This sounds like a mathematical or statistics issue that is more fundamental
than MD. I.e. how a single particle moves in a fixed potential. I don’t think
you need LAMMPS to figure it out. I suggest you write a Python or Matlab
script with your equations and do the analysis and experimentation with
the system there. Once you have figured out its properties, you can code
it in LAMMPS if necessary. But I imagine it would be slower than the Python
script, because you computing your forces via LAMMPS variables.