Browsed by
Category: Equilibrium

The 1-D Cluster Variation Method (CVM) – Simple Application

The 1-D Cluster Variation Method (CVM) – Simple Application

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…

Read More Read More

Statistical Mechanics – Neural Ensembles

Statistical Mechanics – Neural Ensembles

Statistical Mechanics and Equilibrium Properties – Small Neural Ensembles Statistical Mechanics of Small Neural Ensembles – Commentary on Tkačik et al. In a series of related articles, Gašper Tkačik et al. (see references below) investigated small (10-120) groups of neurons in the salamander retina, with the purpose of estimating entropy and other statistical mechanics properties. They provide the following interesting results: Simple scheme for entropy estimation in undersampled region (1), given that only a small fraction of possible states can…

Read More Read More

Statistical Mechanics, Neural Domains, and Big Data

Statistical Mechanics, Neural Domains, and Big Data

How Neural Domain Activation and Statistical Mechanics Model Interactions in Large Data Corpora (Big Data) I was enthralled. I could read for only a few pages at a time, I was so overwhelmed with the insights that this book provided. And I was about twenty-five years old at the time. I had just discovered this book while browsing the stacks as a graduate student at Arizona State (ASU). The book was The Mindful Brain: Cortical Organization and the Group-Selective Theory…

Read More Read More

Visualizing Variables with the 2-D Cluster Variation Method

Visualizing Variables with the 2-D Cluster Variation Method

Cluster Variation Method – 2-D Case – Configuration Variables, Entropy and Free Energy Following the previous blog on the 1-D Cluster Variation Method, I illustrate here a micro-ensemble for the 2-D Cluster Variation Method, consisting of the original single zigzag chain of only ten units (see previous post), with three additional layers added, as shown in the following Figure 1. In Figure 1, we again have an equilibrium distribution of fraction variables z(i). Note that, as with the 1-D case,…

Read More Read More

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.) 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…

Read More Read More

What Makes a Metastable State Happen?

What Makes a Metastable State Happen?

Metastable States – the Meltdown Precursors I’ve just read a recent column by JL, one of the editors from Taipan Daily. He states, in his column “There Will Be Blood in Europe”: Stepping back a bit: What is so frightening right now, not just in Europe but China and America and Japan too, is the presence of fraud-fueled “Lehman 2.0” catalysts threatening to explode. One could say that the 2008 financial crisis was the mother of all wake-up calls. But…

Read More Read More

"What is X?" – Modeling the Meltdown

"What is X?" – Modeling the Meltdown

“What is X?” – Modeling the 2008-2009 Financial Systems Meltdown We’re about to start a detailed walkthrough of applying a “simple” statistical thermodynamic model to the Wall Street players in the 2007-2009 timeframe. The two kinds of information that I’ll be joining together for this will be a description of Wall Street dynamics, based largely on Chasing Goldman Sachs (see previous blogposts for link), and the two-state Ising thermodynamic model that I’ve been presenting over the past several posts. The…

Read More Read More

"The Origin of Wealth" – Revisited

"The Origin of Wealth" – Revisited

The Origin of Wealth – and Phase Transitions in Complex, Nonlinear Systems Once again, after a nearly two-year hiatus (off by only a week from my first posting on this in May of 2010), I’m getting back to one of my great passions in life – emergent behavior in complex, adaptive systems. And I’m once again starting a discussion/blog-theme referencing Eric Beinhocker’s work, The Origin of Wealth. Since this book was originally published (in 2006), we’ve seen an ongoing series…

Read More Read More

The Beauty of Phase Spaces

The Beauty of Phase Spaces

Phase Spaces: Mapping Complex Systems I’ve spent the day on cloud computing. Yes, there will be a course on it at GMU this Fall of 2010. And cloud computing is simply a technology; a means of getting stuff done. In and of itself, I think there are more exciting things in the world — such as phase spaces. One of the classic nonlinear systems is the Ising spin glass model. This system is composed of only two kind of particles;…

Read More Read More

Quick Note: Helmholtz vs. Gibbs Free Energy

Quick Note: Helmholtz vs. Gibbs Free Energy

Using this blog as an online set of research notes (about that which I don’t mind sharing ) — suppose that we try using an equilibrium-based approach of some sort for modeling what we all know is a very non-equilibrium world. Which formulation, Helmholtz or Gibbs, works best for us? Helmholtz free energy is at constant temperature and volume. It is denoted as A, where the defining equation is A = U-TS, where U is enthalpy, T is temperature, and…

Read More Read More