Another method of setting dt uses the Einstein period to scale the time step to a value which is suitable for the simulation. The Einstein period is with the Einstein frequency, given by
where and are the average force and velocity squared acting on each ion with mass m.
Allen and Tildesley suggest that the time step, dt, should be less than the
Einstein period by at least an order of magnitude for simulations of liquids
. A scaling factor of was generally successful
for simulations of Paul traps. However, many-ion Penning trap simulations
required a smaller scaling factor: TrapApp uses
for Penning trap simulations.
As with micromotion scaling, Einstein scaling is intended as
a guide for finding the optimal time step--- it is not a fool-proof method
for achieving stability (for example, if low-frequency terms dominate, the
Einstein-scaled time step may be too large), nor does it yield the largest
possible ``accurate'' time step. Don't be intimidated by the
time step chosen by TrapApp; feel free to experiment and search for
a more optimal time step.