# Algorithms for Long Range Interactions

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## Long Range Interactions & the root of the problem

Formally a potential is defined to be short ranged if it decreases with distance $r$ quicker or similar than $\frac{1}{r^{d-1}}$ where $d$ 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.

## How to mimic bulky systems with long range interactions

The straight cut-off (sometimes including a shift) of the long-range interactions in these systems have been observed to lead to many unphysical artifacts in the simulations. Better approaches currently available are:

• Reaction Field Methods.
• Periodic Boundary Conditions (artificial periodicity): Lattice-Sum Methods
• Hybrids of 2 and 3, eg. LSREF (Heinz2005).
• MEMD – Maxwell Equations Molecular Dynamics (*2)

## Periodic Boundary Conditions

Frequently, periodic boundary conditions are used in simulations in order to approach bulk systems within the limits of currently available computers.

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## Useful references

[Heinz2005] Heinz et al , JCP 123, 034107, (2005). [*2] RottlerMaggs and DunwegPasichnyk,2004

Long Range interactions page is under construction