Long Range Interactions

A potential is defined to be short ranged if it decreases with distance quicker or similar than where is the dimensionality of the system. Electrostatic, gravitatory and dipolar interactions, present in many physical systems, are examples of long range interactions. When long range intgeractions are present in a system, the weight of the interactions comming from far particles is non negligible. This is due to the type of decay of the interaction with the distance: despite the particle-particle interaction decreases with the distance, the number of interactions increases in such way that the total contribution of the far particles may have a weight as large as the one due to the interaction of neighbouring particles.

The limited power of current computers makes impossible simulate macroscopic bulky systems. Small systems have a large surface vs volume ratio and therefore surface effects may govern the physics of the system. When long-range forces are present, the scenario to mimic bulky systems is even worse because we will neglect a substantial part of the long-range interaction.

Then, why we don't wait a little bit until computers become more powerful? Even if Moore´s law was able to hold on indefinitely, we would still need around two centuries to be able to tackle with systems of the size of about one cubic centimeter. Therefore, it is clear that we need to do some sort of approach in order to mimic bulky systems right now.

The straight cut-off (sometimes including a shift) of the long-range interactions have been observed to lead to many unphysical artifacts in the simulations of bulky systems. Although no perfect solution has been found, there exist some approaches to tackle with the problem:

• Reaction Field Methods.
• Periodic Boundary Conditions (artificial periodicity): Lattice-Sum Methods
• Hybrids of the previous two approaches, eg. LSREF (Heinz2005).
• MEMD – Maxwell Equations Molecular Dynamics (see ref.2)

Frequently, periodic boundary conditions are the chosen approach. When periodic boundary conditions are used, an artificial periodicity is introduced in order to emulate the bulky system. The cell system is replicated and the interactions between the particles in the main cell and the particles located in the replica cells is taken into account and added to the interactions between particles of the main cell. For this reason, this kind of methods are also known as Lattice Sum Methods. When one performs this kind of sums by brute force, the method is known as Direct Sum.

Despite it seems very easy to perform a Direct Sum, it is in fact very tricky because this kind of sums have a conditional and very slow convergence, which implies that many terms must be included to obtain a reasonable accuracy for the value of the interactions.