A molecular dynamics simulation involves four main components:

- initialization--- define relevant parameters, assign initial positions and velocities
- equilibration--- allow the system to evolve to the point that it ``forgets'' its initial values, which may have been unrealistic (e.g., random assignment of initial positions might put two ions unreasonably close together)
- evolution--- use a finite difference algorithm to model the time evolution of the system; monitor values of interest (energy, ion's position, etc.)
- analysis--- represent data graphically; perform further numerical analysis of the results

While this basic structure is common to any molecular dynamics simulation,
several factors render ion-trapping simulations different--- and, in general, more
computationally intensive--- from those discussed in Haile's and Allen
and Tildesley's books, which
were taken as primary references.[19,1] First of all,
the Coulomb interaction falls into the category of long-range interactions;
formally, this means that the force term is proportional to
where , and **d** is the dimension
of the space occupied by
the system.[1]
Loosely speaking, long-range forces are those for which the interaction of
every particle with every other particle (as opposed to
only interactions with nearest neighbors or with particles within
a critical radius, ) must be taken into account. Notorious for its
computational intensity, the long-range interaction problem is
generally on a sequential machine.[14] Furthermore, because Paul and
Penning traps involve an external potential which is a function
of space, ``tricks" suggested by Allen and Tildesley for dealing
with the long range potential by using periodic boundary conditions
are not valid.

Another difference between this simulation and general molecular dynamics
(MD) simulations arises from time scale. A small time scale is characteristic of MD
simulations, which typically model phenomenon which fluctuate on the
order of **100**-**1000** ps.[19] However, as mentioned with regard to
Penning trap simulations, ion trapping involves terms which fluctuate
on a range of significantly different time scales.
For example, the position of a single Paul-trapped ion in stability region A consists of a fast micromotion superposed on a slower secular motion. Similarly, the motion of a single ion in a typical Penning trap is described by a very fast cyclotron motion, a slow magnetron motion, and intermediate axial oscillations. In order to see what is happening on the larger time scales
without loosing accuracy with respect to the short time scale behavior,
it is necessary to use a large number of time steps with small
**dt**, where **dt** is the increment between steps. Hence, ion-trapping
simulations are computationally intensive in that they require a large volume
of calculation.

Fri May 12 10:36:01 EDT 1995