Combining Variables in Optimization

In Newton's method one needs to add the coefficients of an equivalence class to compute the potential energy. So one can do the following.

The multiples in the potential function also affect the results for the optimization parameters in Newton's method in nD. In the case of a triangular configuration the parameters can be varied independently by the formulas are the same. This is why all the lengths are equal to one up to a configuration on 4 "atoms" or particles. The extra link for 5 particles results in a more compact cluster.

For the problem with 7 atoms the upper and middle sets of 3 atoms are not in the same equivalent class since their relative positions are different.

Configurations 07

I got the nD version of Newton's method for finding the zeros of a function to improve on a search for the minimum potential energy of a configuration of 7 atoms. With 7 "atoms" there are 21 links to optimize which can be split up into 6 types for the chosen configuration of 2 sets of 3 coplanar atoms and another on the z-axis. The configuration is as follows.

This figure will help to keep track of the arrangement of the atoms and the links.

I completed the post mortem and found an error in the formula used to compute change in position needed for an improved estimate for the zero. Typos are easy to make when entering complex formulas and difficult to find in Excel. I went back and rechecked the calculations for the earlier configurations the new formula. For the configuration of 7 atoms the estimates converged quite nicely to the minimum.

The larger number of links pulls the atoms together more so the sides of the horizontal coplanar sets of atoms have different lengths for ρ and r. The value of κ was chosen to make the equilibrium distance for two atom equal to one. The center plane of three atoms is pushed out from the center somewhat and all the other lengths for nearest neighbor atoms are reduced with the length u of the single atom on the z-axis being pulled towards the center the most.

Configurations 06

  Extending Newton's method to higher dimensions is proving to be a little more difficult but I've had some success. A configuration of seven atoms has 21 pair of links and can be arranged according to this scheme. The three independent parameters chosen are q,z,w with dimensionless counterparts θ,ζ,χ. 

This is the executive summary of the procedure used. To check the formulas I derived for Newton's method for a higher dimensions I did a search for the minimum total potential energy. The value for χ was 1.53580.

A 3D plot shows the surface looks like this:

The surface can be can be approximated by a quadratic form derived from 1st and 2nd order derivatives and can be written like this.

The independent parameters converge fairly quickly to a point slightly displaced from the search value.

It's a little puzzling why this doesn't work out exactly but the formulas derived may be somewhat biased estimators of the minimum.

Supplemental (Nov 9): The quadratic form isn't a perfect match for the potential surface. A least squares fit of the potential surface gave the following set of coefficients.

A plot shows the difference between the two surfaces.

The difference between the quadratic form and the configuration potential may account for the deviation of the computed minimum.

Configurations 05

  Doing a post mortem on the failed Newton's method for six atoms seems to indicate that a term was missing due to the assumption of ceteris paribus. This appears to be the correct procedure.

The values for ρ and ζ converge to the values found by doing a search for the minimum of the sum of the potentials.

Supplemental (Nov 5): This is a problem in the calculus of variations since we want the change in the potential for arbitrary displacements from the initial position. The displacements are the variables here. The changes along the differential axes are directional derivatives and don't provide enough information about the quadratic surface to determine the minimum value. The treatment of differentials in texts can be rather intuitive at times. For example see Price, A Treatise on the Differential Calculus.

Edit (Nov 9): Correct sequence number in the title.

Configurations 04

  For six atoms there is another possible configuration to be considered in which the atoms are arranged as follows. It has four characteristic distances which can be expressed in terms of parameters λ and ρ. The solution of this problem proved to be a little more difficult and a two dimension search done as a check showed that the two parameter version of Newton's method gave an erroneous result. The search again gave an equilibrium value of 0.956 for the two parameters and the same equilibrium potential as the previous configuration for the six atoms.

The potential is approximately parabolic.

One needs keen eyesight to determine what went wrong with Newton's method. A comparison of the actual potential with a surface fit produced this surface representing the difference between the two.

The light blue at the top is bounded by contours of zero potential difference. So the original surface appears to be slightly saddle shaped. Apparently there was a contradiction in the assumptions made for Newton's method.

Supplemental (Nov 4): This configuration is identical with the previous one. It's just seen from a different direction, for example, from along the line through the origin and a point in the direction θ=35.2644° and φ=0° or some equivalent.

Configurations 03

  For six atoms there are 15 pair of links. The atoms can be placed into two nearby planes and this configuration has three sets of identical distances, r, s and t, for the following symmetrical arrangement.

Again there are only two independent parameters choosen to be ρ and σ which converge to the equilibrium value of 0.956.

So this configuration is slightly more dense than the previous configurations involving a fewer number of atoms. Each atom is attracted by the five other atoms to the center of the configuration.

Configurations 02

 For four atoms there are six pairs of links which we can split into two groups in the following configuration represented by the relative distances ρ and λ.

By symmetry the first three positions are assumed to be points equidistant from the origin in the x,y plane and the fourth position is assumed to be on the z axis.

The total potential is just the sum of the number of links times the individual potential for each set of links. Using an extension of Newton's method for two independent parameters two arbitrary separations converge to a distance of κ. Comparing the equilibrium potential with one nearby confirms that there is no change in potential occurs if the parameters are separately changed.

Placing a fifth atom at position -z on the z-axis results in 10 pair of links but there is still only two independent parameters along with an additional potential term for the link between the two atoms on the z-axis.

The equilibrium distances again turn out to be κ.

Is this a pattern?

Configurations 01

  Lately I've been studying the forces between "atoms" and the configurations or assemblies they can form. A simple model for an atom that mimics the short range repulsive force and a longer range attractive force employs inverse first and second power potentials. So for two atoms we can determine the equilibrium distance between them. The potentials can be simplified using a scaling factor, κ, which represents a distance determined by the ratio of the two potential field strengths.  For two atoms only one pair of links can be formed between them.

The relative distance between the two atoms can be represented by a single parameter ρ and we can use Newton's method to find the distance where the forces balance. It turns out the equilibrium distance between the atoms is κ.

The case for three atoms is nearly identical since we can define a plane by their three positions and symmetry suggests that the equilibrium distances will all be the same and can be again represented by the single parameter ρ.

The equilibrium distance is again κ.

The actual forces between atoms is more complicated but the model appears to give qualitatively correct results.