Configuration Variables Along the Phase Space Boundaries for a 2-D CVM

 

Last week’s blog showed how we could get x1 for a specific value of epsilon0, by taking the derivative of the free energy and setting it equal to zero. (This works for the special case where epsilon1 is zero, meaning that there is no interaction enthalpy.) Last week, we looked at one case, where epsilon0 = 1.0. This week, we take a range of epsilon0 values and find their (approximate, to three decimal places) corresponding values of x1. We work through the range of epsilon0 = 0.0 up through 5.0, and find that x1 ranges from 0.5 (when epsilon0 = 0.0), down to 0.007. The following Figure 1 shows these results. The data tables, and screenshots of the actual code outputs, are in the associated PPT deck. Both the code and the PPT are available on GitHub; see the link at the end of this post. The Excel spreadsheet containing the data points, from which Figure 1 was generated, is also in the same GitHub directory; the filename is at the end of this post.

 

 

Configuration Variable Values at the Phase Space Boundaries

We now have two key ingredients for our next step, which is to fill in a data table for the configuration variables as a function of the two enthalpy parameters; epsilon0 (the activation enthalpy) and epsilon1 (the interaction enthalpy). We have separately addressed two distinct cases.

The first case, which we’ve been working on for this post and the previous one, deals with the case where the epsilon0 parameter is allowed to range, but epsilon1 is set to zero. In this case, we can analytically find x1 as a function of epsilon0. I’ve used a detailed table of x1 calculations to get an approximate x1 value when epsilon0 has been specified; that is what we showed in Figure 1. If we wanted more detail, a finer granularity of computation tables would work, or we could use a numeric solution. For now, an approximate result (good to three significant figures after the decimal point) is good enough. Recall that in this specific case, we can compute the values for all other configuration variables (the y(i), w(i), and z(i) values) via a straightforward probabilistic computation. This is because when there is no interaction, the distribution of the remaining configuration variables is completely dependent on the initial x1 and x2 values. (Of course, x2 = 1.0 – x1.)

For the other phase space boundary, we set epsilon0 = 0. This means that there is no activation enthalpy. As a result, x1 = x2 = 0.5. However, we now allow the interaction enthalpy epsilon1 to take on non-zero values, meaning that we do have interactions between neighboring units. We have an analytic solution for this case, giving our configuration variables in terms of h, which is a function of epsilon1. This result was initially reported in Kikuchi and Brush (1967) and replicated (this time with the details that Kikuchi and Brush had omitted) in Maren (2016).

As a result, we can now construct a phase space where the values for x1 and the other configuration variables are known along the edges. Our next job will be to compute the configuration variable values for the interior of this phase space.

We want more than just the actual configuration variables themselves, though. We want to assess the shape of the free energy curve in various phase space areas.

{To be continued.}

 

 

Live free or die, my friend –

AJ Maren

Live free or die: Death is not the worst of evils.
Attr. to Gen. John Stark, American Revolutionary War

 

 

Key References

 

Cluster Variation Method – Essential Papers

• Kikuchi, R. (1951). A theory of cooperative phenomena. Phys. Rev. 81, 988-1003, pdf, accessed 2018/09/17.
• Kikuchi, R., & Brush, S.G. (1967), “Improvement of the Cluster‐Variation Method,” J. Chem. Phys. 47, 195; online as: online – for purchase through American Inst. Physics. Costs $30.00 for non-members.
• Maren, A.J. (2016). The Cluster Variation Method: A Primer for Neuroscientists. Brain Sci. 6(4), 44, https://doi.org/10.3390/brainsci6040044; online access, pdf; accessed 2018/09/19.

 

The 2D CVM – Code, Documentation, and V&V Documents (Including Slidedecks)

• GitHub for 2-D CVM Entropy with No Interaction Enthalpy
• Code: 2D-CVM-simple-eq_no-interaction-calc_v3_2018-10-27.py (Python 3.6; self-contained). Same code. Same code as previous two weeks: I twiddled the starting values and increments of x1 to get a finer detail on epsilon0. Interaction enthalpy epsilon1 still equals 0. Data tables summarizing the results are in the PPT slidedeck identified below.
• Documentation / V&V: 2-D_CVM_No-interaction-enthalpy_2018-11-11.pptx (same PPT as listed in previous two blogs, but with the addition of the data table(s) and the figure showing how x1 depends on epsilon0).
• Excel Spreadsheet: x1-vs-Epsilon0_2018-11-11.xlsx (this contains the summary data table used to generate the figure showing how x1 depends on epsilon0).

 

Previous Related Posts

• Obvious, But Useful (Getting the Epsilon0 Value when the Interaction Enthalpy Is Zero)
• An Interesting Little Thing about the CVM Entropy (With Code)
• Expressing Total Microstates (Omega) for the 1-D and 2-D CVM (Part 2)
• Expressing Total Microstates (Omega) for the 1-D Cluster Variation Method – Part 1
• Wrapping Our Heads around Entropy
• Figuring Out the Puzzle (in a 2-D CVM Grid).
• 2-D Cluster Variation Method: Code Verification and Validation
• The Big, Bad, Scary Free Energy Equation (and New Experimental Results)