Stat Mech II
Home
Table of Contents
First Lecture
Overview
Phase Transitions
Symmetry
Ising Model Def'n
Ising Thermo
1D Ising Model
Motivation
The Big Picture
Exact Solution
Mean Field Theory
Overview
Variational Principle
Non-interacting Spins
Mean-Field Ising Sol'n
Summary + Interp'n
Landau-Ginzburg
Overview and Outline
Probe Fields
What is Free Energy?
Landau Theory
Order Param Fields
Symmetry Arguments
Spatial Textures
Gaussian Model
Applets!
Transport
Kinetic Theory
Diffusion and Viscosity
Boltzmann Equation

Prof. Kivelson's first lecture

We had our first lecture with Prof. Steve Kivelson on the Friday of the first week of class. For the first hour, we discussed lots of course logistics. Then we started discussing physics.

The Ising Model

Prof. Kivelson gave some detailed philosophical motivation for why we're studying the Ising Model. Earlier in the week, Prof. Raghu had given a bit of motivation as well, but it was nice to hear even more motivation, especially since (I'm guessing) we'll be spending so much time on the Ising model.

Prof. Kivelson's motivation for Ising Model

Thermodynamics is a very abstract discipline – so abstract that you either understand it entirely or you don't understand it at all. Now on one hand, the abstract nature of thermodynamics is extremely valuable – it's the reason thermo can describe any system of many interacting parts, regardless of the microscopic details of interaction. Indeed, the beauty of thermodynamics is its generality to all sorts of the physical systems.

On the other hand, though, the abstract nature of thermodyamics makes it very hard for us to wrap our head around the material at all! Rather than figuring out the ‘grand vision’ of thermodynamics from the get-go, we need to start somewhere. We need to ground our understanding in a well-defined model before we go off to generalize to different systems. Once we do know how to solve all sorts of different problems, we can keep coming back to our original model system to refer to old concepts and draw analogies.

So before we dive off into all sorts of crazy materials, it's valuable to just try understanding the Ising Model and see how it behaves. Once we do, then we can generalize to other systems.

Prof. Kivelson also reminded us that the Ising model does not actually represent a real physical system – it's just an abstract mathematical problem. Miraculously, though, we can actually draw analogies to real systems, such as magnets, alloys, or lattice gases.

Another crucial feature of the Ising model is that you can easily simulate it on a computer. After all, each site of the lattice is just a bit which is in one of two states.

Here is a nice simulation that comes up when you google ‘‘Ising model simulation’’.

Solving the 1D Ising Model

We then went on to define the 1D Ising model. Since we already defined the Ising Model in the introductory lecture, I'll just refer back to those notes here.

Afterwords, we explicitly solved the model, but, we went a bit too fast for me to really understand. Thankfully, we did come back and re-derive everything again the following day.

Leave a Comment Below!

Comment Box is loading comments...