Structure of the Simulation

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.