Difference between revisions of "Algorithms for Long Range Interactions"

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(Long Range Interactions & the root of the problem)
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Formally a potential is defined to be short ranged if it decreases with distance <math> r </math> quicker or similar than <math> \frac{1}{r^{d-1}} </math> where <math> d </math> is the dimensionality of the system.  Electrostatic, gravitatory and dipolar interactions, present in many physical systems, are examples of long range interactions.   
 
Formally a potential is defined to be short ranged if it decreases with distance <math> r </math> quicker or similar than <math> \frac{1}{r^{d-1}} </math> where <math> d </math> 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.
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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
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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.  
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.
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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 ==
 
== How to mimic bulky systems with long range interactions ==

Revision as of 01:15, 15 January 2008

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. 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 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.

...

Long Range interactions page is under construction

Links

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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