# Algorithms for Long Range Interactions

** Long Range interactions page is under construction**

## Contents

## Long Range Interactions & the root of the problem

Formally 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. Due to the type of decay with the distance of the interaction, as we go further an furhter the particle-particle interaction decreases but the number of interactions increases in such way that the contribution of the far particles to the total interaction of a particle can have a weight as large as the one due to the interaction of a particle with neighbour particles.

The limited power of current computers makes impossible simulate macroscopic bulky systems. We should always work with very small systems where the ratio area vs volume is large and therefore surface effects modify the behaviour respect to bulky systems. Furtheremore, due to the very small sizes accesible to us, the long-range component of the electrostatic interactions cannot be addresed in an exact manner.

Even if Moore´s law was able to hold on indefinitely, we would need still 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.

## How to mimic bulky systems with long range interactions

The straight cut-off or subsequent shift of the long-range interactions in these systems has been observed to lead to many unphysical artifacts in the simulations. Better approaches currently available are:

- Reaction Field Methods.
- LatticeSum Methods: Direct Sumation, Ewald, P3M, FMMM & Co
- Hybrids of 2 and 3: LSREF (Heinz2005)
- MEMD – Maxwell Equations Molecular Dynamics (*2)

http://en.wikipedia.org/wiki/Intermolecular_force

## How to mimic the effects of Long Range interactions in Simulations

Currently, numerical simulations are playing a main role in explaining and unravelling the rich variety of new, and unexpected behaviours found in recent theoretical and experimental studies about these electrostartic systems. ... http://cmt.dur.ac.uk/sjc/thesis_dlc/node61.html

## Use of Periodic Boundary Conditions in Simulations

Frequently, periodic boundary conditions are used in these 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