Monte Carlo Simulations and Tetrahedra on the 3-Sphere. Kellen Gillispie and Michael Zlotoff
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Our goal is to apply the methods and ideas learned this summer to the Skyrme model, a model of pions. There were two components to this summer?s research that will arise in our work next summer: the application of Monte Carlo methods to investigate properties of nucleons at finite temperature, and determining the (oriented) volume of spherical tetrahedra, which yields the winding number of a configuration of pions. In order to create such a model, it was necessary for us to first become familiar with multidimensional spheres and Monte Carlo methods. First, we created two simulations to solve for the area of triangles on 2-dimensional spheres to acclimate ourselves with multidimensional spheres. After determining that both simulations were valid with a known equation for the area for spherical triangles, we scaled our simulations to solve for 3-dimensional spheres where there is no simple equation. After converting both simulations to 3 dimensions we ran the simulations for 500,000 random vectors and compared the approximations with a program that our mentor uses to calculate volume of spherical tetrahedra. We concluded that our simulations were accurate to within .01% error for this many runs. To better understand Monte Carlo methods, we examined the Metropolis algorithm, a method for simulating a system as its energy fluctuates toward equilibrium. We applied the algorithm to the Ising model and investigated properties of the model on a lattice, each location of which represents magnetic spin. We swept through the lattice many times and use the algorithm to move the system toward equilibrium. The research accomplished this summer has laid the foundation for the Skyrme model, which we hope to begin next summer.