Difference between revisions of "Algorithms for Long Range Interactions"

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== Long Range Interactions & the root of the problem ==
 
== Long Range Interactions & the root of the problem ==
  
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.
+
<onlyinclude>A potential is defined to be short ranged if it decreases with distance <math> r </math> quicker or similar than   <math>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.</onlyinclude> 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.
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
+
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.
+
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 ==
  
The straight cut-off or subsequent shift of the long-range interactions in
+
The straight cut-off (sometimes including a shift) of the long-range interactions have been   
these systems have been  observed to lead to many unphysical artifacts  
+
observed to lead to many unphysical artifacts in the simulations of bulky systems. Although 
in the simulations. Better approaches currently available are:
+
no perfect solution has been found, there exist some approaches to tackle with the problem:
 
* Reaction Field Methods.
 
* Reaction Field Methods.
 
* Periodic Boundary Conditions (artificial periodicity): Lattice-Sum Methods
 
* Periodic Boundary Conditions (artificial periodicity): Lattice-Sum Methods
* Hybrids of 2 and 3, eg. LSREF (Heinz2005).  
+
* Hybrids of the previous two approaches, eg. LSREF (Heinz2005).  
* MEMD – Maxwell Equations Molecular Dynamics (*2)
+
* MEMD – Maxwell Equations Molecular Dynamics (see ref.2)
  
==  Periodic Boundary Conditions  ==
+
==  Our Research: Periodic Boundary Conditions  ==
  
 
Frequently, periodic boundary conditions
 
Frequently, periodic boundary conditions
are used in simulations in order to approach bulk systems within the limits of currently available computers.
+
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'''
 
''' Long Range interactions page is under construction'''
  
 
== Links ==
 
== Links ==
* {{Download|jcerda_web_t1.pdf| Internal Talk for the group in 2006 (by .}}
+
* {{Download|jcerda_web_t1.pdf| Internal Talk for the group in 2006 (by J.J.Cerdà.}}
  
 
== Scientists ==
 
== Scientists ==
Line 45: Line 51:
  
 
== Publications ==
 
== Publications ==
<bibentry> deserno99b, wang01a, dejoannis02a, arnold02c, arnold02d, arnold02b, arnold05b, arnold05a, ballenegger07a, tyagi07a, cerda08d</bibentry>
+
<bibentry> deserno99b, wang01a, dejoannis02a, arnold02c, arnold02d, arnold02b, arnold05b, arnold05a, ballenegger07a, tyagi07a, ballenegger08a,cerda08d, ballenegger09a</bibentry>
  
 +
== Useful references ==
  
== Useful references ==
+
[Heinz2005] Heinz et al , JCP 123, 034107, (2005)
  
[Heinz2005] Heinz et al , JCP 123, 034107, (2005).
+
[2]        RottlerMaggs and DunwegPasichnyk,2004
[*2]        RottlerMaggs and DunwegPasichnyk,2004
 
  
  
  
 
''' Long Range interactions page is under construction'''
 
''' Long Range interactions page is under construction'''
 +
[[Category:Research]]

Latest revision as of 10:06, 11 June 2012

Long Range interactions page is under construction

Long Range Interactions & the root of the problem

A potential is defined to be short ranged if it decreases with distance  r quicker or similar than 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 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): Lattice-Sum 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

Publications


Useful references

[Heinz2005] Heinz et al , JCP 123, 034107, (2005)

[2] RottlerMaggs and DunwegPasichnyk,2004


Long Range interactions page is under construction