# Algorithms for Long Range Interactions

Long Range interactions page is under construction

## 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. 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 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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Long Range interactions page is under construction

## Useful references

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