Single-Point Analytion CVM Solution Involves Solving Set of Nine 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; I'll include a link to it as soon as I post it on my website, http://www.aliannamaren.com.
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) actually increases slightly beyond its nominal expectation (when there is no interaction energy), and goes above 0.25, 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.
Somewhat surprisingly,
z3 (A-B-A triplets) also decreases as
h approaches 0.1. This means that the increase to above-nominal distributions for the CVM variable goes to
z2 (A-A-B). Given that this is an even distribution of A and B units (
x1 = x2 = 0.5), another way to think of the far LHS is when the temperature is very large. (We then have the exponent of a negative interaction energy over a large temperature, and can think of the increased temperature as producing greater "disorder" in the system - moving us away from the highly structured A-B-A-B-A order that would otherwise exist if
y2 (A-B) predominated with no other influence.
