This Really Is Kind of Obvious, But …

 

There’s something very interesting that we can do to obtain values for the epsilon0 parameter.

Let’s stay with the case where there is no interaction enthalpy. In that case, we want to find the epsilon0 value that corresponds to the x1 value at a given free energy minimum. Or conversely, given an epsilon0 value, can we identify the x1 where the free energy minimum occurs?

Turns out that, for this very limited case (no interaction enthalpy), we can easily do this.

We take the derivative of the free energy equation with respect to x1, and set it equal to 0.

The free energy equation that we’ll use has the simple form for the entropy, because – as we’ve just found out – when the interaction enthalpy is zero, all the other entropy terms cancel out. That makes life much simpler!

Here’s our free energy equation – very simple and easy. (We’re ignoring LaGrange factors, etc.)

or

or

We take the derivative with respect to x = x1, and set this equal to zero to obtain

This gives us epsilon0 as

 

 

Graphing the Results

 

It’s easy enough to obtain the minimum in the previous equation. The following results shows us the values in tabular form.

Similarly, we see those same results in graphical form.

We see that when epsilon0 = 1, we have x1 = (approximately) 0.265. (Yes, we could easily get a more precise value … either with a more detailed table or using a computational approach such as Newton-Raphson; not the point here. All that we’re showing is that we CAN get an analytic result.)

So, for any given value of epsilon0, we can find – via a little bit of computational work on top of a fairly simple equation – the x1 value for the corresponding free energy minimum. It all checks out.

Also, in this scenario (with epsilon0 = 1.0, and of course epsilon1 = 0.0; we’re still not allowing any interaction enthalpy), the free energy curve has a nice, obvious, clean minimum.

 

 

Our Next Step

 

We now have easy means to produce the configuration variables for two cases:

• The interaction enthalpy equals zero: epsilon1 = 0, and epsilon0 > 0: This is the case we’ve been addressing here, and
• The activation enthalpy equals zero: epsilon0 = 0, and epsilon1 > 0: I’ve done a lot of work with this, see the previous blogposts, together with their link to a verification and validation document (on arXiv) and also the previously published paper in Brain Science.

With these, we can define the boundaries of the phase space that we want to investigate. We’ll start roughly framing this next week.

 

 

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). New code. Same code as last week, with the addition of a little bit to compute the free energy as a function of x1 for a given value of epsilon0. Interaction enthalpy epsilon1 still equals 0.
• Documentation / V&V: 2-D_CVM_No-interaction-enthalpy_2018-10-29.pptx (contains all the new figures shown in this blogpost and prior one).

 

Previous Related Posts

• 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)