For F ~ 1/r^13 the weight r^(-k) gives r^(-k-13). Any k <= -13 makes the near-origin readout bounded (k = -13 gives a constant leading term; k < -13 sends it to zero), and even then it is a readout: the atoms still feel the full 1/r^13 repulsion, and the timestep is still set by it.
On r > 0 the log-slope of r^-13 is the constant -13 (D_BG[r^-13] = e^-13). That bounded diagnostic describes the singularity; it does not remove it.
Math checked against local simulation utilities. View more demos | All tools