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…
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…
The 2-D CVM Entropy and Free Energy Minima when the Interaction Enthalpy Is Zero: Today, we transition from deriving the equations for the Cluster Variation Method (CVM) entropies (both 1-D and 2-D) to looking at how these entropies fit into the overall context of a free energy equation. Let’s start with entropy. The truly important thing about entropy is that it gives shape and order to the universe. Now, this may seem odd to those of us who’ve grown…
Showing How the Omega Equation is Obtained for the 1-D and 2-D CVM – Part 2: The cluster variation method (CVM) lets us characterize a system in terms of local patterns, and not just the numbers of units in on (A) and off (B) states. It works with a more complex entropy term. The natural question is: How do obtain this more complex entropy? This post continues a discussion begun in the last post, on how we actually get…
The 1-D CVM – A Single Zigzag Chain – Part 1: The cluster variation method (CVM) lets us characterize a system in terms of local patterns, and not just the numbers of units in on/off states. This is likely to be useful for machine learning and AI applications. Up until now, we’ve not told the story of how we actually compute the CVM entropy from the microstates. We’ll do that starting with this post; it will be a handy…
The 1-D Cluster Variation Method – Application to Text Mining and Data Mining There are three particularly good reasons for us to look at the Cluster Variation Method (CVM) as an alternative means of understanding the information in a system: The CVM captures local pattern distributions (for an equilibrium state), When the system is made up of equal numbers of units in each of two states, and the enthalpy for each state is the same (the simple unit activation energy…
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…
Correspondence Between Free Energy, Belief Propagation, and Markov Random Field Models As a slight digression from previous posts – re-reading the paper by Yedidia et al. on this morning on Understanding Belief Propagation and its Generalizations – which explains the close connection between Belief Propagation (BP) methods and the Bethe approximation (a more generalized version of the simple bistate Ising model that I’ve been using) in statistical thermodynamics. The important point that Yedidia et al. make is that their work…
Complexity and Graph Theory: A Brief Note Santo Fortunato has published an interesting and densly rich article, Community Detection in Graphs, in Complexity (Inter-Wiley). This article is over 100 pages long, it is relatively complete, with numerous references and excellent figures. It is a bit surprising, however, that this extensive discussion misses one of the things that would seem to be most important in discussing graphs, and particularly, clusters within graphs: the stability of these clusters. That is; the theoretical basis for cluster…