# Blog

### Simulating Magnetism Part 4: The $q$-State Potts Model

In this post we’ll go over what the $q$-State Potts model is, how the Metropolis algorithm can be applied to it, and see the results of running simulations with it.

### An Interesting Problem

A few years ago one of my professors mentioned an interesting problem to me: take a charged particle located at $\mathbf{x}$ of charge $q$ confined to motion on a ring of radius $r$ with another charged particle at $\mathbf{x}’$ fixed at the bottom of the ring with a like charge $q’$. The charged particle that is allowed to move freely around the ring is subject to both the gravitational force as well as the Coulomb force generated by the electromagnetic interaction of the charges. In this post we’ll cover how to solve it and then animate it.

### Simulating Magnetism Part 3: Running an Ising Model Simulation

In this post we’ll cover how to implement an Ising model simulation using the previously discussed Metropolis algorithm. We will also see an annotated implementation of the Metropolis algorithm with some optimizations that help reduce the compute time. Once we have the simulation data we will plot it and analyze it for various temperatures and lattice sizes to see how the system behaves differently as those variables change.

### Simulating Magnetism Part 2: Markov Chain Monte Carlo and the Metropolis Algorithm

In order to simulate systems we need algorithms with which to do so. There is a certain class of algorithms which are extremely useful for this application called Markov chain Monte Carlo (MCMC) methods. In this post we will cover some of the basic ideas behind MCMC methods and see the Metropolis algorithm which will help us on our quest to simulate ferromagnetic systems.

### Simulating Magnetism Part 1: Statistical Physics and the Ising Model

In physics we are often interested in modeling systems that are too complex to be modeled exactly so we have to turn to statistical modeling to to get an understanding of some of the macroscopic properties of the system without having to worry too much about the exact microscopic composition. Naturally computational physics has helped us understand a lot about statistical physics because of our ability to simulate and analyze systems with today’s ever increasing computational power. In this post we’ll cover some of the necessary concepts of statistical physics to use as a foundation for our journey through simulating magnetism.

### An Efficient Workflow

One of the most important things you can do to be more productive is to make using your environment as efficient as possible. In this blog post I will talk about my current setup and what I have done to increase efficiency.

### Electromagnetic Waves

Electromagnetic radiation is the science of how waves in the electric and magnetic fields propagate through space-time. These waves are made up of EM quanta which we call photons. They can be anything from radio waves and microwaves to x-rays and gamma rays, depending on their frequency/wavelength. In this post let’s derive the wave equations for the electric and magnetic fields from Maxwell’s equations.

### Laurent Series

Laurent series are generalized power series for complex valued functions that allows the use of negative indices (with the $a_{-1}$ coefficient being the residue of the function at the point of expansion). I recently came across this neat example of one that I found illustrates the idea of a disc or annulus of convergence pretty well. A nice interpretation of the radius of convergence is the distance to the nearest singularity in the complex plane.

### Hamilton's Equations

Hamiltonian mechanics are similar to Lagrangian mechanics, and in the end will yield the same equations of motion. The concept of the Hamiltonian is heavily used in many aspects of physics, and most notably in the formulation of quantum mechanics.

### The Principle of Least Action

There’s several different ways one can approach mechanics. Analytical mechanics is the term that encompasses some of these ways including the well known Lagrangian and Hamiltonian mechanics. I will start by explaining what the Lagrangian approach is and derive how we arrive at the famous Euler-Lagrange (differential) equation that describes how the system evolves in time.

### Simple Harmonic Oscillations

We’ve all seen simple harmonic motion, it is the motion you observe when you watch a grandfather clock tick back and forth, or when a spring oscillates by contracting and extending. It is something that is periodic in motion and moves at a frequency determined either by the system or a driving force. In this post we will discuss how we can solve the equations of motion for oscillators.