Algorithms for Long Range Interactions
Long Range interactions page is under construction
Contents
Long Range Interactions & the root of the problem
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 particleparticle 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 longrange forces are present, the scenario to mimic bulky systems is even worse because we will neglect a substantial part of the longrange 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 cutoff (sometimes including a shift) of the longrange 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): LatticeSum Methods
 Hybrids of the previous two approaches, eg. LSREF (Heinz2005).
 MEMD – Maxwell Equations Molecular Dynamics (see ref.2)
Our Research: Periodic Boundary Conditions
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.
Long Range interactions page is under construction
Links
Scientists
Collaborators
 Dr. Vincent Ballenegger, CNRS, Institut UTINAM, Besancon, France
Publications

Wang, Zuowei and Holm, Christian.
"Estimate of the Cutoff Errors in the Ewald Summation for Dipolar Systems".
The Journal of Chemical Physics 115(6277–6798), 2001.
[PDF] (2 MB) [DOI] 
Jason de Joannis and Axel Arnold and Christian Holm.
"Electrostatics in Periodic Slab Geometries II".
Journal of Chemical Physics 117(2503–2512), 2002.
[PDF] (318 KB) [DOI] 
Arnold, Axel and de Joannis, Jason and Holm, Christian.
"Electrostatics in Periodic Slab Geometries I".
The Journal of Chemical Physics 117(2496–2502), 2002.
[PDF] (217 KB) [Preprint] [DOI] 
Arnold, Axel and de Joannis, Jason and Holm, Christian.
"Electrostatics in Periodic Slab Geometries II".
The Journal of Chemical Physics 117(2503–2512), 2002.
[PDF] (267 KB) [Preprint] [DOI] 
Axel Arnold and Christian Holm.
"A novel method for calculating electrostatic interactions in 2D periodic slab geometries".
Chemical Physics Letters 354(324–330), 2002.
[PDF] (425 KB) [DOI] 
Axel Arnold and Christian Holm.
"MMM1D: A method for calculating electrostatic interactions in 1D periodic geometries".
Journal of Chemical Physics 123(12)(144103), 2005.
[PDF] (122 KB) 
Axel Arnold and Christian Holm.
"Efficient methods to compute long range interactions for soft matter systems".
In Advanced Computer Simulation Approaches for Soft Matter Sciences II, volume II of Advances in Polymer Sciences, pages 59–109. Editors: C. Holm and K. Kremer,
Springer, Berlin, 2005.
[PDF] (2 MB) [DOI] 
Sandeep Tyagi and Axel Arnold and Christian Holm.
"ICMMM2D: An accurate method to include planar dielectric interfaces via image charge summation".
Journal of Chemical Physics 127(154723), 2007.
[PDF] (305 KB) [DOI] 
Vincent Ballenegger and Joan J. Cerdà and Olaf Lenz and Christian Holm.
"The optimal P3M algorithm for computing electrostatic energies in periodic systems".
Journal of Chemical Physics 128(3)(034109), 2008.
[PDF] (426 KB) [Preprint] [DOI] 
Juan J. Cerdà and Vincent Ballenegger and Olaf Lenz and Christian Holm.
"P3M algorithm for dipolar interactions.".
Journal of Chemical Physics 129(234104), 2008.
[PDF] (516 KB) [Preprint] [DOI] 
Vincent Ballenegger and Axel Arnold and Joan J. Cerdà.
"Simulations of nonneutral slab systems with longrange electrostatic interactions in twodimensional periodic boundary conditions".
Journal of Chemical Physics 131(9)(094107), 2009.
[PDF] (204 KB) [DOI]
Useful references
[Heinz2005] Heinz et al , JCP 123, 034107, (2005)
[2] RottlerMaggs and DunwegPasichnyk,2004
Long Range interactions page is under construction