Visualizing Configuration Variables with the 1-D Cluster Variation Method

Visualizing Configuration Variables with the 1-D Cluster Variation Method

Cluster Variation Method – 1-D Case – Configuration Variables, Entropy and Free Energy

We construct a micro-system consisting of a single zigzag chain of only eight units, as shown in the following Figure 1. (Note that the additional textured units, with a dashed border, to the right illustrate a wrap-around effect, giving full horizontal nearest-neighbor connectivity.)

Figure 1: Micro-system for the 1-D Cluster Variation Method ensemble (single zigzag chain) with equilibrium-values for the configuration values z(i), where the interaction enthalpy is set to zero.
Figure 1: Micro-system for the 1-D Cluster Variation Method ensemble (single zigzag chain) with equilibrium-values for the configuration values z(i), where the interaction enthalpy is set to zero.

In Figure 1, we have the equilibrium distribution of fraction variables z(i). Note that the weighting coefficients for z(2) = z(5) = 2, whereas the weighting coefficients for the remaining fraction variables (z(1), z(3), z(4), and z(6)) are all 1.

In this illustration, we have a short zigzag chain that wraps around to connect with itself.

This micro-system is at equilibrium, so that:
x(A) = x(B) = 0.5, where x(i) is the fraction of units in state A or state B, respectively.

Further, the pairwise interactions are also at their equilibrium values (for equal amounts of A and B, and for no interaction energy). These additional variables are shown in Figure 2.

Figure 2. Configuration variables in a single zigzag chain for the 1-D Cluster Variation Method.
Figure 2. Configuration variables in a single zigzag chain for the 1-D Cluster Variation Method.

Thus, y(1) = y(3) = 0.25, and y(2), which appears with a weighting of 2, is at 2y(2)=0.5. We have y(1) + 2y(2) + y(3) = 1. The next-nearest neighbor fraction variables w(i) also appear with the same weights.

The z(i) appear with equilibrium values (for interaction energy equals zero, and x(A) = x(B) = 0.5), so that z(1) = z(3) = z(4) = z(6) = 0.125, and z(2) and z(5), each of which are weighted at twice that of the other z(i) variables, are each at 2z(i)=0.25. See Figure 1.

This micro-system is shown for the case where the interaction enthalpy term is set to zero, i.e., there is no preference towards creating pairs of either like or unlike units. This corresponds to the case where h=exp(beta*epsilon/4) = 1. See Analytic Single-Point Solution for Cluster Variation Method Variables (at x! = x2 = 0.5).

The reduced free energy for this system is given by

CVM_1-D_Free-Energy-Eqn_crppd

where

CVM_1-D_Free-Energy-Eqn_eq2-v2_crppd

We note that the last two terms, which are Lagrangians, of course are at zero in the equilibrium case. Also, the enthalpy term is at zero. In this particular case (equilibrium at x(A)=x(B)=0.5), we have the simpler equation

CVM_1-D_Free-Energy-Eqn_eq3_crppd

We get an exact value for the reduced free energy with this micro-system, with the particular constraint that the system is at equilibrium and the enthalpy is zero. This is

CVM_1-D_Free-Energy-Eqn_eq4_crppd

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