Development of the Ion Trapping Simulation

The skeleton of the program, as shown in Figure 2, 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 assigment 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 rendered this simulation 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 [1][9]. 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 is the dimension of the space occupied by the system [1]. Loosely speaking, long-range forces are those for which every particle interacts with every other particle (as opposed to only with nearest neighbors or only with particles within a critical radius, ). Notorious for its computational intensity, the long-range interaction problem is generally on a sequential machine [8]. 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 deals with the time scale. As mentioned with regard to Penning trap simulations, ion trapping involves terms which fluctuate on a very small time scale. A small time scale is characteristic of MD simulations, which typically model phenomenon which fluctuate on the order of - ps [9]. However, the ion-trapping simulation also involves behaviors which fluctuate on a much larger time scale. In order to see what is happening on the larger time scale without loosing accuracy with respect to the short time scale behavior, it is necessary to use a large number of time steps with small , where is the increment between steps.