Essence of Critical Phenomena

Slides

See the slides as a Google Slide

Essence-of-of-Critical-Phenomena

Video

Play with the Ising Model

This is a NetLogo Model. If you download the NetLogo application, this model is included. But there is no need for that, you can simply try that here. The model is embedded just below this note, so if you cannot see it please try it on your own computer. Please read the “Model Info” section of the module (you need to scroll down in the module). Have fun playing with the module!

References

Ising Model On Wikipedia and Scholarpedia

MIT – The Net Advance of Physics: ISING MODELS : web.mit.edu/redingtn/www/netadv/Xising.html

The Fate of Ernst Ising and the Fate of his Model: arXiv:1706.01764 [physics.hist-ph]

A glimpse of Glauber dynamics for simulating Ising Model: bit-player.org/2019/glaubers-dynamics

A good video lecture on Ising model From enumeration to Cluster Monte Carlo Simulations: youtube.com/watch?v=kvf7aUPZCWk

The classic reference for the particular renormalization transformation we do in this unit is Leo Kadanoff’s book, Statistical Physics: Statics, Dynamics, and Renormalization — Chapter 14 covers the clever decimation technique where the new lattice is a 45-degree rotation of the old. It’s a lovely introduction to the ways in which smart people flailed around at a problem they didn’t quite yet know how to solve.

Ising Model Practical

The Ising model simulation is by Bernd Nottelmann (with some nice interface design by A. Peter Young)

Douglas Ashton’s YouTube of life at the critical point is at https://www.youtube.com/watch?v=MxRddFrEnPc

We dealt with the Ising model on the 2D lattice, but what about the Ising model on arbitrary graphs? You can also take a look at a story about renormlization in “self-similar” (fractal) network structures:  http://rsif.royalsocietypublishing.org/content/9/74/2131?sa=X&ved=0CDcQ9QEwEGoVChMIqsL9tpT1xgIVwu0UCh2bRwOT (and many references therein) — or, if you really want to dig down into the different ways of modelling the Ising model on arbitrary graphs, consider the Linked-Cluster Expansion http://tuvalu.santafe.edu/~simon/wortis-1974.pdf

A great deal of work on the Ising model on arbitrary graphs is done by simulation (see the first link, the “Ising Model practical” for how this works).

Exercises