Exploring Periodic Boundary Conditions: A Study on 2D and 3D Spheres and Continuum Limit

Physics students do math as they wish :)

In this project we will analyze different systems and come up with a through introduction to the mathematical theories behind PDEs governing this weird phenomena. We will also implement a numerical solver for the PDEs and use it to simulate the systems. This will be done mainly in LaTeX and Python. Figures all made in LaTeX PGFPlots. We will create Figures by Ipe and Tikz. We will use GitHub for version control and collaboration.

Sections

1. Abstract
   • Write a short summary of what we did.
2. N Particle system
   • Derive the ODE for the N particle on a circle simple case (fully linear)
   • Explain Circulant Matrices and how they can be used to solve the ODE
   • Use the Symmetry argument to describe the harder N-particle case
   • Approximate the ODE to get the modified linear ODE and solve it analytically
   • Implement the solver in Python and compare it to the analytical solution. Find the error.
3. Continuum Limit
   • Derive the PDE of propagation of a wave on a circle
   • Explain Periodic Boundary Conditions and how they can be used to solve the PDE
   • Solve it completely based on initial conditions. Use Fourier Series.
   • Create beautiful figures of the solution.
4. 3D N-Particle Case (Regular Polyhedron)
   • Derive the ODE for the N particle on a regular polyhedron simple case (fully linear)
   • Try to get symmetry argument to work
   • Explain the importance of Neighbours and how to use it to solve the ODE.
   • Solve the equation numerically.
   • Statistical Mechanics of the system
5. Longitudinal Waves on a Sphere (3D Continuum Limit):
   • Derive the wave equation on a sphere
   • Try to solve it Analytically
   • General Solutions? Numerical Fourier Series in 2D?
   • Green Function?
   • Periodic Boundary Conditions
   • Can we use it to send information?
6. References
   • check the references and add them to the bibliography
   • Make sure all the references are cited in the text
   • Make sure all the figures are referenced in the text
   • Make sure all the equations are referenced in the text
   • Refrence GitHub repo for code
7. Clean up
   • Make sure all the figures are in the figures folder
   • Make Jupyter Notebooks for all the code
   • Check the Correct Formatting of The Report.