Summary:
The partition function of the Ising model is presented and investigated and the road is paved to the famous and incredible solution found by Onsager in 1944.
Besides explicit calculations of the partition function for small lattice sizes a simulation is created and physical quantities such as magnetization and heat capacity are introduced and their behaviour is studied near the phase transition. Results from the partition function are compared to results from the simulations.
Content:
0:00 How do magnets work?
0:37 Spins
1:05 Introduction to the hardest sum
3:08 Summer of Math Exposition
3:45 Magnetization (Experiment)
4:15 Show of hands
5:20 Curie temperature (Experiment)
5:59 Phase transition inside the spin model
7:40 Expectation values (Dice experiment)
8:56 Magnetization as expectation value
10:53 The 3x3 partition function
14:33 The computation with Mathematica
16:35 The simulation of the 3x3 model
19:29 The ergodic theorem
20:42 The comparison: Partition function vs. Simulation
22:05 Phase transition
22:53 The heat capacity (Experiment)
26:01 The heat capacity as expectation value
27:13 The comparison: Partition function vs. Simulation 2
28:43 Let's make the lattice larger - Onsager's solution
31:40 Solution 1: Algebraic solution
33:00 Solution 2: Combinatorical solution
34:00 Words of Gratitude
Correction:
24:03 The energy values at the axis are too small by a factor of two. They should read 36kJ, 72kJ,...
References:
The link to the book:
title: Introduction to statististical physics
author: Kerson Huang
publisher: Taylor & Francis
https://www.taylorfrancis.com/books/mono/10.1201/9781439878132/introduction-statistical-physics-kerson-huang
My permission for the reproduction in the video can be seen here:
https://www.dropbox.com/s/a12bicfcutjon7w/9781420079029%20-%20Martin.pdf?dl=0
The link to Gandhi Viswanathan's Blog post:
https://gandhiviswanathan.wordpress.com/2015/01/09/onsagers-solution-of-the-2-d-ising-model-the-combinatorial-method/
youtube resources:
https://youtu.be/geyC6FfMFf8 @BlenderMadeEasy
https://youtu.be/ZQIk3MTExaU @5MinutesBlender
https://youtu.be/1aUWip4DSQo @5MinutesBlender
https://youtu.be/JYyUMMboZFk @RyanKingArt
The icon for the "show of hands" was obtained from
https://iconscout.com/contributors/jemismali
The layout for the computer was purchased at
https://www.vecteezy.com/vector-art/5113229-computer-keyboard-button-layout-template-with-letters-for-graphic-use-vector-illustration
The material is created with
blender: https://www.blender.org/
audacity: https://www.audacityteam.org/
pycharm: https://www.jetbrains.com/pycharm/
gimp: https://www.gimp.org/
blendermath: unpublished python library that I use as a manim-like interface to create the animations
pyautogui: https://pyautogui.readthedocs.io/en/latest/
is a library that was used to automatically write the mathematica script, presented for the computation of the 3x3 partition function
And last but not least the link to the mathematica notebook that can calculate the partition function for small lattice sizes:
https://www.dropbox.com/s/flvmf0qylcab7j7/IsingModel_Auto.nb?dl=0