Single-Point Analytic Cluster Variation Method Solution: Solving Set of Three Nonlinear, Coupled Equations

The Cluster Variation Method, first introduced by Kikuchi in 1951 (“A theory of cooperative phenomena,” Phys. Rev. 81 (6), 988-1003), provides a means for computing the free energy of a system where the entropy term takes into account distributions of particles into local configurations as well as the distribution into “on/off” binary states. As the equations are more complex, numerical solutions for the cluster variation variables are usually needed. (For a good review, see Yedidia et al., Constructing free energy approximations and generalized belief propagation algorithms.

When allowed to stabilize, the system comes to equilibrium at free energy minima, where the free energy equation involves both an interaction energy between terms and also an entropy term that includes the cluster variables. This computation addresses a system composed of a single zigzag chain.

I have computed an analytic solution for representing one of the cluster variables, z3, as a function of the reduced interaction energy term:

 
 
 

The equation details are presented in a separate Technical White Paper. (Note added June 25, 2014: This White Paper can be found as: The Cluster Variation Method I: 1-D Single Zigzag Chain: Basic Theory, Analytic Solution and Free Energy Variable Distributions at Midpoint (x1 = x2 = 0.5).

This pattern of CVM variables follows what we would expect.

The point on this graph where h=1 (the x-axis is 10) corresponds to h = exp(beta*epsilon)=1. Effectively, beta*epsilon => 0. This is the case where either the interaction energy (epsilon) is very small, or the temperature is very large. Either way, we would expect – at this point – the most “disordered” state. The cluster variables should all achieve their nominal distributions; z1=z3=0.125, and y2=0.25. This is precisely what we observe.

Consider the case of a positive interaction energy between unlike units (the A-B pairwise combination). The positive interaction energy (epsilon>0) then suggests that a preponderance of A-B pairs (y2) would destabilize the system.

We would expect that as epsilon increases as a positive value, that we would minimize y2, and also see small values for those triplets that involve non-similar pair combinations. That is, the A-B-A triplet, or z3, approaches zero. We observe this on the RHS of the above graph. This is the case where as h = exp(beta*epsilon) moves into the positive range (0-3), we see that y2 and z3 fall towards zero. In particular, z3 becomes very small. Correspondingly, this is also the situation in which z1 = z6 becomes large; we see z1 taking on values > 0.4 when h > 2.7.

This is the realm of creating a highly structured system where large “domains” of like units mass together. These large domains (comprised of overlapping A-A-A and B-B-B triplets) stagger against each other, with relatively few instances of “islands” (e.g., the A-B-A and B-A-B triplets.)

Naturally, this approach – using a “reduced energy term” of beta*epsilon, where beta = 1/(kT), does not tell us whether we are simply increasing the interaction energy or reducing the temperature; they amount to the same thing. Both give the same resulting value for h, and it is the effect of h that we are interested in when we map the CVM variables and (ultimately) the CVM phase space.

At the LHS of the preceding graph, we have the case where h=exp(beta*epsilon) is small (0.1 – 1). These small values mean that we are taking the exponent of a negative number; the interaction energy between two unlike units (A-B) is negative. This means that we stabilize the system through providing a different kind of structure; one which emphasizes alternate units, e.g. A-B-A-B …

This is precisely what we observe. The pairwise combination y2 (A-B) increases beyond its nominal expectation (when there is no interaction energy), and goes towards 0.5, notably when h is in the range of 0.1 and smaller. Also, as expected, the value for z1 (A-A-A triplets) also drops towards zero – triplets of like units are suppressed when the interaction energy between units is positive.