Lennard-Jones Singularity: 1/r^12
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, causing forces to explode and requiring dt ~ r^13 for stability.
How NNC Helps
Classical (k=0)
F ~ 1/r^13 (diverges at r=0)- - Force grows as r^-13 during close approach
- - Timestep must shrink proportionally
- - Close encounters cause stalls or explosions
NNC (k=-1)
F_nnc = F * r ~ 1/r^12- - Regularization factor r^(-k) = r
- - Reduces divergence rate by one power
- - Lower peak forces in controlled comparisons
Note: Unlike 1/r or 1/sqrt(r) singularities, the 1/r^12 singularity is not fully canceled by k=-1. The regularization can lower peak forces by reducing the divergence rate, but hard close encounters can still stall and must be validated per setup.
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Benchmark Context
22.3%
Reported Closer Approach
from CASCADE 21-sim suite
k=-1
NNC Transform
regularization factor = r
OpenMM
Physics Domain
molecular dynamics
Real-World Applications
Drug Discovery
Simulating protein-ligand binding requires accurate modeling of close atomic contacts. NNC can reduce extreme forces, but longer timesteps still need energy-conservation checks.
Materials Science
Studying phase transitions and defect dynamics involves atoms passing through close encounters. NNC regularization can reduce blowup severity rather than remove the singularity.
Computational Chemistry
Reaction dynamics simulations require traversing transition states where atoms are in close proximity. NNC can make these regions easier to probe, but stability is system- and timestep-dependent.
Why Only 22.3%?
The 1/r^12 singularity is much steeper than 1/r or 1/sqrt(r). The NNC regularization with k=-1 provides factor of r, reducing 1/r^12 to 1/r^11 - still a strong singularity.
For complete cancellation, we would need k=-12, but this would over-regularize other parts of the physics. The k=-1 choice is a benchmarked partial regularizer, not a general proof of stable molecular dynamics.